On Well-Ordering(s) and Sets Dense in the Reals, Infinity
Ross A. Finlayson
reals, concepts about the specificities of a "previous" and "next"
points on the real number line broaden and solidify.
Where the rationals and irrationals, eg numbers of the form a/b for
integer a, b as rational numbers and otherwise as irrational numbers
are complements the union of which is the set of real numbers and as
well each is dense in the real numbers, then a naive rationalization of
a given point, as a real or scalar number, and its immediate
neighborhood in terms of the finite set of points that are having the
least difference, with having rationals and irrationals alternate on
the real number line is that: a naive rationalization of the immediate
neighborhood of a point on the continuous real number line.
The irrationals may well be divided into two sets, each dense in the
irrationals and complements whose union is the irrationals: the
algebraic and transcendental irrational numbers, where the algebraic
irrationals are real solutions or roots of a polynomial with integer
coefficients that are not otherwise rational, and the transcendentals
are then the other irrationals.
Then, there are considerations of other ways to divide the sets that
are thus far to be determined to be separate, in that their
intersection is null, and complementary in that their ultimate union is
the set of reals which have the property of completeness and on the
line continuity, and dense in the reals where between any two
definitely valued numbered reals there are infinitely many members of
these sets: rationals, algebraic irrationals, and transcendentals.
To that end it is useful to determine further categorizations of these
subsets of the reals that share these three properties of being
nonintersecting, complementarily forming the reals, and each dense in
each other and the reals.
One possible consideration is that there are many ways to reorganize or
compose from the above sets' definitions other definitions of sets that
can be said to comprise the reals. Where the above definitions might
seem to be the most distinct in the sense of immediacy, there are
further subcategorizations of the transcendentals as for example
Mahler's S, T, and U transcendentals, if they are disjoint and dense in
the reals. As well, something like the rationals can be separated
into, for example, the rationals with even versus the rationals with
odd denominator, each dense in the reals with their union the
rationals. As well, where definitions as above form a tree structure
rooted at the reals, various subsets could be composed together in
their definition with for example the transcendentals unioned with the
rationals or rationals with even denominator unioned with the
irrationals: there are many ways to determine two, three, or more of
the nonintersecting, dense, completing subsets of the reals to be of
many various definitions.
Thus where a naive rationalization of the reals as alternating
rationals and irrationals from their properties of being
nonintersecting, dense(in each other and the reals), and completely
forming in their union the entire set of reals, those properties hold
for many other combinations of sets with those properties.
That naive rationalization might be along the lines that along the real
number line that rationals q in Q and irrationals p in P alternate as
so:
....pqpqpqpqpqpqpqpqpqp...
with the notion that given a selected distinct rational q, that the
immediate neighbor in that contiguous sequence is _not_ an element of
Q, and is in this case an element of P, the only other set considered.
Towards that rationalization if there are, for example, rationals q in
Q, algebraic irrationals in a in A, and transcendental irrationals t in
T, then that naive rationalization follows the the exact specific
number that is greater than a given q in Q and less than any other
specific number is _not_ an element of Q, but that it be either an
algebraic irrational or transcendental number. This is where the real
numbers are well-ordered by fiat or theorem. _If_ the next number is
an a, then the pattern thus follows:
...qatqatqatqat...
else
... qtaqtaqtaqta
Here, there should be a replacement of those letters that represent
specific set with the N, C, D properties for nonintersecting
(disjoint), completing, and dense, with a given number of digits that
represent how many of those sets are deemed to exist.
...012012012012...
...021021021021...
One problem with that is that the rationals, for example, can be broken
into, for example, numbers with even and odd denominators, or numbers
with denominators equal to 0, 1, 2 (mod 3), or 0, 1, ..., n-1 (mod n).
Thus the rationals can be divided into n many NCD subsets, for n E Z+.
The algebraics can as well be broken into various definitions of
subsets with these properties, and the transcendentals may be possibly
divided into further subsets in these ways.
When there are four or more subsets, then the specific "next" element
is not predetermined by previous elements. That is because where there
is the negative condition that a successive element _not_ be of the
same categorization of the current, nor as possible previous, there are
more possibilities under those constraints implied by each's density
within each other.
...012301230...
It would seem that that order would be fixed for any definition of
these sets numbered 0, 1, 2, and 3 with the NCD property. That is not
to say the order could not be:
...01320132...
...02130213...
...02310231...
...03210321...
...03120312...
but once a given permutation of (0, 1, 2, 3) was determined, it would
hold through all successions of those values. Equivalently, the sets
could be relabelled but the order would always be:
...01230123...
It might seem that for n > 3 that the order could vary, the fugue. By
fugue I mean that type zero elements would be each n'th element, but
the others would be arbitrary in their permutation except for that the
immediate neighbors of the type 0 elements would differ.
...0123013203120321023102130...
If you've read this far, then you might consider that the rationals and
irrationals alternate in the reals. Then again, by the same naive
rationalization algebraic and transcendentals would alternate, etcetera
etcetera for any pair of disjoint sets dense in each other, as ordered
fields, whose complement is the reals.
The vague fugue continues, and further methods to subcategorize NCD
sets would be a way to further explore this question: in the
well-ordered reals, what is the next?
It is regularly claimed that there is no "next" or "previous" due to
abscence of well-ordering. Where there is well-ordering, there is.
Besides the moduli of the denominator of the rational, also may be
considered various combinations to do with the coprimality of that
modulus of the numerator. For example: even and odd denominators and
even and odd numerators for the odd numbers. Combinatorially, the
number of ways to divide the rationals into these disjoint sets each
disjoint, dense in each other, and complementarily forming as their
union the rationals, explodes.
The algebraic irrationals have even further combinatoric possibilities
as a function of any number of finite integers as the order of the
polynomial increases past two, the least order of a polynomial with
possibly algebraic as opposed to necessarily rational roots. Indeed,
the least order to represent the polynomial the roots of which is a
given algebraic irrational is one possible jumping-off point to
asymptotically compare that variety.
The transcendentals again, as roots of power series for example with no
finite order, or more exploredly as non-algebraic irrationals, see
categorizations such as Liouville's and Mahler's S, T, and U types, are
readily subdivided.
That is all so, yet once again the real numbers are rationals and
irrationals, or algebraics and transcendentals, and where any of these
N, C, D sets is everywhere discontinuous, where the goal here is to
determine a specification of a point and finitely many of its nearest,
least different neighbors, all the possible nonintersecting,
completing, dense sets alternate.
That is a function of their density and disjointness, when infinitely
many of the contiguous elements of a well-ordered sequence of the reals
has that due to the density of each within the reals that deductively
finitely many of the contiguous elements would seem to _intuitively_
alternate. To consider finitely many elements of the sequence might
indeed lead to a deductive breakdown, where as there are infinitely
many ways to divide the reals into NCD sets that any finite sequence
would only have some few elements, and perhaps only one of each. Yet,
where the reals are comprised (a union) of only some finitely many
sets, then a finite sequence could contain one of each, again in each
subsequence of n elements one of the n subdivisions.
This discussion is self-contained among my various other arguments,
it's specifically about this. Some choose to not even address the
concept.
That's irrelevant, here are some questions: in what ways may the
rationals or generally algebraics, or transcendentals or generally
irrationals, be divided into nonintersecting (disjoint) sets each dense
in their union? Is it possible to describe and parameterize all
possible ways that they can be so subdivided?
Between any two rationals a/b and c/d, there is an irrational. Between
any two irrationals you might define as a decimal (ie Dedekind or
Cauchy), there is a rational.
(Infinite sets are equivalent.)
Infinite sets are equivalent for other reasons, eg induction shows that
an infinite set is inexhaustible, the binary case is sufficient and
certain composable monotonic mappings avoid contradictory functions,
and there can only be one proper class. Some theories do have a set of
all sets.
This consideration then of how to informedly guess what the "next"
element in the well-ordered reals would be in terms of its
characteristic properties besides being infinitesimally different and
"one-sided" at this point seems to be that into however many sets you
subdivide the reals, that many alternatives cycle.
Consider the Nyquist limit, and the hyperintegers. Consider sampling a
real by flipping a coin infinitely many times, and how that might be a
sample of finitely or infinitely many reals at once.
I think we all learned in high school math that the rationals and
irrationals can not be said to alternate on the real number line,
perhaps with little justification, that's trusted. It was covered in
passing.
So, what do they do?
There are lots of rational numbers.
1. In what ways may the rationals or generally algebraics, or
transcendentals or generally irrationals, be divided into
nonintersecting (disjoint) sets each dense in their union?
2. Is it possible to describe and parameterize all possible ways that
they can be so subdivided?
3. So, what do they do?
These deductions may be valid, the ones expressed here are not
particularly controversial, they are couched in inquisitive terms, but
then the question arises: what would be the use of such analysis?
You might know that I talk of iota as the "least positive real", ie,
the actual number right after zero in the well-ordering of the reals is
named iota. As well, iota would be the difference between any given
real number and its previous and next in that nonstandard real number
model, and its successive elements are named "2iota", "3iota",
etcetera. In that sense 0iota = 0 and n*iota converges to 1 as n
diverges, per the natural/unit equivalency function.
Speaking of iota itself, where zero is rational, iota is only known to
be _not_ rational. Where zero is algebraic, iota is _not_ algebraic.
For any of the NCD sets that contain zero, except for trivially
changing the definition about zero, iota does not have the property
that defines that set.
In "trivially changing" the definition, for example dividing the reals
into rationals besides zero and irrationals and zero, then iota, or
"the next element after zero in the well-ordering of the reals", would
appear to be non-zero and as well rational, where again the negation
stems from the density, complementarity, and disjointness of those two
specified sets. There could also be deductive curiousities about
subdivisions of the rationals. For example if the rationals are
divided into reduced fractions with even and odd denominators, then
iota is simply irrational, and zero is algebraic. The more "simple"
definition would have more meaning.
There is to be no defining the sets by their containment of iota,
iota's definition itself rests upon those sets' definition without
iota.
Iota exists if you well-order the reals.
Returning to the notion of the rationals and irrationals and their
place in the reals, the "simple" delineation or categorization of the
reals seems prima facie to be either the separation into rationals and
irrationals or the separation into algebraics and transcendentals. In
either case the progression is as so:
...01010101010101...
Not much use can be made of a presupposed fact that the rationals and
irrationals alternate in the well-ordered reals, because those sets of
numbers are fields. It is only that that fact can be presupposed for
applications that do not conflict with their characteristics as NCD
sets for the reals in the well-ordering of the reals, via fiat in ZFC,
where any set can be well-ordered.
So, in well-ordering the reals, the "next" element after zero is the
infinitesimal, irrational, and real: iota.
Where that may be so, simply assigning a name to each element of the
well-ordering of the unit interval, there still remains a deductive
impasse about the density of these sets that comprise the reals, and
the notion of the interspersal of each within each other on the finite
scale of contiguous elements of the sequence derived from well-ordering
the set of reals.
Where due ot their density, in addressing only the rationals and
irrationals, each is dense in the reals thus:
i. between any two rationals a/b and c/d there exist infinitely many
irrationals
ii. between any two irrationals x E N^N and y E N^N (more precisely x E
P and y E P, P < N^N, many sequences converge to the same value) there
exists infinitely many rationals
Then a question is whether in the examination of a finite subsequence
(of length > 2) of the well-ordering whether there exists in any finite
subsequence of the well-ordering an irrational between each pair of
rationals and an irrational between each pair of rationals. If there
exists only irrationals on some finite subsequence, then not all
irrationals can be defined by Dedekind or Cauchy, and those definitions
are insufficient to define each real number. If each finite
subsequence that contains irrationals contains rationals then the
rationals and irrationals are equivalent.
There is no finite well-ordering sequence between any two rationals a/b
and c/d, is thus the well-ordering principle contradictory in
assumption? Well-ordering is a consequence of Choice without which the
reals are not a set.
So, you can divide the reals into infinitely many nonintersecting,
dense in the reals sets whose complement is the reals, but the reals
are only rationals and irrationals. That's not to say they aren't
algbraics and transcendentals, they're rationals and irrationals.
Infinite sets are equivalent (equipotent, equipollent).
Under what conditions can it be said the reals are well-orderable? In
ZFC, any set is well-orderable. If any set of reals dense is
well-orderable then so is the complete superset and any subset. Is not
that obvious to you?
If a set is well-orderable, then its elements can be iterated.
Infinite sets are equivalent.
Regards,
Ross Finlayson
--
"The key, of course, is to synthesize a context around each and all of
the quotes."
|-|erc
the solution to all your problems, 100 years of trying to connect the dots is here!
How many digits of a random sequence have that digit, and every preceding digit up to that digit
occur in order in the right spot on (a member of) the computable list?
Sequence = <6545345.....................................................................................>
|<------how many digits---------->|
UTM(row,col) mod 10
{
1 <333333333..>
2 <300000000..>
3 <699999999..>
4 <654314314..>
..
}
Herc
--
No act is more patriotic than speaking out when your government
is doing the wrong thing in your name. This is not your right
but your sacred duty. And none are more treasonous than those
who would silence these voices.
-------------------------------------s-o-s------------------------------------
John Savard
wrote, in part:
>In some recent discussion about the well-ordering of sets dense in the
>reals, concepts about the specificities of a "previous" and "next"
>points on the real number line broaden and solidify.
An excellent post, but I'm afraid that the chances that the fellow named
Muckenheim will read it, and see where he has gone wrong, are not high.
John Savard
http://home.ecn.ab.ca/~jsavard/index.html
John Savard
wrote, in part:
>Under what conditions can it be said the reals are well-orderable? In
>ZFC, any set is well-orderable. If any set of reals dense is
>well-orderable then so is the complete superset and any subset. Is not
>that obvious to you?
>
>If a set is well-orderable, then its elements can be iterated.
>
>Infinite sets are equivalent.
Infinite sets are not equivalent. The set of real numbers cannot be
placed in a one-to-one correspondence with the integers; the set of
rational numbers can be placed in a one-to-one correspondence with the
integers.
The rationals can be well-ordered, but when they are well-ordered, they
are not in order according to their numeric magnitude.
Theoretically, we expect from set theory that a well-ordering of the
reals should exist, but no one has yet constructed such a well-ordering;
we do not know how to spell out an order to put the reals in that is
well-ordered. We only know many different well-orderings of sets with
cardinality aleph-null, some with very high ordinality, and hence
properties different from the usual numerical ordering of the integers.
John Savard
http://home.ecn.ab.ca/~jsavard/index.html
Ross A. Finlayson
Hi,
Why thank you.
I think that the Cantorian transfinities are like training wheels that
only go in a circle. They help in the understanding of their mechanics
with some aspects of the "INFINITE" set but it's difficult to get
anywhere. Removing the training wheels leads to its own difficulties,
and freedoms.
I was worried when my dad took the training wheels off my bicycle.
They were only on for a week after I heard they should be on for
longer. I was six years old. It worked out, I never hurt myself too
bad on my bicycles. I've been through four bicycles: banana seat,
Raleigh, 10-speed, and Trek, my father's old bicycle.
I don't necessarily agree with Mueckenheim, or Adkins, among the many
people with registered disagreements about infinite sets, although I
have in the past suggested the consideration that there are only
rational numbers, or infinite sets like the naturals contain an
infinite element. I also qualify those statements and use them to
amuse myself and hopefully others.
Those are things that come to mind when you think about infinity for a
couple years, particularly in the context of mathematical infinity and
even the state-of-the-art of set-theoretical mathematical infinity.
A lot of people have problems with the Cantorian proofs about the
uncountability of infinite sets, and I can see why: if you well-order
a set (eg the canonical ordering operator application) then
inductively, for each element, of one infinite set there one unique
element of the other.
The powerset result is kind of deep, to reconcile induction and the
reductio ad absurdum, I believe, or proof through contradiction, or the
powerset result, you get to quite a few avenues of the infinite as
zero, for example, or negative one, and the specific and specialized
identity of numbers and their application. For mathematicians, the
subfield of set theory particularly as it is accepted as the foundation
and explanation of the deepest roots of mathematics is to
mathematicians like the nature of god is to priests or the nature of
the universe to physicists, or the nature of being to humans.
While that's so it's _mathematics_ and thus immediately accessible,
communicable, irrepudiable, and permanent.
Thus, I saw fit to get it out of my system, so I can enjoy mathematics.
It's better to discover new mathematics.
Regards,
Ross Finlayson
--
"I know... I just love hearing it."
Ross A. Finlayson
No, infinite sets are equivalent.
The reals and any subset of the reals are totally ordered (by their
numeric magnitude). I guess I think a proper well-ordering of the
reals is its natural total order.
Sure it is.
It's confusing to think of the total ordering of the reals and the
properties of the previous and next points in the total well-ordering
of the reals. There are everywhere and only real numbers within the
reals, thus in the total ordering there must be a previous and next for
each. There is not necessarily much to be said about these adjacent
values except they're real numbers and different. One thing considere
in the thread is what it means for the reals to be divided into two
disjoint, dense sets whose union is that set, and how the implication
remains they alternate, or not.
One reason why to consider they do not is that the density property of
having infinitely many of each and the other dense set type only holds
for a finite, non-infinitesimal interval, and that on the contiguous
subsequence of the total well-ordering "scale" that the density
property doesn't hold, but when you deem infinite sets equivalent
that's not a problem anyways.
I thought the notion of well-ordering the reals is that each set in its
ordering, for a finite interval, has a least element. If that is so,
then it is apparent that where each of its subsets is well-orderable
that the sequence formed x E X < (y E X != x), the value x from set X
that is less than any element y in X other than x, the sequence formed
from the least element, and then recursively from X \ x, is the
sequence of the elements in order.
It is obvious that assigning x a name to begin with from, say, X =
R(0,1), that the name of x is not 1/2 nor 1/4 nor 1/8 nor 1 millionth,
it's an infinitesimal, but the set is only of reals so it's named iota.
When you look at EF + REF, eg the constant function f(x)=1 defined on
the naturals, then it's similar to a step function or rather square
wave, where as the unit square is halved and cut to thirds and fourths
etcetera, in the extreme the split that leaves a point- or zero-width
doubles its area. That makes some sense in talking about Banach Tarski
ball cutting, in the analog to volume with the continua. For example a
circle, or rather disc, should be able to be divided into two
everywhere discontinuous shapes or point-sets and via only translation
form two identical discs, with perhaps a missing point in one or both
copies.
Anyways, back to well-ordering the reals, from any given point you can
determine a well-ordering, but the "definite" value for any element of
the set is unknown except for that if the well-ordering begins at a
closed interval endpoint then that element of the sequence is definite.
The others are indefinite in their real, scalar value, in that they
have a real, scalar value, and it is hidden, and for any finite
subsequence of the total well-ordering sequence no other definite
values can be known, and each is less than any definite value of the
interval besides the closed endpoint.
Regards,
Ross Finlayson
--
"Also, consider this: the unit impulse function times
one less twice the unit step function times plus/minus
one is the mother of all wavelets."
bri...@encompasserve.org
> The reals and any subset of the reals are totally ordered (by their
> numeric magnitude). I guess I think a proper well-ordering of the
> reals is its natural total order.
Sure. If you believe that every subset of any totally ordered set has
both a smallest and a largest element then it follows that every
totally orderd set is well ordered.
You're a fellow who believes that N contains oo.
You're a fellow who believes in two real numbers, infinitesimally
close to one another so that there are no other numbers in between.
So one would expect you to fall for this fallacy, hook, line and sinker.
John Briggs
John Savard
wrote, in part:
>No, infinite sets are equivalent.
Are you familiar with Cantor's diagonal proof? It really is a proof, you
should know, even if much else about infinite sets is still not fully
established.
>The reals and any subset of the reals are totally ordered (by their
>numeric magnitude). I guess I think a proper well-ordering of the
>reals is its natural total order.
An ordered set is well-ordered when there is a smallest element in every
subset of that set. What is the smallest number in the subset of the
reals "all real numbers greater than 3"?
The answer is that there is none; any real number different from 3
differs from it by a constant amount. Hence there are other real numbers
between it and 3.
John Savard
http://home.ecn.ab.ca/~jsavard/index.html
John Savard
wrote, in part:
>Why thank you.
I made the mistake of assuming it was an orthodox discussion of the
issues without properly reading it.
>I think that the Cantorian transfinities are like training wheels that
>only go in a circle. They help in the understanding of their mechanics
>with some aspects of the "INFINITE" set but it's difficult to get
>anywhere. Removing the training wheels leads to its own difficulties,
>and freedoms.
If you assume the contrary of statements known to be true, you will get
to interesting places, but they won't be very useful.
John Savard
http://home.ecn.ab.ca/~jsavard/index.html
Jesse F. Hughes
> I think that the Cantorian transfinities are like training wheels that
> only go in a circle. They help in the understanding of their mechanics
> with some aspects of the "INFINITE" set but it's difficult to get
> anywhere. Removing the training wheels leads to its own difficulties,
> and freedoms.
The training wheels you removed are the restrictions of mathematical
rigor and sound deduction.
But enjoy your freedom, you deep thinker, you.
--
"Now I'm informing all of you that the people arguing against me are EVIL,
yes they are real, live EVIL people as mathematics is that important, so
it's important enough for Evil itself to send minions like them."
-- James Harris on Evil's interest in Algebraic Number Theory
Lee Rudolph
>"Ross A. Finlayson" <r...@tiki-lounge.com> writes:
>
>> I think that the Cantorian transfinities are like training wheels that
>> only go in a circle. They help in the understanding of their mechanics
>> with some aspects of the "INFINITE" set but it's difficult to get
>> anywhere. Removing the training wheels leads to its own difficulties,
>> and freedoms.
>
>The training wheels you removed are the restrictions of mathematical
>rigor and sound deduction.
>
>But enjoy your freedom, you deep thinker, you.
And don't forget to wear your helmet!!!
Lee Rudolph (if it's too late to prevent brain damage, in this case)
Ross A. Finlayson
The set N is the set of finite natural numbers. If there are
ubiquitous naturals then N is the set of all sets. The set of sets
that are rootsets of the empty set is the empty set, the contents of
which is the empty set. That doesn't change that the powerset of the
empty set is not the empty set, because the empty set is deemed a
subset of every set. That's where the powerset is order type is
successor except for the special case of the the ur-element, the only
set in the set theory that is also not a set, the ur-paradox that is
not. The set of all sets is its own powerset.
So, I don't necessarily believe that N contains oo: N = oo, and N E N+1
( N E P(N) ), and oo+1=oo.
There are only real numbers on the real number "line", that is to say
the set of real numbers. In a total well-ordering of the reals (or
rationals or irrationals), it's fair to say that for a given definite
real there is a previous and next real number, because the set contains
only real numbers. You can say that there exists a well-ordering of
the reals and I showed above in a simple and direct manner how there
could thus be a total well-ordering.
So, you say there is a fallacy and imply that "a total ordering is a
well ordering" is false. A total ordering on the reals has no greatest
and least element, it diverges towards infinity towards the negative
and positive. Neither does the set of natural numbers, it diverges
towards infinity in its total and well-ordering, nor does the set of
all integers. So that's one reason why that the least or greatest
would apply to a set of the reals with the infimum or supremum, the
greatest lower bound or least upper bound. Alternatively any subset of
the reals has a total ordering and thus a well-ordering.
The reasons I say that is not blind refutation and denial of
"orthodoxy" but informed logical progression using common vocabulary
and first order logic, coincidentally to work with the axiom free
theory that is strong enough to represent the natural integers and
having no non-logical axioms free from incompleteness, in these
modular, logical components.
That is to say, I think you understand what I say and why about that if
you can well-order the reals then its total ordering is a well
ordering. I don't have a hidden agenda, I just want to establish among
us that the truth value of that statement is true, that it is a theorem
of the logic (logical theory) in use, and that is thus usable as a
mathematical fact.
So, via well-ordering with the ability to enumerate the total ordering,
then there is determined a sequence of real numbers, in order, with
none missing between them. They're real numbers because there are only
real numbers in the real numbers. They're enumerable because of their
total well-ordering. So there is a sequence of real numbers and to
give names to one of them, iota, is a way to impart structure to the
sequence, in defining names and rules of operations upon these sequence
elements, representing contiguous elements, for their consideration and
discussion at leisure.
Deduction ensues and it can be seen that any finite sequence contains
at most one "definite" real number, the rest being "indefinite". Iota
might be non-constant, it's the least infinitesimal. There are smaller
infinitesimals, or perhaps not. Infinity happens to be larger than any
finite number.
John, what do you think about EF? Orthodoxy says the domain of EF is
countable, thus with measure zero. I disagree, because x/x = 1. Is
the impulse function a function? I guess I hope you would elaborate
and go into detail on your opinion on infinite sets in a set theory.
Ross A. Finlayson
a well-ordering, where the well-ordering is a total ordering, or are
Dedekind cuts or Cauchy sequences insufficient to represent all reals?
It's not contradictory to say that any finite interval of the reals
they are well-orderable and that you can give names to the elements in
the sequence without knowing the scalar value besides its difference in
terms of sequence index from the lower boundary of the interval.
Consider the entire set of integers, positive and negative, they can be
ordered in a way as so: 0, 1, -1, 2, -2, ..., with considering the
negative sign an extra bit in terms of absolute value. Then again it
could also be 0, -0, 1, -1, 2, -2.... Similarly, for the reals: 0,
iota, -iota, 2iota, -2iota, ..., compared to 0, iota, 2iota, in this
context and not to be confused with the square root of negative one 0,
i, 2i, 3i, ..., or perhaps more compactly 0., 1., 2., 3., .... That
gets into the monotonic and everywhere non-monotonic functions.
Heh, that's a pun about compactly, about the natural numbers and the
definition of compactness. 0^0 = 1. The direct sum of infinitely many
copies of N is _defined_ to be not the empty set, un-mechanistically.
Infinite sets are equivalent!
Here are my problems that I have yet to resolve to my own satisfaction:
the arbitrary radix antidiagonal, although that's not so bad, because
x=x, and the binary coded powerset antidiagonal, although there is
convergence (re: "Antidiagonal, Infinity", there's always one more,
induction vs induction failure, finitism, uncertainty). The nested
intervals result about mapping the naturals to the reals, or
Cantor/Megill style, Cantor's first proof, does not have a problematic
issue for me at this point. I use induction so believe that infinite
sets are equivalent. ("Set of all sets." "Class of all classes."
"Half a shadow play.")
I really think the primary objective of calling infinite sets
equivalent is to enable the rigorous footing, among others, of analytic
results to do with concepts like the equivalency function: EF, the
natural/unit equivalency function.
That's because in consideration of the continuous and discrete, that
some things are continuous and others discrete, there is probably
utility in deriving solutions that are related to relations of those
types of things. So, I am casually seeking a real world phenomenon to
predict, in that way.
Another consideration is number-theoretic extensions, way beyond half
of the integers are even.
Another notion is that acceptance that infinite sets are equivalent
would lead to reform of measure theory, or rather, a rational basis for
the reevaluation of measure theory and its foundations. That would
strengthen it and help bring some parts of it, basically working with
what is called "measure zero", from the realm of unapplied, or
unapplicable, "pure" mathematics to useful mathematics for solving real
world problems.
This is getting beyond the simple notion that any set that has a total
ordering, for example the complex numbers, that that total ordering is
a well-ordering.
So: the total ordering is a well-ordering.
Regards,
Ross Finlayson
Timothy Little
> So, are the reals well-orderable where their total (linear) ordering
Their usual linear ordering is not a well-ordering. That's not
surprising, it also applies to the rationals and the integers.
The rest of your post was rambling nonsense, snipped.
- Tim
Ross A. Finlayson
No, the point is that the total ordering of the finite or semi-infinite
interval comprising a set is a well-ordering. Any well-ordering is a
total ordering. If you can well-order the reals is not that
well-ordering a total-ordering? As there are only two main total
orderings on the reals "<" and ">" is not thus one or both of them a
well-ordering?
The total ordering on any finite interval is necessarily a
well-ordering and the (a) name of the least element greater than zero
is iota.
As well, any semi-infinite, or infinite to the left or right on the
number line, interval's total-ordering, where the endpoint is least
element, is a well-ordering.
It is shown how a simple total ordering of all reals, for example
interleaving elements of the well-orderings of the positive and
negative reals, is a well-ordering of all reals.
That the set of reals is well-orderable is a consequence of the Axiom
of Choice in Zermelo-Fraenkel set theory with the Axiom of Choice or
ZFC.
So, the set of reals on the unit interval, or any other finite
interval, is well-orderable and any total ordering of the real numbers
from that interval is a well-ordering.
Many other total orderings of the reals could be considered. For
example, divide the set into however many nonintersecting, dense,
completing subsets, order those arbitrarily, and apply after that
each's total ordering.
Thus are provided a variety of examples of well-ordering the reals, and
descriptions of a well-ordering of the non-negative reals with the
initial elements of the sequence thus generated 0, iota, 2iota,
etcetera.
If you reject that logic, you apparently see no utility in analytic
results among the continuous and discrete, nor non-standard measure
theory, or for that matter non-standard analysis.
Besides your flippant non-response, do you have a point? What do you
think set theory is? Do you consider set theory to be applicable as
the foundation of mathematics, or do you perhaps prefer a hybrid with
numbers as primary objects, or perhaps category or type theory? I know
little about your position on set theory and the foundations of
mathematics. Basically I've never heard of you. Do you have anything
to say? You appear to have been a regular poster to "rec.arts.sf.*"
some years ago more than sci.math, there's nothing wrong with that, in
the past year it can be seen that you write mostly to sci.math. Please
dont feed the trolls. You introduced a discussion about Gosper's
suggestion that 1/3 of rationals have even denominator. I introduced
where I think half of the integers are even, and another where it
apears that half of the binary sequences have equal zero- and
one-density, and recently am on about the cumulative hierarchy as
manifestation of the integers. Are you a Platonist? Does being a
Platonist mean there's only one true theory? If so are you _not_ a
Platonist? What's your point? What are your mathematical beliefs?
Also, the above is not nonsense, although I do note a missing word:
"of". I made some mistakes in the above, about finite or semi-infinite
versus infinite "interval", and if you can otherwise not make sense of
those words I am not apologetic. Others derive some meaning from it,
perhaps you could find some to translate for you into terms you
understand.
For the reals on the unit interval, a total ordering is a
well-ordering. For any semi-infinite interval, there is a total
ordering that is a well-ordering, one of "<" and ">".
For the set of all reals there exists a total ordering composed with
"|x| < |y|" and "positive < negative" or "negative < positive" thus
that that total ordering is a well-ordering.
It's simple to say that there's a total ordering that is a
well-ordering for any set because all well-ordering are total orderings
and all sets are well-orderable.
Regards,
Ross Finlayson
--
"I suggest that you not reject it."
Timothy Little
> Any well-ordering is a total ordering. If you can well-order the
> reals is not that well-ordering a total-ordering?
That is correct. Every well-order is a total order, by definition.
However, not every total order on an infinite set is a well-order.
> As there are only two main total orderings on the reals "<" and ">"
> is not thus one or both of them a well-ordering?
No. There are infinitely many total orderings. Under the Axiom of
Choice, at least one of those is a well-ordering.
> do you have a point?
Hopefully, showing you how to fill the hole in your logic. It is up
to you whether that purpose is pointless or not.
> Also, the above is not nonsense,
Many of the phrases and propositions individually made sense. Some of
them were true. However, they were logically disconnected in a manner
that made nonsense of the post taken as a whole.
> For the reals on the unit interval, a total ordering is a
> well-ordering.
At least one total ordering is, of the infinitely many total
orderings, but not all of the total orderings are well-orderings.
> For any semi-infinite interval, there is a total ordering that is a
> well-ordering, one of "<" and ">".
Why one of those two? Why not one of the infinitely many other total
orderings?
> It's simple to say that there's a total ordering that is a
> well-ordering for any set because all well-ordering are total
> orderings and all sets are well-orderable.
It's simple to say, and in ZFC it's true. It doesn't imply that all
total orderings are well orderings.
- Tim
Ross A. Finlayson
The concept I want to promote is that the total ordering "less than" or
"<" for any finite open, closed, or half-open interval is a
well-ordering of the reals.
That is counter to the general argument that for a given real there is
no "previous" nor "next", instead, there is a previous, and next, and
then the idea is to assign names to the elements of the sequence that
are derived from the well-ordering of [0,1] as follows: 0, iota,
2iota, ....
When there is a bijection f between the naturals and set X then it is
simple to determine a total ordering between N and X, if the inverse of
f is f' then for x and y in X then if f'(x)<f'(y) then x "<" y in that
total ordering where N has a natural total ordering with ordinals.
When people consider a total ordering of the rationals they generally
consider that if x<y then x "<" y. Here I could use better usage of
the conventional term R to denote a relation. Anyways the perhaps most
obvious total ordering of the rationals is based upon the numeric
magnitude, or scalar value, due to trichotomy of the reals. Another is
defined by any bijection from the naturals to the rationals. Now for
some when they get a finite, open interval of the rationals, they might
have stated that the linear ordering was not a well-ordering, where
here it is seen that it is.
When there is a notion to well-order the reals, then a given total
ordering of the open unit interval of the reals that has a least
element or "avoids infinitely descending chains" of some form causes a
dilemna for some: if you combine the claim that there is no "least"
real on the open unit interval with that there is no bijection between
the naturals and reals on the open unit interval, then there is not a
total ordering that is a well-ordering, and thus there could be no
well-ordering, because a well-ordering is a total ordering. Yet, a
contradiction ensues from that there is a well-ordering for any set.
If instead the linear or trichotomous total ordering is deemed a
well-ordering, via the notion that the least element of the unit
interval is iota because the open unit interval is a subset of the
closed unit interval, where the total ordering is a well-ordering
simply with zero being the last element and proceeding elements named
integral multiples of iota, then it is seen that the linear total
ordering of the closed unit interval is a well-ordering for it and any
of its subsets. The open unit interval is a subset of the closed unit
interval.
In that sense it is a provisioned example of a well-ordering of any
finite interval of the reals: the linear or trichotomous total
ordering. Here trichotomous could mean "greater than" or "less than."
With a semi-infinite interval, that is infinite to the left or right on
the number line but not both, then the trichotomous total ordering with
one or the other of ">" or "<" is a well-ordering.
With the infinite interval or set of all reals, consider the two
semi-infinite intervals of [0,oo) and (-oo, 0]. Each is well-orderable
by a "main" or trichotomous total ordering, eg (0, iota, 2iota, ...)
and (0, -iota, -2iota, ...). Then, what seems to be the obvious
well-ordering of (-oo, oo) is (0, iota, -iota, 2iota, -2iota, ...).
That arose in earlier discussion as a function from the naturals to the
reals that would not engender contradiction in terms of "Cantor's first
proof" or "Cantor/Megill style", nested intervals, in the construction
of a bijection between the naturals and reals. That is to say, if f(0,
1, 2, ...)= (0, iota, 2iota, ...) and that is a bijection from the
naturals to reals, then Cantor's first proof does not say otherwise,
because that function would not fit the assumptions of the proof, thus
that it does not apply in its otherwise contradiction.
Consider Megill's computer "verified" proof about the rationals, with
the focal "for any q>0 in Q (the rationals), there exists x such that
q>x>0." Also consider that that focal statement is falsifiable.
It would seem then that from the closed unit interval's trichotomous
total ordering being a well-ordering, that it is also a well-ordering
of the open unit interval.
Do you know any total orderings of the reals besides those using
trichotomy? If there are none, is not at least one of those a
well-ordering?
Regards,
Ross Finlayson
Timothy Little
> The concept I want to promote is that the total ordering "less than"
> or "<" for any finite open, closed, or half-open interval is a
> well-ordering of the reals.
Why would you want to promote a concept that leads to contradiction?
I prefer my reasoning to be free of contradictions.
> the idea is to assign names to the elements of the sequence that are
> derived from the well-ordering of [0,1] as follows: 0, iota, 2iota,
> ....
So, iota is supposed to be real? The reals are closed under
multiplication, so iota*(1/2) is real. How does iota*(1/2) compare
with iota and with 0?
> Here I could use better usage of the conventional term R to denote a
> relation.
If you're talking about an ordering on the reals that isn't the usual
"<", it would be a very good idea to use some symbol other than "<".
Probaly not R, since that's the usual ASCII symbol for the set of
reals. Other ASCII symbols I've seen used for relations as binary
operator: "~", "#", and "<[".
> Anyways the perhaps most obvious total ordering of the rationals is
> based upon the numeric magnitude, or scalar value, due to trichotomy
> of the reals.
It's also by far the most useful. It has nice properties with
arithmetic, such as x<y => x+z<y+z for all z. It also has the useful
property that given the usual numeric representation of rationals x
and y, it is easy to decide whether or not x<y.
> Another is defined by any bijection from the naturals to the
> rationals.
For such bijections, you lose the properties that make the order a
useful and interesting relation to study.
> Now for some when they get a finite, open interval of the
> rationals, they might have stated that the linear ordering was not a
> well-ordering, where here it is seen that it is.
This is word soup. Care to clarify it? Who are "some"? With what
ordering is their interval defined? Where is it seen that "it is" a
well-ordering? You've only shown that rational well-orderings exist,
not that all total orderings are well-orderings.
> if you combine the claim that there is no "least" real on the open
> unit interval with that there is no bijection between the naturals
> and reals on the open unit interval, then there is not a total
> ordering that is a well-ordering
More word soup. All you have shown is that a bijection with naturals
induces a well-ordering. You have not shown that a well-ordering
induces a bijection with naturals.
> If instead the linear or trichotomous total ordering is deemed a
> well-ordering
A well-ordering cannot be "deemed". It has a definition, which can be
checked. It is either satisfied, or not. In this case, it is not.
Deeming it to be a well-ordering leads to contradiction.
> the focal "for any q>0 in Q (the rationals), there exists x such that
> q>x>0." Also consider that that focal statement is falsifiable.
It is not, since for any q, you can choose x = q/2.
> Do you know any total orderings of the reals besides those using
> trichotomy?
Infinitely many. For example: let q:N->Q be an enumeration of all
rationals. For any real x, define f_q:NxR -> N so that f_q(n,x) is
the n'th smallest j having q(j) in the upper dedekind cut for x.
Define a relation ~q on R^2 by setting (x ~q y) iff there exists n
such that f_q(n,x) < f_q(n,y) and f_q(m,x) = f-q(m,y) for all m < n.
Then ~q is a total order on R.
If q1 and q2 are different enumerations of the rationals, then ~q1 and
~q2 will be different total orders of R. Hence there are infinitely
many different total orders on R.
- Tim
Piotr Sawuk
> reals, concepts about the specificities of a "previous" and "next"
> points on the real number line broaden and solidify.
>
> 2. Is it possible to describe and parameterize all possible ways that
> they can be so subdivided?
as you hinted, there are already infinitely many possibilities to divide
rational numbers into sets dense in R: by using only powers of each
prime-number as the denominator. also algorithmically one could divide
real numbers into disjoint sets dense in eachother and in R, and since
uncountabe infinitely many algorithms do exist, so that partition would
contain uncountabe infinitely many such sets of real numbers each dense
within eachother. my guess is that theoretically it would be possible
to create more than alef2 sub-sets of R (such that no bijection does
exist from the set of those sets to the real numbers) together with a
well-ordering such that each such set is dense in all sets with a
higher order of that well-ordering, and each of them is dense within
the set of real numbers. if my guess would reflect the truth (and
thereby prove that indeed all infinities are equal even though no
injective function does exist from alef2 to alef1), how would that
then change your theories?
>
> Under what conditions can it be said the reals are well-orderable? In
> ZFC, any set is well-orderable. If any set of reals dense is
> well-orderable then so is the complete superset and any subset. Is not
> that obvious to you?
>
> If a set is well-orderable, then its elements can be iterated.
of course iterated by ordinal numbers. the whole point of well-ordering
aribatary sets is that even though you could create disjoint subsets
and put some well-ordering on each such subset, you simply can not write
down any algorithm such that found well-ordering does match the order
you already have given on that big set. it's like leaning out of the
window towards infinity: awful things would happen if you fell through,
it's a good thing that you only can see a small part of infinity, it
is a good thing that you can only reach an even smaller part of it, you
really should stop leaning out of that window and rather do something
productive, but finite in nature, instead...
--
Better send the eMails to netscape.net, as to
evade useless burthening of my provider's /dev/null...
P
Ross A. Finlayson
linear ordering, normal meaning by scalar comparison) is a
well-ordering is founded on the notion that the real number line is a
sequence of points, and that where it is enumerable it is necessarily
via names that reflect that addressing and addressal of the points on
the line.
We have by fiat (a consequence of axiomatization) that the reals are
well-orderable, in ZFC. Where the reals are well-orderable, then it
stands to reason that there is a well-ordering on the set. The most
"obvious" ordering on the reals is the normal, linear, total ordering.
Do you know any well-orderings of the reals?
I search for "well-ordering", "well-orderings" and "reals" and "reals
are well-orderable" and "well-orderings on the reals" and there are a
variety of academic results.
I read somebody say the reals' well-orderings have complexity Sigma-1-2
in Goedel's constructible universe and thus the well-ordering is not
Lebesgue measurable, and do not understand that. I don't yet know what
some few of those phrases mean and if they can't be explained
completely in a few paragraphs then I suspect them meaningless. If
something is independent of ZFC, then it's of little use to a
Platonist. Please describe Sigma and Pi of Goedel's constructible
hierarchy. Those are closely held.
If we consider the normal total ordering, referred to above as the
total ordering, then one consideration is to define a sequence that
represents that ordering for some subset of the reals.
V = L.
When the set of reals, to begin in the unit interval, contains any and
all reals >= 0 and <= 1, that ordering relation of <, =, and > is
transitive, or trichotomous. What that means if for any a, b, and c if
a < b and b < c then a < c. A subset of the unit interval contains
elements that obey the same ordering. For example if a, b, and c are
in X, Y, Z, ... where R[0,1] > X > Y > Z > ... then in each of the
subsets of the unit interval a < b and b < c => a < c.
If you define the reals to be a sequence of points on the real number
line then the obvious well-ordering is the sequence of points. That
does not necessarily contradict the definition of rationals as ratios
of integers or algebraic numbers as solutions of polynomials with
integer coefficients, those numbers exist as a sequence of points on
the real number line regardless of whether you know which one they are,
besides naming each. Where that is so, they are well-ordered. Where
the reals on the unit interval are well-ordered in that way, their
well-ordering is the normal, linear, total ordering.
A well-ordering is always a linear, or total, ordering. Where the
reals on the unit interval are defined as a sequence of points from
zero to one then the beginning of that sequence is the least element of
the total ordering, and as is seen above each subset has its own least
element in the total ordering and thus well-ordering. Is that not so,
where the set is the normally ordered sequence of points on the line?
If not, why not? If so, good.
Consider these definition of subsets of the reals and their least and
totally ordered succeeding elements:
[0,1] -> (0, iota, 2iota, ...)
(0,1) -> (iota, 2iota, 3iota, ...)
[1/2, 1] -> (1/2, 1/2 + iota, 1/2 + 2iota, ...)
(1/2, 1) -> (1/2 + iota, 1/2 + 2iota, 1/2 + 3iota, ...)
[pi-e, 1] -> ( pi-e, pi-e + iota, pi-e + 2iota, ...)
If the points on the line are labelled in that way, then that's a
well-ordering of the non-negative reals, and coincidentally the normal
ordering.
Regards,
Ross Finlayson
Ross A. Finlayson
That's interesting, for you to divide the subsets into more than
aleph_1 many. I hadn't thought of that. How is that done?
It's an interesting notion there that there would be aleph_2 ways to
divide the reals into sets with that characteristic. I'm not sure
about how to proceed showing that.
I partially agree with you about that infinite sets are equivalent. A
key concern is to be able to provide to others the rationale or logic
that they can use themselves or share with others so they can feel
rational and socially acceptible in terms of mathematical discourse in
their belief that infinite sets are equivalent. I proffer a set theory
with logical axioms, ubiquitous ordinals, and ur-element.
I think one term I have used that was unclear was "dense", referring to
the density property. It's about saying that the sets are dense "in
each other" and the reals, instead of just saying they are dense in the
reals and nowhere continuous which is the condition. The rationals are
not dense in the irrationals, for example, in the irrationals there are
no rationals to be in place in the normal ordering between any two
irrationals. The rationals are dense in themselves. In the reals,
each of the rationals and irrationals is dense and as well the
rationals and irrationals are disjoint and their union is the reals.
When it's written that the sets in question are dense in each other,
that means that any pair of irrationals that is defined as a sequence
of rationals is having infinitely many rationals between them.
That's about an inductive impasse between that on the "macro-" scale,
viewing the reals as rationals and sequences of rationals, between any
pair of rationals there are infinitely many rationals, that on the
"micro-" scale, it's difficult to say into which sets the reals are
divided thus that in terms of their propensity or, heh, "perspacity",
it is possible to describe or specify which real number is "next" after
zero or greater than zero and less than all other positive real
numbers, the least real number.
It might be intellectually easy to say something along the lines of
that: the well-ordering exists, it's the same as the normal ordering,
after zero is iota, and because zero is rational iota is irrational
(because it's a real number and all real numbers are rational or
irrational and _no two elements of the same NCD subset of the reals may
be neighbors_). Then again it's trivial to lump zero in with the
irrationals and say that iota is not an element of that set. A more
troubling progression is that iota is not an element of any of those
sets. The hyperreals are the reals, because the reals are continuous.
By the same token, lumping zero with the irrationals would make the
function defined on that set continuous at zero instead of everywhere
discontinuous, thus, the "trivial" set can be outlawed refining the NCD
condition to include everywhere discontinuous, nowhere continuous, or
NC2D.
So, you shouldn't have to go to the hyperreals or non-standard models,
because the real number line is continuous.
Thank you for your input. If you would please further develop the
notion of how many sets can be generated that would be useful for
proving your point.
Regards,
Ross Finlayson
Timothy Little
> We have by fiat (a consequence of axiomatization) that the reals are
> well-orderable, in ZFC. Where the reals are well-orderable, then it
> stands to reason that there is a well-ordering on the set. The most
> "obvious" ordering on the reals is the normal, linear, total ordering.
> Do you know any well-orderings of the reals?
Not any constructive ones. It is pretty easy to use the Axiom of
Choice to "define" one mathematically, but not in a particularly
useful way.
> I read somebody say the reals' well-orderings have complexity
> Sigma-1-2 in Goedel's constructible universe and thus the
> well-ordering is not Lebesgue measurable, and do not understand
> that.
I understand the concept of Lebesgue measureability, but not what it
means to have complexity Sigma-1-2.
> If you define the reals to be a sequence of points on the real
> number line then the obvious well-ordering is the sequence of
> points.
Unfortunately the reals are not a usual sequence of points since they
are uncountable, and a sequence is a map with domain N. The
definition of a sequence can be generalized to arbitrary ordinal sets,
and there is an uncountable ordinal that would suffice. However, you
won't find a map that preserves order.
This shouldn't be too surprising, since it's not specific to the
reals. The same is true of the integers and the rationals.
> Is that not so, where the set is the normally ordered sequence of
> points on the line? If not, why not?
The points aren't a sequence. It's not even true of the rationals,
which are countable, let alone the reals. Even the integers don't
form a sequence in their usual order. In all three cases, certain
subsets can be viewed as a sequence, but you either have to choose a
different ordering or leave an infinite number of elements out.
> Consider these definition of subsets of the reals and their least and
> totally ordered succeeding elements:
>
> [0,1] -> (0, iota, 2iota, ...)
I asked before but don't recall seeing a reply: where does iota/2
appear in this list? Any real number can be multiplied by 1/2 to give
another real number.
- Tim
Ross A. Finlayson
of that, a/2b, is also a rational positive number that is less than
a/b. Neither of them is iota.
If you consider that infinitesimals, which some equate to zero, are
positive, then as the real numbers are continuous, then the
infinitesimals in the neighborhood of zero are real numbers, because
the real numbers are continuous, and not just an Abelian group, ring,
and field, with a multiplicative identity. If there exists positive
and non-zero infinitesimals in the field of the real numbers and it is
a field then the real numbers also include a "point at infinity", or
otherwise the multiplicative inverse of each element of the reals is
contained in the reals.
What's 1/0?
If the reals include iota and are a field then iota's multiplicative
inverse is oo, or I, iI = 1, 2iota's multiplicative inverse oo/2,
etcetera. Where there are problems defining iota there will probably
be issues defining half infinity.
Where that is so then there is the consideration of iota/2. One notion
is that the division of iota is undefined, or iota/x = 0 for x > 1.
The idea and definition of sorts of iota is that it is the least
positive real number.
You ask what iota/2 is, my considered response is that it is plainly
not to be divided by two, if the result of iota/2 were to be itself be
positive then the original term was mislabelled and was not actually
iota.
That leads into the obvious notion that the infinitesimals do not exist
in the real numbers and that the integral multiples of iota are a
constant zero.
While that is so the reals are continuous, and indeed there are points
between zero and any 1/n for finite integer n, and that returns to the
notion that there are positive, non-zero infinitesimals. The real
number line is continuous. It's nice to know that there exist real
numbers larger than zero.
One thing to consider is that iota, an infinitesimal, is an element of
the reals, but that it is not in the same field or ring structure as
the elements defined as solutions to polynomials with finite integer
coefficients that are not complex or continued fractions of finite
integers, or transcendentals defined as a converging value of an
algebraic, that the set of reals is comprised of two or more field
structures, perhaps indeed one for each element of the normal field,
and that that would apply to other similarly typed structures.
If somebody argues that there are no infinitesimals in the reals, then
I say there are infinitesimals in non-standard models of the reals, and
then say that they are the real numbers.
What do you think about that, the implicit fields with different
operations? I've been complaining about Dedekind and Cauchy's
insufficiency. If you talk about "non-computable" reals, so have you.
If the normal ordering of the non-negative reals is a well-ordering,
then that's a particularly useful way!
Regards,
Ross Finlayson
--
"Skolemize: your model is countable."
Timothy Little
> The idea and definition of sorts of iota is that it is the least
> positive real number.
It's a nice idea, but it doesn't work. The reals have a definition,
and iota doesn't fit it. If you want to define a set with something
like iota in it, go right ahead but you'll have to call it something
other than "the real numbers" since that name is already taken.
> One thing to consider is that iota, an infinitesimal, is an element
> of the reals [...]
"One thing to consider is that iota, an infinitesimal, is an element
of the Finlayson numbers [...]"
Fixed. Happy to help.
> If somebody argues that there are no infinitesimals in the reals,
> then I say there are infinitesimals in non-standard models of the
> reals, and then say that they are the real numbers.
There's a reason they're called non-standard. The standard term I've
always heard used is "hyperreals". You can play with them if you
want, but do try not to confuse them with the standard reals.
Unfortunately the hyperreals don't contain an "iota" element either,
and the total ordering on positive hyperreals is not a well-ordering.
Even more unfortunately, there cannot be a well-ordered set that
contains an order-isomorphic copy of the reals. Can you see why?
Here's a hint: it has something to do with the definition of a
well-order on a set being an order in which *every* nonempty subset
has a least element. If even *one* nonempty subset has no least
element, then the ordering is not a well-order.
- Tim
Chris Menzel
> We have by fiat (a consequence of axiomatization) that the reals are
> well-orderable, in ZFC. Where the reals are well-orderable, then it
> stands to reason that there is a well-ordering on the set.
It doesn't merely "stand to reason", it follows by definition of the
term "well-orderable".
> I read somebody say the reals' well-orderings have complexity
> Sigma-1-2 in Goedel's constructible universe and thus the
> well-ordering is not Lebesgue measurable, and do not understand that.
> I don't yet know what some few of those phrases mean and if they can't
> be explained completely in a few paragraphs then I suspect them
> meaningless.
Behold the root of all your problems.
Ross A. Finlayson
Hi Chris,
Yeah, it definitely stands to reason that if a well-ordering on the
reals exists then the reals are well-orderable.
About the succinct definition and why Goedelian hierarchical complexity
should be explainable tersely or else it's suspect, that's not a
problem.
Would you please summarize what those things are? Is it difficult to
summarize those things? Does their symbolic representation or brief
plain language statement require eighty pages of historical background?
Are there varying definitions of those things in use? Are those
things in constant and current change today and bound to be obsolete by
the time they could be transcribed?
They speak of complexity, I would think they have to do with
algorithmic complexity and O, "big O", and Theta notation.
Anyways, I hope you would describe those things as a working definition
will help us and others avoid confusion about them and their use.
Regards,
Ross Finlayson
Ross A. Finlayson
No, I just define real numbers to be all those on the number line, as
they were defined before Dedekind and Cauchy.
I think the hyperreals are just the reals and the hyperintegers are
just the integers. Here's one reason I think that: the reals are
continuous between zero and one, or more generally: the reals are
continuous on the real number line. Their continuity in being
continuous and the superset of all sets dense in the reals leads to
more structure than their status as a set, group, ring, and field. For
the integers, the sum over the natural integers n of 1/2^n equals the
limit of that sum, as Ullrich once said.
I think Dedekind/Cauchy might be insufficient to describe all of the
real numbers. If you have non-computable real numbers, can you define
them with cuts of the rationals? If not, are there consequently then
real numbers for which the Dedekind/Cauchy definition is not useful and
thus Dedekind/Cauchy is _not_ the definition of a real number?
Infinitesimals have infinitesimals, the point of iota is that it among
values less than1/n for any finite integer you name and it is greater
than zero.
The reason it's a real is that any value between zero and one is a real
number. Here then is consideration of how iota is between zero and one
on the real line. Consider the normal ordering, if a=iota, b=1/2, and
c=1/2+iota, then a < b < c and a < c.
That's similar in a way to comparing complex numbers, a+bi, by first
comparing the real component and then the imaginary component. Here,
the "reals with iota multiples" are a sum or difference of a real and
a non-negative integral multiple of iota.
The problem with that is that "infinitesimals have infinitesimals", and
as well if some value of the reals is less than 1/n yet positive then
there are infinitely many values of the reals less than 1/n yet
positive. There are values less than 1/n yet positive else the
rationals would comprise the reals.
With the notion of "extending" the reals with a ring at each field
point of iota-values, embedding rings but not fields of infinitesimals
in the reals, the difference between any two being iota, in that set
each of the subsets has a least element.
Is that not so? Why or why not?
I'm interested in your reasoning about well-orderings order-isomorphic
to the reals. Besides the things we discuss here, I have not yet
guessed or deduced what you mean. Please state that line of reasoning,
as it is presumably "well-known".
In the real numbers, there has to be something between zero and one:
each and every of the _points_ on the line.
Regards,
Ross Finlayson
Barb Knox
"Ross A. Finlayson" <r...@tiki-lounge.com> wrote:
[snip]
>I think Dedekind/Cauchy might be insufficient to describe all of the
>real numbers. If you have non-computable real numbers, can you define
>them with cuts of the rationals?
Yes. But for a non-computable real there is no *computable* Cauchy sequence
of rationals for it. Just because every single rational is computable does
NOT mean that every subset of the rationals is.
[snip]
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
Timothy Little
> No, I just define real numbers to be all those on the number line,
> as they were defined before Dedekind and Cauchy.
In other words, you want to replace a very good formal definition
which actually allows one to do rigorous mathematics, with a sloppy
geometric one that doesn't. There's a very good reason why Dedekind's
and Cauchy's definitions are held in high regard.
Even so, if you're referring to this vague definitionless set of
"points on the number line", give them a name other than "reals". As
I said before, that name is already taken.
> I think Dedekind/Cauchy might be insufficient to describe all of the
> real numbers. If you have non-computable real numbers, can you
> define them with cuts of the rationals?
Yes.
> With the notion of "extending" the reals with a ring at each field
> point of iota-values, embedding rings but not fields of infinitesimals
> in the reals, the difference between any two being iota, in that set
> each of the subsets has a least element.
It is not so. In a well-ordered set, *every* nonempty subset must
have a least element. That includes sets like {1/n: n in N}. This
set has no least element. Adding elements like iota doesn't help,
because you can always choose a subset that doesn't include any of the
elements you added.
Well-ordering is a much more restrictive property than you think.
- Tim
Ross A. Finlayson
Hi,
Yes, that's correct: _if_ the normal ordering of the non-negative reals
was a real ordering, _then_ the normal ordering of the non-negative
reals would be a well-ordering of any non-empty subset of the
non-negative reals. For me to claim that the normal ordering is a
well-ordering then, obviously, I would need to show that the normal
ordering is a well-ordering of _any_ non-empty subset of the
non-negative reals.
I'm not very familiar with the notion of the non-computable real, but I
think that I can draw an analogy to the notion that the normal ordering
is a well-ordering. If the set of rational 1/n for natural n is
considered in its normal ordering, besides that the negative normal
ordering would be a well-ordering for 1/n, then it's kind of like
saying: there's an infinite sequence with a beginning, and it's
considered reversed thus that it has a beginning and no end. (I can
see why the negative normal ordering would not be a well ordering of
1/n, but only for some subset of the natural integers with no least
element, of which there are none.) So, one or the other of the normal
or negative (or reverse) normal ordering is a well-ordering of 1/n.
That's blithe.
This is where the "negative" normal ordering is just the opposite of
the normal ordering, "<" replaced with ">" in the regular usage.
If I think that the normal ordering the reals of 1/n for natural n then
I should be able to state the least element of that set, which in the
limit is zero and is ever non-zero and positive. Where for the reals I
think (it is my opinion) and believe that as the set of reals is a set
of points on the real number line that the normal ordering is a
well-ordering and proceeds from zero as natural integral multiples of
iota, the notion of iota as the least positive real (and irrational)
does not allow iota to be the least positive member of the set defined
as the range of 1/n for natural n. Where that is so, the same
deduction that would define iota leads to the consideration that
inductively each subset of in this case the unit interval would be
comprised of iota values that are transitively linearly ordered, that
while indefinite in terms of indefinability as rational numbers, and as
well in terms of their natural integer multiplier, coefficient, or
index, are yet real numbers in the normal ordering. As well, each has
its own natural index, iota is reserved to be the least positive real.
If the normal ordering leads to that any (every) _subinterval_ of the
unit interval has a least element, then any _subset_ does as well in
the normal ordering.
Regards,
Ross Finlayson
--
"A model is to a theory as a class is to a set."
Timothy Little
> For me to claim that the normal ordering is a well-ordering then,
> obviously, I would need to show that the normal ordering is a
> well-ordering of _any_ non-empty subset of the non-negative reals.
Indeed. Now you know what you need to do, and you haven't done it.
Your task includes, among other things, demonstrating that a natural
number N exists such that 1/N is smaller than every other member of
{1/n:n in N}.
I suspect that will be very difficult for you:
Suppose such N exists. Then N+1 is also a natural number, and so
1/(N+1) is in the set, but 1/(N+1) < 1/N. Thus 1/N is not the least
element after all. Our supposition (that the set had a least element)
was false. How sad.
[More paragraphs of word soup snipped, being irrelevant to the problem]
- Tim
Ross A. Finlayson
Any well-ordering is a total ordering, so any well-ordering of the
reals comes from the set of total orderings of the reals.
One obvious total ordering of the reals is the normal ordering.
Here, in the consideration of the reals, it is appropriate to start
with the unit interval of reals, in determining total and
well-orderings on the unit interval of reals, and then from those,
where there is a well-ordering on all of the reals, then the
well-ordering of the reals is obviously a well-ordering of the (set of
elements of the) unit interval of the reals.
Besides the obvious normal ordering of the reals, other total orderings
of the reals imply the definition of a comparator, eg "<[". If there
is a function f between the naturals and reals of the unit interval
then the comparator is obvious the total ordering on the naturals of
f', in this context the inverse of f instead of its first derivative or
other meaning.
Now, I don't agree with that there doesn't exist a function between the
naturals and unit interval of reals, I say that infinite sets are
equivalent, in acceptance that many do not accept that fact. So
anyways the function f I use is the Equivalency Function, where "<["
has the same meaning of "<", that is to say, if the Equivalency
Function is a bijection between the naturals and unit interval of
reals, then the normal ordering of the reals is a well-ordering of the
unit interval of reals, as is the reverse EF and sufficient
compositions of EF.
Putting EF aside, as it is not accepted by everybody, then I hope you
could help me define total orderings on the unit interval of reals that
at no point are compositions of the normal ordering. What's a total
ordering, not well-ordering, on the reals besides the normal ordering
or anything constructed from the normal ordering?
For the unit interval of rationals, any bijection from the naturals to
the rationals is an obvious total ordering of the unit interval of
rationals. If you do not allow bijections between the natural integers
and reals of the unit interval, then there is not an obvious total
ordering beside the normal ordering and "compositions" of the normal
ordering. That is to say, if you define an invertible function with
the domain being the naturals that explicitly gives for each element of
the unit interval of reals a natural to indicate their total ordering,
then there exists a bijection between the naturals and unit interval of
reals.
Before, you indicated that the "sequence" or "function" would instead
have as an index or domain the infinite ordinals, or perhaps
hyperintegers. This is where the ordinals are well-ordered, there is a
minimal element in the set theory, the empty set. Is that the way it
is?
My idea is that if you define a well-ordering of the reals, then I
would challenge myself to present real numbers that were either
indistinguishable under your definitions or illustrative of the lack of
a total ordering and thus a well-ordering.
When the normal ordering is a well-ordering, then that would not be
possible because the normal ordering of the reals is a well-ordering,
the reals are totally ordered under the usual definition of "less than"
or "greater than".
Now to go back to the previous post, about the discussion of the
well-ordering of the unit interval by its normal, total ordering in as
necessary a non-standard model of the real numbers: if the infinite
sequence (0, iota, ...) represents the least two elements of the
non-negative reals, and each successive real in its normal ordering,
then the non-negative reals are well-ordered by their normal ordering.
Regards,
Ross Finlayson
Timothy Little
> I think we agree here that the reals are well-orderable.
We do, in the context of ZFC. (Which is the context I use when I
don't specify a context)
> Any well-ordering is a total ordering, so any well-ordering of the
> reals comes from the set of total orderings of the reals.
Quite true.
> One obvious total ordering of the reals is the normal ordering.
It is indeed an obvious total ordering.
> Here, in the consideration of the reals, it is appropriate to start
> with the unit interval of reals, in determining total and
> well-orderings on the unit interval of reals,
That sounds fair enough.
> Besides the obvious normal ordering of the reals, other total orderings
> of the reals imply the definition of a comparator, eg "<[".
Sounds good.
> If there is a function f between the naturals and reals of the unit
> interval then the comparator is obvious the total ordering on the
> naturals of f', in this context the inverse of f instead of its
> first derivative or other meaning.
There is indeed a function f from the naturals to the reals of the
unit interval. However, there isn't a bijective one.
> Now, I don't agree with that there doesn't exist a function between
> the naturals and unit interval of reals,
If you mean there is a bijection, you're assuming a statement that
leads to a contradiction. I don't usually bother talking with people
who accept contradictions in the context of classical logic. They'll
believe anything. But I've got plenty of time, so I'll keep playing.
> What's a total ordering, not well-ordering, on the reals besides
> the normal ordering or anything constructed from the normal
> ordering?
I described an infinite set of total orderings of the reals in an
earlier post, which was not defined using the normal ordering on the
reals. Maybe you didn't read it carefully enough.
> If you do not allow bijections between the natural integers and
> reals of the unit interval, then there is not an obvious total
> ordering beside the normal ordering and "compositions" of the normal
> ordering.
Just because there's not an "obvious" one doesn't mean there isn't
one.
> Before, you indicated that the "sequence" or "function" would instead
> have as an index or domain the infinite ordinals, or perhaps
> hyperintegers.
An uncountably infinite ordinal.
> This is where the ordinals are well-ordered, there is a minimal
> element in the set theory, the empty set. Is that the way it is?
Yes, that's right.
> My idea is that if you define a well-ordering of the reals, then I
> would challenge myself to present real numbers that were either
> indistinguishable under your definitions or illustrative of the lack
> of a total ordering and thus a well-ordering.
Sorry, I can't give you a definition of a well-ordering for the reals
that will be of much use. I think any well-ordering of the reals is
necessarily nonconstructive, though I'm not certain of this.
> Now to go back to the previous post,
Good idea, since so far you have completely avoided addressing the
definition of a well-ordering.
> if the infinite sequence (0, iota, ...) represents the least two
> elements of the non-negative reals, and each successive real in its
> normal ordering, then the non-negative reals are well-ordered by
> their normal ordering.
Oh, just when I thought you were actually going to go back to the
previous post, you just come out with more word soup and unsupported
assertions. Boring.
{1/n:n in N}: that's a nonempty subset, so if the normal ordering is a
well-ordering, it must have a least element. Did you find the natural
number n such that 1/n is its least element?
- Tim
ken quirici
In the normal ordering, what is the least member of the set of all
reals greater than 3?
Thanks.
Ken
Ross A. Finlayson
1/m > 0, for no n does there not exist n+1 thus that 1/n <= 1/(n+1).
In the limit as n diverges 1/n = 0, and for any finite integer n: 1/n >
0.
Where that's agreed upon, there is this notion that the reals are
points on a line, beads on a string. As discrete elements if you
consider the reals to be a contiguous sequence of points between zero
and one, then somewhere between zero and one in the normal progression
a single point is the first element of the reals of that interval to
satisfy 1/n for some finite positive integer n. No particular
"definite" value of the reals can be considered to be that value, but
some "indefinite" element, an element of the contiguous, totally
normally ordered sequence of reals is that value, because the set of
all real numbers includes all subsets of the real numbers.
An obvious problem with that is that in the normal ordering if you did
select any finite integer n then there would be infinitely many values
of the sequence preceeding it, where the sequence is not to be doubly
infinite, that being an opposite of semi-infinite.
So a problem with, or reason to abandon the notion that, the numbers
can be defined in that way is that it implies the infinite set of
finite integers contains an infinite element, which while conceptually
feasible in terms of ubiquitous naturals and the set of all sets
containing itself, is still not a clear issue.
In that sense the normal ordering progression could contain only
indefinite values, with some notion that they were the same numbers as
the set of definite values. That is similar to the consideration of
the open interval only containing indefinite values: (iota, 2iota,
...), except that as "iota" could not be "rational", as one is the
least positive integer the first element of the sequence 1/n would be
the least positive rational instead of iota, the first positive real.
Here my goal is to avoid confusion and crystallize points of dissent,
towards reconciliation and a broader firmament.
In the consideration of dissent on points, the unit interval of reals,
totally, contains everywhere reals. The notion that there exists for
each real number x a real number y greater than x and less than every
other real, except for the upper boundary, that the reals are
contiguous, sequential, as points besides being continuous as the real
number line, is fraught with vagaries on the borderlands of sound, or
rather, practiced, definitions.
So, of the reals, I consider that there is not a "next" after zero, and
also that there is.
Where there is some least positive (infinitesimal transcendental) real
called iota, and integral multiples of iota are representative of the
real numbers and necessarily not translatable to "definite" real
numbers, except in terms of scalar infinities (x/2x = 1/2), then the
set of reals is a contiguous point set on the real number line, besides
being a field.
If the reals were defined in that way, perhaps obviously in not the
standard model, with the multiple definition of rationals and
irrationals and as well contiguous points in a sequence, then the
normal total ordering would be a well-ordering for any finite interval,
where they are not defined in that way, it is not so.
Obviously, in standard practice, it is said that it is not so and I'm
aware of that. I develop some lines of argument as to what
considerations would take place in a redefinition that supports the
reasoning otherwise, which is justified for various deductively
determined reasons, and consequences of such reasoning.
Regards,
Ross Finlayson
Ross A. Finlayson
>
> In the normal ordering, what is the least member of the set of all
> reals greater than 3?
>
> Thanks.
>
> Ken
Hi Ken,
Do you mean the least element of the set of all reals greater than
three?
That's what it is.
It's followed by the least element of the set of all reals greater than
the least element of the set of all reals greater than three.
Ken, you probably want a standard reply: the reals are not
well-ordered, in that their normal ordering is not standardly
considered a well-ordering of the reals. So, standardly, there is no
least element of the set of all reals greater than three.
Now, for you to have written to this thread that discusses a variety of
non-standard considerations of the real numbers and the number system
within the framework of general foundations of mathematics, perhaps you
have an idea that there are non-standard answers for your question, or
rather, there are others who would consider the non-standard cases.
Anyways, it is what it is. What do you expect the answer to be to that
question? Do you have a prepared reply? That could be unfair to you,
you could just be ignorant of this issue and have stumbled upon this
thread and had the sudden query of the orderings of the reals in the
neighborhood of three. Why do you ask? What is your answer to your
question?
The idea being presented here is that the normal ordering of the reals
is considered as a well-ordering of the non-negative reals, or
reversely the negative reals, or for either any finite subset of the
reals. Extensions to the structures adequate to define the reals in
this way are considered. The answer is 3+iota.
Regards,
Ross Finlayson
ken quirici
> ken quirici wrote:
> >
> > In the normal ordering, what is the least member of the set of all
> > reals greater than 3?
> >
> > Thanks.
> >
> > Ken
>
> Hi Ken,
>
> Do you mean the least element of the set of all reals greater than
> three?
>
> That's what it is.
>
> It's followed by the least element of the set of all reals greater
than
> the least element of the set of all reals greater than three.
You have a symbolic representaion of 'least element of the set of all
reals greater than three' below, so I'll wait til I get down there!
>
> Ken, you probably want a standard reply: the reals are not
> well-ordered, in that their normal ordering is not standardly
> considered a well-ordering of the reals. So, standardly, there is no
> least element of the set of all reals greater than three.
>
Yes, exactly.
> Now, for you to have written to this thread that discusses a variety
of
> non-standard considerations of the real numbers and the number system
> within the framework of general foundations of mathematics, perhaps
you
> have an idea that there are non-standard answers for your question,
or
> rather, there are others who would consider the non-standard cases.
>
> Anyways, it is what it is. What do you expect the answer to be to
that
> question? Do you have a prepared reply? That could be unfair to
you,
> you could just be ignorant of this issue and have stumbled upon this
> thread and had the sudden query of the orderings of the reals in the
> neighborhood of three. Why do you ask? What is your answer to your
> question?
>
My answer to the question, as you have already guessed, is that there
is NO least real greater than 3.
> The idea being presented here is that the normal ordering of the
reals
> is considered as a well-ordering of the non-negative reals, or
> reversely the negative reals, or for either any finite subset of the
> reals. Extensions to the structures adequate to define the reals in
> this way are considered. The answer is 3+iota.
> Regards,
>
> Ross Finlayson
Thanks for your reply Ross.
Actually, I had not realized that there is a theoretical framework
extant, in which this real number 3 + iota exists. Let me ask in turn
two questions of you:
1. Do you consider that this 3 + iota is a real, or rather do you
consider that it is a member of some superset of the reals? I suppose
this reduces to the question, is IOTA a real, or part of some superset
of the reals, since this superset, if it's what you're proposing, I
would assume would be closed WRT 'arithmetic'.
Now I'm really dredging, using my WAY-BACK machine to college -
but is what you're doing assigning a value 3 + iota to the limit of the
[Cauchy?] sequence that supposedly, in ordinary real arithmetic,
approaches but never equals 3? Where iota <> 0?
2. If iota is claimed to exist, <> 0, and be a real, then the closure
of the reals under ordinary arithmetic would imply that 3 < 3 + iota/2
< 3 + iota, thus contradicting the assumption that 3 + iota is the
LEAST
real > 3.
So, that's my perspective on your previous post, which
stated, rather baldly to my I-thought-was-well-educated collection of
assorted math facts and fallacies, that 3 < 3 + iota (or at least I
extended it to state) and there is no j < iota such that 3 < 3 + j.
Is my perspective hopelessly deluded, from your perspective? Basically,
I can't imagine how you can propose this real, or super-real if that's
what you're proposing, iota, and some sort of consistent arithmetic.
Altho I have to admit it seems kind of alluring in a funny kind of way.
Another way of looking at it may be that you think the interval from 3
to 3 + iota is empty? Is that part of your proposal? Well at least I
realize you weren't just trying to whatever that french expression is,
epater les bourgeois, or whatever - shake up the establishment, more
or less.As represented here in sci.logic (and of which I am a
fellow-traveller).
Thanks.
Ken
Ross A. Finlayson
ordering seems a worthwhile theoretical excursion is that a
well-ordering of the reals is proven to exist, but no examples are
provided.
Thus there is consensus that a well-ordering of the reals exists, so a
notion is to look directly under the the collective nose.
One possible construction of a well-ordering of the reals might come
from the discussions of surreal numbers. John Horton Conway, a
prolific and quite excellent mathematician who I think of most for his
sphere packing book written with N.J.A. Sloane, derives a system of
what he calls the surreal numbers in his book "On Numbers and Games."
The surreals are constructed in a similar manner as naive enumerations
of the rational numbers as a closure of their field for inductively
increasing members of the natural integers. There is the notion of a
time component, with new sets of surreal numbers generated each "day",
and on day omega new rationals are yet being generated and as well
irrationals numbers are first, which leads to a conundrum where omega
is not a finite natural integer, regardless of whether it is a member
or element of the set of all finite natural integers.
Could thus the surreal numbers be totally ordered by their generation?
Yes, they can, although it is unclear if all real numbers are
generated.
As that would be at least a total ordering, then there is the
consideration whether it could be a well-ordering, as any well-ordering
is necessarily a total ordering. If it's not a total ordering, it's
obviously not a well-ordering, being a total ordering is a necessary,
but not sufficient, condition to be a well-ordering.
It seems obvious that the total ordering of the surreals by their
generation day and index is not a well-ordering.
Then, in consideration of the reals and a well-ordering of the reals,
besides facile rationalizations of the normal, linear, total ordering
as a candidate for a well-ordering, another line or direction of
enquiry is which properties a well-ordering specifically of the real
numbers would have to have besides just the characteristics of a
generic well-ordering because of the properties of the real numbers as
being comprising each point on the line and being at once an infinite
field with multiplicative identity.
So, besides that I want to show some characteristics of the normal
ordering of the reals to help explain EF, the natural/unit equivalency
function, where sets dense in the reals are Megill's inadvertent
hostages, I hope to see that a redefinition of the reals that is at
once that of the field of the reals as basically the closure of the
hyperintegers or p-adics is as well a coincident, non-conflicting,
definition as a contiguous sequence of points, and geometrically,
towards new mathematical subject categories to do with relations among
the continuous and discrete.
So, in the consideration that a mapping between the naturals and reals
that is non-constant and either strictly monotonic or everywhere
non-monotonic and divergent is not precluded from being a bijection by
Cantor's first proof of the unaccountability of the reals, such a
function would as well be, as necessarily qualified, a well-ordering of
the reals.
It thus does not seem unjustified to consider the real numbers for what
they are in terms of search for a well-ordering of the real numbers, as
a set, that is their normal ordering.
Ken, I've read some of your posts to sci.math and sci.logic over the
past month or so.
I promote my theory with no non-logical axioms, where excluded middle
applied to the minimal (and dual and maximal) ur-element enables a set
theory with a set of all sets and resolution of Cantor, Russell, and
Burali-Forti, towards a set theory that can be consistent and complete
and compatible with a concrete T.O.E., "Theory of Everything."
There are only, and everywhere, real numbers between zero and one.
Regards,
Ross Finlayson
Timothy Little
> About 1/n, 1/N, 1/n for n E N, inductively for m > n there exists 1/n >
> 1/m > 0, for no n does there not exist n+1 thus that 1/n <= 1/(n+1).
> In the limit as n diverges 1/n = 0, and for any finite integer n: 1/n >
> 0.
Does this word soup have any relation to the question you need to
answer, that is, that {1/n:n in N} has a least element? It seems to
me that you know that it has no least element, and are willing to
write all sorts of mathematical formulas to avoid actually saying it.
[Snip drivel about "indefinite" elements that are somehow in a set
defined in terms of the natural numbers without actually corresponding
to any natural number]
> except that as "iota" could not be "rational", as one is the least
> positive integer the first element of the sequence 1/n would be the
> least positive rational instead of iota, the first positive real.
I don't care whether "iota" is rational, algebraic, transcendental,
indefinite, or running for President. Your claim was that the usual
ordering on the reals (with or without extra elements) was a
well-ordering. The set {1/n:n in N} demonstrates that it is not.
> So, of the reals, I consider that there is not a "next" after zero,
> and also that there is.
I don't have a problem with considering a set that includes the
rationals with their usual ordering, has a total ordering, has a
"next" element after zero, and indeed has a "next" element after every
element of the set. E.g. QxN with lexicographic ordering.
However, it is not the set called the reals (your terminology
problem), and no such set can be well-ordered (your mathematical
problem).
If you want to pursue your quest, you're going to have to give up the
rationals.
- Tim
ken quirici
> Perhaps one reason the well-ordering of the reals by their normal
> ordering seems a worthwhile theoretical excursion is that a
> well-ordering of the reals is proven to exist, but no examples are
> provided.
>
> Thus there is consensus that a well-ordering of the reals exists, so
a
> notion is to look directly under the the collective nose.
>
Hi Ross,
I'm not sure what you mean above. My first take on it is that you mean
you're really just POSTULATING that the normal ordering (3< 3.1,
4.57483 < 4.58483, etc.) is a well-ordering of the reals simply to
allow the POSSIBILITY of a well-ordering of the reals to be
entertained.
If that's what you mean, sounds good to me.
> One possible construction of a well-ordering of the reals might come
> from the discussions of surreal numbers. John Horton Conway, a
> prolific and quite excellent mathematician who I think of most for
his
> sphere packing book written with N.J.A. Sloane, derives a system of
> what he calls the surreal numbers in his book "On Numbers and Games."
>
> The surreals are constructed in a similar manner as naive
enumerations
> of the rational numbers as a closure of their field for inductively
> increasing members of the natural integers.
I didn't understand the naive enumerations of the rational numbers you
mention above. Is it something like the enumeration of the rational
numbers n1,d1,n2,d2,n3,d3,.... where you 'interleave' numerators and
denominators? (I didn't follow 'as a closure of their field for
inductively increasing members of the natural integers').
>There is the notion of a
> time component, with new sets of surreal numbers generated each
"day",
> and on day omega new rationals are yet being generated and as well
> irrationals numbers are first, which leads to a conundrum where omega
> is not a finite natural integer, regardless of whether it is a member
> or element of the set of all finite natural integers.
>
> Could thus the surreal numbers be totally ordered by their
generation?
> Yes, they can, although it is unclear if all real numbers are
> generated.
>
What do you mean by 'omega' above? Is it the number of elements in
the natural numbers? If so, 'day omega' is troublesome, at least for
me. Also, how do you generate the numbers? Also, there seems to be
a switch from surreal to rational to irrational which left me
hopelessly
befuddled.
> As that would be at least a total ordering, then there is the
> consideration whether it could be a well-ordering, as any
well-ordering
> is necessarily a total ordering. If it's not a total ordering, it's
> obviously not a well-ordering, being a total ordering is a necessary,
> but not sufficient, condition to be a well-ordering.
>
> It seems obvious that the total ordering of the surreals by their
> generation day and index is not a well-ordering.
>
It seemed a well-ordering to me (with a caveat about day omega and my
general befuddlement)!
> Then, in consideration of the reals and a well-ordering of the reals,
> besides facile rationalizations of the normal, linear, total ordering
> as a candidate for a well-ordering, another line or direction of
> enquiry is which properties a well-ordering specifically of the real
> numbers would have to have besides just the characteristics of a
> generic well-ordering because of the properties of the real numbers
as
> being comprising each point on the line and being at once an infinite
> field with multiplicative identity.
>
> So, besides that I want to show some characteristics of the normal
> ordering of the reals to help explain EF, the natural/unit
equivalency
> function, where sets dense in the reals are Megill's inadvertent
> hostages, I hope to see that a redefinition of the reals that is at
> once that of the field of the reals as basically the closure of the
> hyperintegers or p-adics is as well a coincident, non-conflicting,
> definition as a contiguous sequence of points, and geometrically,
> towards new mathematical subject categories to do with relations
among
> the continuous and discrete.
>
Way over my head - many terms I am completely unfamiliar with -
obviously not your problem.
> So, in the consideration that a mapping between the naturals and
reals
> that is non-constant and either strictly monotonic or everywhere
> non-monotonic and divergent is not precluded from being a bijection
by
> Cantor's first proof of the unaccountability of the reals, such a
> function would as well be, as necessarily qualified, a well-ordering
of
> the reals.
>
ditto (to my above remark)
> It thus does not seem unjustified to consider the real numbers for
what
> they are in terms of search for a well-ordering of the real numbers,
as
> a set, that is their normal ordering.
>
> Ken, I've read some of your posts to sci.math and sci.logic over the
> past month or so.
>
and?
> I promote my theory with no non-logical axioms, where excluded middle
> applied to the minimal (and dual and maximal) ur-element enables a
set
> theory with a set of all sets and resolution of Cantor, Russell, and
> Burali-Forti, towards a set theory that can be consistent and
complete
> and compatible with a concrete T.O.E., "Theory of Everything."
>
OK, more over my head.
> There are only, and everywhere, real numbers between zero and one.
Don't follow this.
> Regards,
>
> Ross Finlayson
same to you.
Basically interesting stuff but again much of it in realms of math
and/or logic that I am completely ignorant of.
However the notion of ordering the reals by a DIFFERENT but even more
clever ordering than the usual one is very interesting.
Thanks.
Ken
Ross A. Finlayson
A well-ordering is necessarily a total ordering so the well-ordering(s)
are a subset of the total ordering(s).
I'd like to know more about the well-orderings of the reals because for
any well-ordering of the reals, it is a well-ordering of each subset of
the reals, and thus, for example, a well-ordering of the rationals.
One of the points I want to make clear to you is that it's possible to
consider that the only total ordering of the reals that is a
well-ordering is a monotonic function from the naturals to each element
of an interval of the reals, the normal or "usual" ordering. That's
because if you use infinite ordinals to apply the total ordering, then
the finite ordinals of those as a subset of the indicies of the total
ordering would have to be well-ordered for the complete total ordering
to be a well-ordering, and the only way known to me for the naturals to
biject to the reals on the unit interval is a monotonic mapping or
undecomposable composition of monotonic mappings.
That is to say, because by the definition of well ordering each subset
so ordered is itself well-ordered by that well-ordering, in application
to infinite ordinals the subset restricted to indices of finite
ordinals would as well have to be a well-ordering, and the only way to
biject the naturals and reals necessarily must avoid the consequence of
the Cantor/Megill style theorems. Here, Cantor means about his first
proof with the nested intervals and infinite convergent sequences, and
Megill means Norm Megill and his computer-verified theorem that the
reals are uncountable unless considered by their normal ordering that
he showed to us here.
A theorem of ZFC set theory is that there is a well-ordering of any
set.
A well-ordering of the reals is in your estimation R x O for some set
of ordinals O including an "uncountable" ordinal and all lesser
ordinals. Some subset of O is N, and thus for some subset of R R_n the
well-ordering R_n x N is a subset of R x O and R_n x N is a
well-ordering of R_n and each subset of R_n. Here, the Cartesian
product X x Y is indicating a function from Y to X.
As the reals are restricted to the unit interval, then R_n is some
infinite subset of the unit interval that is well-orderable because of
a bijection between it and N, and because R_n is a subset of R and N is
a subset of O.
So, when there is a well-ordering of the unit interval then some subset
of the unit interval is well-ordered by a function between that set and
the natural integers. That could be f(n)=1/n, and for n in N there is
always a greatest element in the normal ordering, it is well-ordered as
f', the inverse. What I'm trying to figure out is why it couldn't be
that or what other considerations there would be, yet thus far in this
progression R_n could be a random scattering of points on the unit
interval. Indeed, from what I understand of the Pi and Sigma Goedelian
complexity definitions a well-ordering of the reals would be some
random scattering of points, in terms of the normal ordering.
While that's so, with using the normal ordering it's possible that R_n
= R, where infinite sets are equivalent. Infinite sets are equivalent.
There are many discussions and questions about why the normal ordering
of the reals, a total ordering, is _not_ a well-ordering. I wondered
it myself some years ago, here's a link to another consideration, with
the result among the participants that the normal ordering of the reals
is _not_ a well-ordering.
http://mathforum.org/discuss/sci.math/m/587599/588095
http://groups-beta.google.com/groups?q=%22well-ordering%22+reals
Here's a consideration, the normal ordering is the well ordering for
the set of all positive integers x of x/n for finite integer n, eg
integral multiples of a half, a third, etcetera. That is where the
domain is x={1, 2, ..., n}, for some finite integer n, the normal
ordering is a well-ordering for f(x)=x/n, and it is so inductively for
each n+1 in N. That is not the same as that it is true for the
complete set of natural integers, it is thus in that way a
non-inductive result.
I guess I'm back to looking for a way that the set of non-negative
reals is as well a contiguous sequence of points from zero, indefinite
except for zero, and showing that each non-empty interval on the reals
contains at least one real number.
Yeah, that's what I need: showing that each non-empty interval on the
reals contains at least one real number. I think that's a pun, but it
could be one of those "plain language multiply-implicative" statements.
It does because it's non-empty.
So, basically I'm taking a theory that defines the numerical system up
to the reals, and then adding that the normal ordering is a
well-ordering, then considerations what further definitions on the
reals make that true without invalidating other consequences of the
regular or standard definition of the reals that are known to hold true
for reasons of ubiquity and real-world applicability.
To that end, I wonder what I should avoid. What happens if,
non-inductively, the normal ordering is used as a well-ordering? Here,
non-inductively means, similarly to the sense as it is used above,
that it is still true that our example 1/N has a lesser element than
1/n for 1/(n+1), but that that doesn't show that the sequence of the
normal ordering doesn't contain only elements of the range of that
function.
What are the consequences of well-ordering?
Regards,
Ross Finlayson
Timothy Little
> In ZFC, there is at least one well-ordering of the reals.
As we both agree, and doesn't need to be repeated every time.
> A well-ordering is necessarily a total ordering so the
> well-ordering(s) are a subset of the total ordering(s).
Yes. More pointless repetition.
> I'd like to know more about the well-orderings of the reals because
> for any well-ordering of the reals, it is a well-ordering of each
> subset of the reals, and thus, for example, a well-ordering of the
> rationals.
I'd like to know more about it too.
> One of the points I want to make clear to you is that it's possible
> to consider that the only total ordering of the reals that is a
> well-ordering is a monotonic function from the naturals to each
> element of an interval of the reals, the normal or "usual" ordering.
It's possible to consider, yes. As in, wonder what consequences
follow and whether it might be true or false. However, reasonable
people stop considering it when it is shown to be self-contradictory.
> Here, the Cartesian product X x Y is indicating a function from Y to
> X.
You're confused. There is a relationship between a bijection X<->Y,
and some *subset* of X x Y, but the cartesian product does not
indicate a function.
> As the reals are restricted to the unit interval, then R_n is some
> infinite subset of the unit interval that is well-orderable because of
> a bijection between it and N, and because R_n is a subset of R and N is
> a subset of O.
There's no guarantee that any well-ordering of R will map N into the
unit interval, but there does exist a well-ordering for which this
property is true.
> So, when there is a well-ordering of the unit interval then some subset
> of the unit interval is well-ordered by a function between that set and
> the natural integers. That could be f(n)=1/n
It could indeed.
> Indeed, from what I understand of the Pi and Sigma Goedelian
> complexity definitions a well-ordering of the reals would be some
> random scattering of points, in terms of the normal ordering.
Yes, very likely.
> While that's so, with using the normal ordering it's possible that
> R_n = R, where infinite sets are equivalent.
In ZFC, infinite sets are not equivalent. If you want to devise a
system of mathematics in which they are, go right ahead. Just don't
expect anyone but you to be interested in it. I'm here talking about
ZFC, as it seemed you were too.
"Ross A. Finlayson wrote:
> In ZFC, there is at least one well-ordering of the reals."
> the result among the participants that the normal ordering of the reals
> is _not_ a well-ordering.
At least we're making some sort of progress.
> So, basically I'm taking a theory that defines the numerical system
> up to the reals and then adding that the normal ordering is a
> well-ordering
Which theory? So far, you've been talking about ZFC. If you are
really talking about some other theory, you'll have to present or at
least give references to the axioms and inference rules of that
theory.
In ZFC, if you add "the normal ordering is a well-ordering" to the
definition of the reals, then you get a trivial system that states
that every well-formed mathematical statement, even contradictory
ones, is true. I don't think you want that.
> To that end, I wonder what I should avoid.
Don't include the rationals in the set you want to well-order with
their usual ordering. The trouble is, once you do that you end up
with something that doesn't look anything like a good model for a
geometric line.
In fact, it is pretty easily shown that if every element in some set
has a "previous" point in some order, that order can't be a
well-order.
- Tim
Ross A. Finlayson
It's a theorem of ZFC, in this case I guess it was Zermelo himself who
proved this, that every set has a well-ordering.
So, it is known the set of reals, where the collection of all real
numbers is a set, has a well-ordering.
The idea of considering the normal ordering and why or why not and
under what circumstances it could be correctly called a well-ordering
partially comes about from wanting an _example_ of a well-ordering of
the reals.
The "Axiom of Choice", AC, basically says that a "choice function"
exists for every non-empty set, and the choice function returns an
element of the set, where the domain is the entire set and the range is
one element of the set. Then, if you subtract that element of the set
to get some proper subset, then it has a choice function, and you can
get an element from it. Repeated ad infinitum, that constructs a list
of those elements, return values from the choice function, that
sequence represents a well-ordering of the initial set, and thus
obviously each of its subsets. There exists x E X, x E X, , not x E
X-x, choice reduces to the quantifier.
Basically then, I'm looking for the choice function that for any set of
non-negative real numbers returns an element less than any other in the
set. For the set of positive real numbers I say there is one and that
the value returned is iota. There's that, and that for any set X of
non-negative reals numbers there is some least real number. That
contradicts with the true notion that for any positive real x, that
that number can be halved or otherwise divided by y > 1 and that x/y is
less than x. So, I say that the value returned, for example iota,
which while necessarily being a real number, is indefinite in that it
represents the necessarily existent point, less than any other, on a
sequential progression of the contiguous reals between zero and one.
That leads to the requirement of proof that the unit interval comprises
some sequence of reals with the normal ordering. That's exactly
equivalent to the statement that the non-negative reals are
well-ordered, by their normal ordering. It is necessary to not
presuppose that which is to be proven. However, it would be
acceptible, to begin, by presupposing the opposite and showing it
always contradictory.
That the normal ordering is a well-ordering (for the non-negative reals
or the unit interval) is the same thing as that there exists a choice
function to select not just an element but the least element, in the
normal-ordering, that there is a least element in the normal ordering.
As we agree, with the standard definition of the reals or set dense in
the reals there is _not_ a least element of the reals. That's because
the standard definition of the set of non-negative reals is not the
sequence of points from zero onwards that is continuous on a straight
line or ray. Where the set of reals is a contiguous sequence of
points, there is a least element by the normal ordering of each
subsequence and the normal ordering is a well-ordering.
One possible notion to address is that the non-negative reals are a
contiguous sequence of points or they would not be complete.
At one point I described that a set might be "ordering-sensitive", that
the sets' elements vary based upon the sequence of choice functions
used. I consider why the reals might be such a set as they are
continuous. That has to do with the two-sidedness of discrete points
or endpoints on the lines, and the one-sidedness of interior or rather
non-endpoints. Ken, that's that kind of conjecture, or, yes, perhaps
even postulate. Dedekind/Cauchy provides _examples_ of real numbers.
That the normal ordering of the reals would be a well-ordering is
conceptually similar in some ways to accepted theory about the
well-ordering of the reals: there is not a way to tell the "value" of
the next real in the ordering except that it is that. If the normal
ordering was a well-ordering, there is not a way to tell the "value" of
the next real number, only that it is a real number and less than each
other remaining.
With the normal ordering coinciding with a well-ordering, it is easier
to show infinite sets equivalent. Indeed, the range of EF is basically
the normal ordering (of the unit interval), but EF can use limit. EF
is a putative building block of bijections between the naturals and
reals.
This is where I think infinite sets are equivalent, which is _not_
standard, although I have proven it to myself. "Set of all sets, class
of all classes", and "no classes in set theory", those are little
mantras. So, I promote a non-standard set theory, and neither en- nor
disourage its use by you. I use it.
I guess the conclusion here is that if the unit interval of reals is a
contiguous sequence of points, then the normal ordering is a
well-ordering.
Regards,
Ross Finlayson
Timothy Little
> Basically then, I'm looking for the choice function
The Axiom of Choice says that such a function exists. It does not
guarantee that any finite expression will unambiguously describe it.
> So, I promote a non-standard set theory, and neither en- nor
> disourage its use by you.
You haven't promoted any non-standard set theory. The only one you've
mentioned is ZFC, which is standard. You're even using results from
ZFC to try to "prove" things. If you were using a non-standard set
theory, it would make no sense to use results from ZFC.
What is this theory: what are its axioms and inference rules?
- Tim
Ross A. Finlayson
I'm talking about the null axiom or axiom-free set theory with no
non-logical axioms, ubiquitous ordinals and an ur-element that is the
unique proper class and a set. In it, the powerset is successor is
order type.
By "no non-logical axioms", that means it revolves around tautology and
the excluded middle. Besides regularity, the axioms (non-logical
axioms) of ZFC are theorems. I guess it's Hilbertian.
Then, I think that's strong enough to model the integers, and
concurrently everything relevant about them.
A partial summary to whit is noted in the listing of "Claims" last year
on sci.logic.
I use ZFC in these discussions because as long as I don't make use of
regularity then it's largely the same thing, and in general people who
read sci.logic have been introduced to ZFC, the Zermelo-Fraenkel axioms
of set theory with as well the axiom of Choice: Z, F, C.
You might not be aware of this, I claim infinite sets are equivalent
and a variety of participants here have discussed that for some time,
with some agreement that what I say is acceptible when I'm not trying
to apply it to all foundations of mathematics, which I do.
I don't say "infinite sets are equivalent" in ignorance of, say,
Cantor, although I did originally. There are hundreds of pages of
discussions of arguments about the equivalency of infinite sets.
Antidiagonal is discussed vis-a-vis dual representation, and
uncertainty, as is the powerset result. Nested intervals are
confronted with monotonic mapping and sets dense in the reals, and to
some extent this discussion. Measure theory is claimed to require
retrofit. The dearth of results in terms of transfinite cardinals with
regards to anything besides transfinite cardinals is noted.
I don't find paradoxes acceptible (acceptable, acceptible). A paradox
is a _contradiction_. Infinite sets are equivalent. If I accept that,
then by reason I would have to not accept that contradictory to it,
basically throwing away infinite cardinals. I think what is _gained_
from that is a foothold of sorts for the consideration of analytic
results of functions with the domain being the naturals and the range
being the unit interval of real numbers.
As well, model expansion and higher order logic is shown to be the same
problem as Burali-Forti, which is the same problem as Cantor, which is
a similar problem to Russell. Skolemize: your model is countable.
Where everything is an ordinal, and the ur-element represents at once
null and U, the universal set, then the powerset is, in a way, the
order type and successor, or it has the same ordinal value. Besides
ubiquitous ordinals, it might be restrictable to ubiquitous naturals.
Then with f(x)=x+1 with the domain being the naturals, the result is
the naturals, or empty set. That has to do with dual representation.
About uncertainty, the ur-element is one or both of empty set and
universal set, and it doesn't matter which because the conclusion is
the same.
I realize that might not seem clear. It's not meant to be otherwise.
So, I've put forth little bits and pieces of logical progression, some
assembly is required.
V = L.
Anyways, in terms of "Well-Ordering(s) of Sets Dense in the Reals", I'm
trying to figure out a way to define the reals as not only the field
with multiplicative inverse that they are but also the sequence of
points that they may be. Unless otherwise noted, assume I use ZFC.
It would be coincidental that the normal ordering would satisfy the
Goedelian requirements for a well-ordering, I rather expect it would,
with a perhaps brusque interpretation.
What's iota?
Regards,
Ross Finlayson
ken quirici
Just a quick reply - thanks for your careful and well-considered
response. I have done some research on the web - mainly in
mathworld.wolfram.com and there is SO MUCH out there I do not know -
that in fact has been added to the mathematical/logical discourse
since I last dipped into it. The world sure does change, and in this
case, in this field, it has sure got more interesting.
So the Axiom of Choice and the well-ordering of every set are
equivalent. So the reals are well-ordered. So nobody has found an
ordering that is a well-ordering on the reals yet. So you are trying
to find some way of using an extension of the normal ordering to
attain that objective.
I really only have a few ideas to add to the pot:
one could add to the reals objects which are the 'results' of every
Cauchy sequence (I take it a Cauchy sequence is any sequence that tends
to a specific finite real number?) of the reals - is that sort of what
you're postulating or thinking about? I wonder whether it has a
different [cardinality? ordinality?] than the reals? It's really I
guess the set of all countable subsets of the reals. What the heck is
the cardinality of that?
I wonder if the uncountability of the reals is equivalent to:
they are NOT well-ordered
the Axiom of Choice only applies to a countable infinity of sets
(using this defn of the Axiom of Choice:
given a [countably infinite] set of sets, one can define a set which
has one member from each set - which is on the face of it a very useful
axiom)
Also I'm wrestling with cardinality, ordinality, and defining
well-ordering from that direction.
Really interesting stuff.
Thanks.
Ken
Ross A. Finlayson
You probably want a standard answer.
I don't have one for you, infinite sets are equivalent. That means
that they all have the same cardinal number. The transfinite cardinals
are not meaningful to me in the same way as they are standardly used to
generate towers of themselves.
That MathWorld is really good. Eric Weisstein's academic treasure
troves were so good that they were co-opted by the Wolfram Institute
for their web-site after Eric made a deal with CRC Press. Also the
planetmath.org and wikipedia are pretty good, don't miss
math-atlas.org, mathpages, Math Forum with Dr. Math, there are many
interesting things there. There are many more accessible resources
for self-directed learning in mathematics on the Internet than there
were ten years ago, not to mention there are some more known
mathematics since ten years ago. You might want to get a copy of
Abramowitz and Stegun's "Handbook of Mathematical Functions", is NIST
ever going to get that webpage done? I report errata. Access to a
good university library is much more useful for in-depth research into
the mathematical subject categories.
I think you probably want to read Rudy Rucker's "Infinity and the Mind"
if you haven't, although he is a bit psychedelic. He discusses
"standard" ordinals and cardinals in an accessible way, it's feasible
to ignore the other parts, or just read them. It's good reading if
you're actually interested in the theories of infinite sets and
mathematical foundations to do with the infinite.
Here, you should read this, it's a post to the FOM e-mail list from
Harvey Friedman about an AMS panel on the Continuum Hypothesis, and
other subjects including computer verification of proofs and proof
space search, in proof space construction.
http://www.cs.nyu.edu/pipermail/fom/2005-January/008756.html
Me, I'm more about the Upanishads, Zen, Sufism, Genesis, the Qabalah
and Gematria, and Classical Philosophical Reasoning: Infinity and the
Void. It's convenient where mathematical foundations concur with
existential phenomenology questions.
That's a joke. (That's a joke.)
Anyways, yeah, you're right, I'm trying to figure out a way that the
structure of the real numbers is extended, but not modified, to
consider that any set of real-numbers is well-ordered by their normal
ordering. In not modifying their regular meaning and implication, the
indefinite elements of the ordering can not be claimed to have a value
of any given scalar, only that they are real numbers and for some
finite sequence of contiguous elements among the reals the finite
sequence does not include two definite points, the infinite sequence
does. The infinite sequence contains a parameterized interval's
numbers, for non-negative reals and the normal, or usual, or canonical,
ordering.
I mass 70 kilos.
Regards,
Ross Finlayson
H. Enderton
Yes. But it is a theorem that, working in ZFC, you cannot
give a specific *definable* example of one. That is, for any
formula that you think might define a well-ordering of the reals,
there is a model of ZFC in which it doesn't. (A result due to
Sol Feferman, back in the early days of forcing.)
To digress, there is a definable well-ordering of the *constructible*
reals (i.e., the reals in Goedel's class L). But that's only a drop
in the bucket.
--Herb Enderton
Ross A. Finlayson
Hi,
Thanks, Herb. I think you can provide a lot of insight into this.
Forcing, or coconsistency, dependent consistency, is the notion that
some things are proved relative to the consistency of their theory.
That is, both are consistent, or neither, for example Con(ZFC) <=>
Con(ZFC + IST), where IST is Nelson's Internal Set Theory. A large
collection of those results would be decided were ZFC shown
inconsistent. ZFC is generally assumed to be consistent.
The "reverse mathematics" idea is to use the _least_ set of axioms to
prove a result. For example, instead of using all of the axioms ZFC,
only use the axioms you need. That kind of axiomatization basically
would help limit the repercussions of Not Con(ZF), where some subset of
the axioms of ZF would probably remain consistent, _were that to be the
case_.
Dr. Enderton, I hope you would explain some more of the meaning of
Feferman's result about well-ordering the reals. We could ask him.
http://math.Stanford.EDU/~feferman/
It would probably be a good idea if I wanted to know to research him
before contacting him. Then, I'd invite him to come inveigh on
sci.logic.
Herb, I think V = L. So do some others, so that is not quite so awash
as claiming infinite sets are equivalent in terms of consensus, a
plurality of people agree that V = L, that the universe is the
constructible universe. Does that apply to the Goedelian-constructible
reals? Does Feferman's result only thus apply if V =/= L?
For the constructible case, you relate that there is a "definable"
well-ordering: is there an example?
I was interested to read some of the descriptions on the web page for
the UCLA Logic Colloquium last week or so. One reads in Caicedo's
abstract:
"We show that fine structural inner models for mild large cardinal
hypotheses but below a Woodin cardinal admit forcing extensions where
bounded forcing axioms hold and the reals are projectively
well-ordered."
If "the reals are projectively well-ordered", is that interpretable as
"the reals are well-ordered" or "the normal ordering of the reals is a
well-ordering on the positive reals"?
I'm glad you wrote, it gave an excuse to ask you.
The normal ordering is a _total_ ordering of the reals. The choice
function of a well-ordering, as you describe, only returns a
real-number without any idea of what it _is_, except for being a real
number.
What do you think about well-ordering the non-negative reals and the
normal ordering as a candidate well-ordering? The provably existent
well-ordering constraints for the reals are semantically vague. In
what ways can you conceive that those definitions can be bent, and not
broken, and allow the normal ordering to be a well-ordering?
Regards,
Ross Finlayson
--
"On further review of the metric system, I mass 85 kilos."
Timothy Little
> Besides regularity, the axioms (non-logical axioms) of ZFC are
> theorems.
OK, so every theorem of ZFC that doesn't use regularity is a theorem
of your system. In the absence of any better information from you,
I'll have to make do with that.
> A partial summary to whit is noted in the listing of "Claims" last year
> on sci.logic.
All I found in Google for that subject was some random-looking
numbered bullet points not even close to a partial summary of anything
like a set theory. To be quite frank, it seems to me you're flailing
about looking for a system that gives you a theorem you like, without
actually knowing how to define a system in the first place and without
recognizing any other consequences of what few definitions you have.
Then you want to call what you end up with "the reals", without
qualification and without even knowing whether the set has properties
anything like the notion a continuous line of numbers with the usual
arithmetic properties.
> I realize that might not seem clear. It's not meant to be otherwise.
It's meant to be unclear? That's rather rude.
> Anyways, in terms of "Well-Ordering(s) of Sets Dense in the Reals", I'm
> trying to figure out a way to define the reals as not only the field
> with multiplicative inverse that they are but also the sequence of
> points that they may be. Unless otherwise noted, assume I use ZFC.
You haven't said what you mean by a "sequence of points" yet. Do you
want the rationals in it, with their usual arithmetic properties?
- Tim
Timothy Little
> one could add to the reals objects which are the 'results' of every
> Cauchy sequence
The reals *are* the results of every Cauchy sequence -- that's one of
their definitions. Every Cauchy sequence of reals has a real as the
limit.
> the Axiom of Choice only applies to a countable infinity of sets
That's not the Axiom of Choice, that's the Axiom of Countable Choice.
You can do set theory with ZF+CC, but (IIRC) the proposition that the
reals have a well-ordering is undecidable in that system.
- Tim
Timothy Little
> What do you think about well-ordering the non-negative reals and the
> normal ordering as a candidate well-ordering?
The normal ordering on the non-negative reals is a failed candidate.
What is so difficult for you to understand about this?
Is a well-ordering even what you want?
- Tim
ken quirici
Thanks for the reply. I'll look into those refs, thanks there as well.
I wish I hadn't let so long go by without refreshing my college logic &
math. Oh well.
Ken
Ross A. Finlayson
Hi Tim,
If the reals are well-orderable, then I have some considerations of how
to make use of those well-orderings.
Consider Cantor's first proof of the uncountability of the reals, can
you think of any well-ordering that would not suffer its conclusion of
uncountability, ie, not containing each element of the reals, and thus
being empty for some non-empty subset, and not containing a least
element? I know you say to just use the "uncountable ordinals". Is
that a corollary, that there can be no well-ordering of a dense set of
reals?
Where the normal ordering is a well-ordering, both of those problems
are resolved.
That's not to say that you can use the indefinite values for anything
besides enumerating the elements towards a choice function leading to a
well-ordering.
Similar arguments hold for the rationals, and other sets dense in the
reals, as do for the reals, as a set dense in the reals.
I'm interested in this Caicedo's discussion about the projectively
extended reals, latching onto an ephemeral quote that they're
"well-ordered", where that generally implicitly means "by their normal
total ordering", where it is generally accepted that the reals are
"well-orderable". I'm not claiming he says that without further
agreement, only that it was written in a casual sense. So I'm
wondering about just the _statement_ that the reals, in some way,
shape, or form, are well-ordered.
http://www.logic.univie.ac.at/~caicedo/papers.html
There there is some discussion of the "super-real" fields that were
passingly mentioned earlier here. I am unfamiliar with them.
What's 1/0? From http://mathworld.wolfram.com/FieldAxioms.html:
"a * a^-1 = 1 = a^-1 * a IF a =/= 0" (emphasis mine)
So, as long as you don't attempt to divide by zero, the field axioms
appear to be preserved. Does x/x = 1?
Tim, I'm looking for a well-ordering of the reals. If the definition
of the real numbers is amended thus that the normal properties hold
true for "definite" values, yet for these "indefinite" values they are
a contiguous sequence of points, then the normal ordering is a
well-ordering.
Perhaps instead I should only concern myself with the sequence of
points. That reduces into the same problem of showing the real numbers
to be a sequence of points. In a way, that's consideration that the
multiples of n, ..., 2, 1, 1/2, ..., 1/n for finite n are each
sequences of points. For each finite n it is a set not dense in the
reals, as a union for all of the integers it is.
The contiguous sequence that would be dense does not contain
identifiable elements that obey the general properties of the reals,
except for possibly one of those elements, the definite element. Yet,
where they exist, they would be real numbers because there are only
real numbers between zero and one.
It's like a jigsaw puzzle where each piece is numbered on the back,
with perhaps a direction. Where you can only flip them all over at the
same time, either way it's possible to reconstruct the puzzle, but you
can only see the picture on one side. If you want to draw a continuous
line between point a and point b on a piece of paper, generally that's
accomplished by drawing a continuous line from point a to point b.
Continuity is beyond the limits of precision, just barely.
I guess I need to learn more about generally accepted results about
well-ordering the reals, so I can utilize those results. Where V = L,
it may be feasible to do that.
About the or "my" theory, the ur-element is at once empty and
universal, it has multiple aspects, and infinite sets are equivalent.
Do you use transfinite cardinals for anything? I'm aware that standard
measure theory has that any countable domain has measure zero,
including sets dense in the reals.
In 2's complement computer logic, the 32 bits 0xffffffff equals
negative one, or (int) UINT_MAX == -1. Why would anybody ever think
that zero equals infinity? Perhaps it's because they use mathematical
logic to prove it to themselves, as I do.
That aside, this is a discussion of well-orderings of the reals and
other sets dense in the reals. It's shown in ZFC that there is at
least one well-ordering, yet no one wants to provide one because then
you could index that well-ordering by the natural integers.
Look directly to the normal ordering, and assume that it is not
contradictory to do so for reasons yet to be developed, or fiat, and
that it is not allowed to conflict with the "usual arithmetic
properties" because they're useful to solve real-world problems.
While you're at it, found a non-standard measure theory based on that.
If you don't care to consider that, there is some ordering of the reals
for which there is a least element of any subset of those numbers. If
you do consider it, then that's what it is.
Do infinitesimals exist among the real numbers? Short answer: no. Do
the hyperreals contain any element that is not an element of the reals?
Short answer, there are everywhere reals between zero and one.
Does the generic extension of N contain any elements not in N? No.
Regards,
Ross Finlayson
George Cox
>
> That aside, this is a discussion of well-orderings of the reals and
> other sets dense in the reals. It's shown in ZFC that there is at
> least one well-ordering, yet no one wants to provide one because then
> you could index that well-ordering by the natural integers.
No you couldn't, you'd run out of integers. (I'm not sure what a
"natural integers" is.)
Timothy Little
> Consider Cantor's first proof of the uncountability of the reals,
> can you think of any well-ordering that would not suffer its
> conclusion of uncountability, ie, not containing each element of the
> reals, and thus being empty for some non-empty subset, and not
> containing a least element?
Cantor's first proof says that any set having an ordering that is
linear, dense, unbounded, and complete must be uncountable. Any other
orderings it might have (including well-orderings) are irrelevant.
> Is that a corollary, that there can be no well-ordering of a dense
> set of reals?
If an ordering is dense on *any* set (reals or not), then it cannot be
a well-ordering. The set might have a well-ordering (and always does
in ZFC), but every well-ordering will be different from every dense
ordering.
> So I'm wondering about just the _statement_ that the reals, in some
> way, shape, or form, are well-ordered.
In ZFC, they have a well-ordering. Their usual ordering is not a
well-ordering.
> If the definition of the real numbers is amended thus that the
> normal properties hold true for "definite" values, yet for these
> "indefinite" values they are a contiguous sequence of points, then
> the normal ordering is a well-ordering.
No, it is not. The normal ordering on the rationals is not a
well-ordering. Adding extra elements, whether they be irrational,
"indefinite", or any other description you want to give them, will not
change that basic property. A well-ordering requires that *every*
nonempty subset with that order, including all subsets without the
extra elements, be well-ordered itself.
> Perhaps instead I should only concern myself with the sequence of
> points.
There are many possible sequences of points. Which do you call "the"
sequence?
> I guess I need to learn more about generally accepted results about
> well-ordering the reals,
Such as, for example, what a well-ordering is.
> It's shown in ZFC that there is at least one well-ordering, yet no
> one wants to provide one because then you could index that
> well-ordering by the natural integers.
This staement is false in two respects:
1) Plenty of people would like to provide one. However, any such
well-ordering is (I think) impossible to finitely construct.
2) In ZFC, a well-ordering of a set is not equivalent to a mapping
from the naturals to the set. Even if they could construct such a
well-ordering, it wouldn't be indexable by the natural numbers.
> Look directly to the normal ordering, and assume that it is not
> contradictory to do so for reasons yet to be developed, or fiat, and
> that it is not allowed to conflict with the "usual arithmetic
> properties" because they're useful to solve real-world problems.
Asserting a contradictory set of propositions in classical logic (by
fiat or anything else) is an exercise in idiocy since you end up with
everything being true, no matter how ridiculous.
> If you don't care to consider that, there is some ordering of the
> reals for which there is a least element of any subset of those
> numbers. If you do consider it, then that's what it is.
In ZFC, there is such an ordering. In fact, there are infinitely
many.
> Do infinitesimals exist among the real numbers? Short answer: no.
Good.
> Do the hyperreals contain any element that is not an element of the
> reals? Short answer, there are everywhere reals between zero and
> one.
Not a good answer. The correct answer is "yes, infinitely many of
them".
> Does the generic extension of N contain any elements not in N? No.
What are your definitions here? With the usual definitions, the
integers are an extension of N, -1 is an integer, and -1 is not in N.
Without a definition, your statement was yet more word soup.
- Tim
Ross A. Finlayson
There aren't enough numbers?
Well, it's obviously going to be more than zero, it's obviously going
to be more than one... are we out of numbers yet? Two, three, four, on
we go.... Four billion, four billion and one, four billion and two....
Geez, this is going to take forever. Four billion three, four billion
four....
Hey I just thought of something: the natural integers are infinite.
They go on forever. There is no end to them. They're inexhaustible.
Remove one at a time, in order, there's always at least one more, for
example the next one. There's always one more!
The definition of the natural integers is generally that of the Peano
axioms, where in the original I believe one was the least natural
integer, thus that they're the positive integers. Today it's generally
accepted that zero is the least non-negative integer, that the naturals
are the non-negative integers. They're often used as a part of a
formalism of induction.
There are theories with a set of all sets, which would be its own
powerset, for example NFU where they divide the sets into "large" and
"small", I just say that a set is large enough when it's infinite.
The rationals and irrationals are each everywhere discontinuous, and
they're disjoint, they're dense in the reals and their union is the
reals. How about that?
Regards,
Ross Finlayson
--
"In a few minutes, a monogrammed coffee mug came to seem to him the
most wonderful imaginable possession on earth." - Wouk
Ross A. Finlayson
"Generic extension" normally applies to a model, I talk about the
generic extension of a set as just the set in the generic extension.
It was recently discussed with regards to IZF and perhaps even CZF.
There are only reals between zero and one, and everywhere reals, real
numbers, where are the infinitesimals?
An infinitesimal is sometimes defined as being less than 1/n for any
finite n. In the hyperreals infinitesimals can be greater than zero in
terms of their scalar magnitude. Internal Set Theory has notions of
infinitesimals, and is conconsistent with ZFC. Cantor was against
infinitesimals. I don't figure to add any elements to the real
numbers, as they are complete or on the line continuous.
Paraconsistency is a notion of a logic where an inconsistency doesn't
invalidate every other true statement. Burali-Forti is difficult to
avoid. While that is so, it is not so bad for "not void."
If the normal ordering of the positive reals was a well-ordering, then
it would be.
(0, iota, 2iota, ...)
For positive and non-positive reals that kind of shorthand of a
well-ordering would be along the lines of
(0, iota, -iota, 2iota, -2iota, ...)
I wonder about the statement that the reals, in some way, shape, or
form, are well-ordered.
About the jigsaw puzzle analogy, it captures some of the notions of the
normal ordering as sequenceable but not all. It's still kind of
useful. That's about the sequence of contiguous, next to each other,
points on the real number line, from having a point being the
intersection of two lines on a plane, as it has been for several
thousands of years.
Do you use transfinite cardinals for normal, day-to-day things? If so,
is it to talk about transfinite cardinals?
Here's an idea: compare Zeno's paradox to uncountable sets. Consider
a straight line there and back.
About constructing the well-orderings of the reals, there is probably a
body of work describing exactly their necessary structure. It appears
to have to do with this Goedelian Kolmogorov complexity in terms of the
randomness. Yet, Enderton says that from L, the constructible
universe, is "definable" a choice function. What are Feferman's
"natural well-orderings", particularly of the reals?
Anyways, in my theory with no non-logical axioms, ur-element as
Thing-in-Itself and Being and Nothing, and ubiquitous ordinals and
naturals, the normal ordering of the positive reals is a well-ordering,
and infinite sets are equivalent.
Regards,
Ross Finlayson
--
"Have you seen my pen?"
"Say hello to my little pen."
Ross A. Finlayson
There aren't enough numbers?
Regards,
Ross Finlayson
most wonderful imaginable possession on earth." - H. Wouk
Ross A. Finlayson
There aren't enough numbers?
Regards,
Ross Finlayson
most wonderful imaginable possession on earth." - Wouk
Timothy Little
> Paraconsistency is a notion of a logic where an inconsistency
> doesn't invalidate every other true statement.
Yes, I studied paraconsistent logics a bit, but not as much as I'd
have liked to. You said your set theory was based on (among other
things) the Law of the Excluded Middle. All the paraconsistent logics
I'm aware of deny that law.
> If the normal ordering of the positive reals was a well-ordering, then
> it would be.
>
> (0, iota, 2iota, ...)
(I think you mean "non-negative reals" here)
I think that's an unsupported assertion. I see no reason why the
least element of the set greater than iota must be 2iota. In fact, I
don't see what guarantee there is that 2iota even exists! If
(1/2)*iota doesn't exist, why must (2/1)*iota?
Unless of course, the "2iota" notation you use here is just another
arbitrary label, and not denoting multiplication.
> Do you use transfinite cardinals for normal, day-to-day things? If
> so, is it to talk about transfinite cardinals?
No, it's usually to talk about functions.
> Anyways, in my theory [...] the normal ordering of the positive
> reals is a well-ordering, and infinite sets are equivalent.
Good luck with that. I hope that one day you recognise the
differences between what you've got now and an actual set theory.
- Tim
George Cox
>
> George Cox wrote:
> > "Ross A. Finlayson" wrote:
> > >
> > > That aside, this is a discussion of well-orderings of the reals and
> > > other sets dense in the reals. It's shown in ZFC that there is at
> > > least one well-ordering, yet no one wants to provide one because
> then
> > > you could index that well-ordering by the natural integers.
> >
> > No you couldn't, you'd run out of integers. (I'm not sure what a
> > "natural integers" is.)
>
> There aren't enough numbers?
That's right. There are only aleph_0 integers. There are 2^{aleph_0}
reals *however* you order them. Well-ordering the reals doesn't change
their cardinality. And aleph_0 < 2^{aleph_0} as Cantor showed. So,
assuming that "natural integer" means "integer" (and it doesn't make any
difference if it means "natural number"), "you could index that
well-ordering [of the reals] by the natural integers" is false
George Cox
>
> George Cox wrote:
> > "Ross A. Finlayson" wrote:
> > >
> > > That aside, this is a discussion of well-orderings of the reals and
> > > other sets dense in the reals. It's shown in ZFC that there is at
> > > least one well-ordering, yet no one wants to provide one because
> then
> > > you could index that well-ordering by the natural integers.
> >
> > No you couldn't, you'd run out of integers. (I'm not sure what a
> > "natural integers" is.)
>
> There aren't enough numbers?
>
> Well, it's obviously going to be more than zero, it's obviously going
> to be more than one... are we out of numbers yet? Two, three, four, on
> we go.... Four billion, four billion and one, four billion and two....
> Geez, this is going to take forever. Four billion three, four billion
> four....
>
> Hey I just thought of something: the natural integers are infinite.
> They go on forever. There is no end to them. They're inexhaustible.
> Remove one at a time, in order, there's always at least one more, for
> example the next one. There's always one more!
Yes, but there are only aleph_0 of them which is not enough to count the
reals however you order the reals.
H. Enderton
>> >In ZFC, there is at least one well-ordering of the reals.
And then I wrote:
>> Yes. But it is a theorem that, working in ZFC, you cannot
>> give a specific *definable* example of one. That is, for any
>> formula that you think might define a well-ordering of the reals,
>> there is a model of ZFC in which it doesn't. (A result due to
>> Sol Feferman, back in the early days of forcing.)
And then Ross wrote
>It would probably be a good idea if I wanted to know to research him
>before contacting him. Then, I'd invite him to come inveigh on
>sci.logic.
To see Feferman's paper, a good place to start is Baumgartner's
review in JSL, vol. 37 no. 3, pp. 612-613. (Feferman has been
working on other topics for the last 40 years or so.)
And also I wrote:
>> To digress, there is a definable well-ordering of the *constructible*
>> reals (i.e., the reals in Goedel's class L). But that's only a drop
>> in the bucket.
To which Ross replied:
>Herb, I think V = L. ... Does Feferman's result only thus apply
>if V =/= L?
Feferman's result gives a model of ZFC, but not a model of V = L.
Ross again:
>For the constructible case, you relate that there is a "definable"
>well-ordering: is there an example?
Here is the idea: Whenever real x is constructed at an earlier
stage than real y, then we put x less than y. If x and y are
constructed at the same stage (say L_alpha+1), then we compare
the formulas that define x over L_alpha and the formulas that
define y over L_alpha and compare them alphabetically. If x has
a defining formula that is less than any defining formula for y,
then we put x less than y. (The formulas involve parameters, so
one needs to be well organized.)
Ross also wrote:
>If "the reals are projectively well-ordered", is that interpretable as
>"the reals are well-ordered" or "the normal ordering of the reals is a
>well-ordering on the positive reals"?
The former. The latter is impossible.
And Ross wrote:
>What do you think about well-ordering the non-negative reals and the
>normal ordering as a candidate well-ordering?
Losing candidate. The usual ordering of the reals fails to be a
well-ordering.
--Herb Enderton
Ross A. Finlayson
I'm rather ignorant about the paraconsistent logic. When I first heard
about it, some years ago shortly after which I wrote to sci.math about
reading about it, it was in the context of multivalent logics, or
multi-valued logics, in the thread titled "Base", for it has besides
the T and F an indeterminate third truth value, generally U, where that
is often ascribed to Kleene, after Lucasiewicz.
In terms of its intuitionistic slant and excluded middle, that is
considered. In the null axiom or axiom-free set theory, the excluded
middle on the ur-element, the ur-paradox of the assertion of existence,
does and does not apply.
It's considered five years ago in that beginner's post about
multivalent logic, or many other posts on sci.math.
I admit as well to not being as well-studied as I would like, ergo I
study. We here actually basically know what we're discussing, or
"talking about". Being as well a voracious learner and enjoying
knowledge for its own sake, you might understand why I'm very
particular about my own personal logical theory of everything and
demand my own input. This group is full of people who love to have
mathematical knowledge and share it with others, each for their own
reasons. Mathematical logic is somewhat vast, and is hopefully
compressible to three or four pages.
Well, enough of that business. When you say you talk about transfinite
cardinals and functions, do you mean along the lines of the functions
from reals to reals, or computational complexity and Turing, ie
asymptotic finite combinatorics?
Do you use that lambda calculus, theory of functions and types,
Coquand, Luo, Pierce, Martin-Lof? I heard of it, type theory and
lambda calculus, read a couple of those Pierce books. It's cumbersome
to apply it in computer programming unless you're writing a compiler,
where it can be useful.
Basically I want to learn more about you. Do you have a math degree or
is your learning self-directed? It's easy to see what words you write
here, and the rapidity of assimilation of material. What's your
philosophy about mathematics, or mathematical discussion?
When you use the transfinite, as you say, is it transfinite induction
you use?
Regards,
Ross Finlayson
Ross A. Finlayson
I posted my previous reply to using "Google Groups Beta 2", it replied
several times on posting with "Temporary Server Error". Please ignore
those duplicates. What do you expect? The project manager of Google
Groups promotes "Space Elevators." (That's an unfair jibe.) There is
censorship going on at Google Groups. That's partly their prerogative,
and partly not. Either that, or they're losing posts from the
database. That's about posts I've already seen before on Google that
are currently inaccessible from their search interface, for example
posts available on Math Forum not available with search for _exact key
phrases_ or more generic search terms.
Read the original! Which one was sent first?
There's a terminology for the "countable" part of a well-ordering,
which is a sequence instead of a set, or that part which corresponds to
countable ordinals, it's called the initial segment.
I have little opinion of transfinite cardinality, except that infinite
sets are equivalent, you know that, especially when the normal ordering
is a well-ordering where Cantor's first result and similar Megill style
proofs about not bijecting the naturals and reals for density in the
reals do not apply.
In fact that's one of the things that I've stated: a bijection of the
naturals and non-negative reals must be compositions of everywhere
monotonic functions like EF, pieces order-isomorphic to the normal or
reverse orderings of real numbers.
Anyways, trivialities aside, what you write is standard and largely
accepted, and few disagree. I'm among the few who disagree, not on
general principles but in terms of mathematical logic.
Back to well-orderings and the reals, defining a well-ordering on the
reals goes a long way towards showing them countable because of the
notion that it might be an invertible function, in terms of the
bijection between the ordinals and the reals that is the well-ordering.
Infinite sets are infinite!
Don't forget the induction part of the transfinite induction.
Four billion six, four billion seven....
Regards,
Ross Finlayson
Ross A. Finlayson
Here are some more links with quick takes on the ordinals:
http://www.dcs.ed.ac.uk/home/pgh/ordinal-notations.html
http://www.dpmms.cam.ac.uk/~wtg10/ordinals.html
Those are probably somewhat "standard", but enlightened.
You could easily find them yourself, in searching for "natural
well-ordering" there are some interesting results. I don't vouch for
them but they're interesting.
Regards,
Ross Finlayson
George Cox
>
> Mathematical logic is somewhat vast, and is hopefully
> compressible to three or four pages.
Eh?
Timothy Little
> This group is full of people who love to have mathematical knowledge
> and share it with others, each for their own reasons.
Sure; I've been posting (off and on, mostly off) for about 12 years.
> Well, enough of that business. When you say you talk about transfinite
> cardinals and functions, do you mean along the lines of the functions
> from reals to reals
Not reals to reals, specifically. It's more universal and embedded
than that. For example, I might talk about a difference in properties
between countable vs uncountable unions of sets in measure theory and
probability.
Countability/uncountability is a property of sets that is usually
easily checked in terms of the arithmetic on their transfinite
cardinals. That's certainly much easier than explicitly finding a
bijection!
Transfinite cardinals are a labour-saving device, much as the natural
numbers are.
An analogy: if you have some seats and want to know if each guest is
going to be able to get a seat, you might (if you were both stupid and
innumerate) ask for a list of all their names and try every
combination of 1:1 matching names with seats -- because, if they don't
match up one way they might match up in another. A smarter way would
be to ask for a count of the guests and count the seats and see if the
number of seats is greater than or equal to the number of guests.
That is an application of finite cardinals.
If you have a pair of sets and want to know if every element of one
set can be paired with another, you could try to explicitly find a
surjection, discarding one function if it doesn't work and trying
another. Or, you could be smart and compare their cardinal numbers.
That's an application of transfinite cardinals.
> Do you use that lambda calculus, theory of functions and types,
Only very occasionally, and rather rusty.
> Basically I want to learn more about you. Do you have a math degree or
> is your learning self-directed?
Both.
> What's your philosophy about mathematics, or mathematical
> discussion?
Whatever looks interesting or useful. I'm not searching for Truth,
and probably wouldn't recognize it anyway.
> When you use the transfinite, as you say, is it transfinite induction
> you use?
Only very, very rarely. I don't have much call to use infinite
ordinals explicitly. Well-ordering just isn't that useful a property
for the majority of sets I use.
I do have some curiosity about sequences defined over larger ordinals
than omega though.
- Tim
Timothy Little
> Anyways, trivialities aside, what you write is standard and largely
> accepted, and few disagree. I'm among the few who disagree, not on
> general principles but in terms of mathematical logic.
As I see it, it's a system. It has theorems and proofs, and many of
them look useful and interesting. Is it True? I don't care.
I'll happily investigate other systems so long as they look equally
interesting or useful. Being as useful and interesting as ZF (without
being mostly equivalent to ZF) is a very high bar though.
> Don't forget the induction part of the transfinite induction. Four
> billion six, four billion seven....
Actually, I think the the fun part of transfinite induction is:
"... Infinity. Infinity plus one, infinity plus two, ..."
- Tim
George Cox
>
> Timothy Little wrote:
> > Ross A. Finlayson wrote:
> ordering
> > > is a well-ordering[?]
> >
> > Their usual linear ordering is not a well-ordering. That's not
> > surprising, it also applies to the rationals and the integers.
> >
> > The rest of your post was rambling nonsense, snipped.
> >
> >
> > - Tim
>
> No, the point is that the total ordering of the finite or semi-infinite
> interval comprising a set is a well-ordering. Any well-ordering is a
> total ordering. If you can well-order the reals is not that
> well-ordering a total-ordering? As there are only two main total
> orderings on the reals "<" and ">" is not thus one or both of them a
> well-ordering?
What do you mean by _main_ total orderings? There are an infinity of
total orderings on a (non-empty) real interval. At least one of them
will be a well-ordering. Neither < or > is a well-ordering.
Consider the interval (0, 1) = {x:0 < x < 1}, consider the non-empty
subset (0, 1) of it. (0, 1) has no least element. If (0, 1) was
well-ordered by < it would have a least element because that's part of
the definition of well-ordered: every non-empty subset has a least
element.
Ross A. Finlayson
Perhaps this is mistaken: in measure theory the only concerns of
transfinite cardinality are countability vis-a-vis the cardinality of
the continuum. Probability's use of transfinite cardinality is based
upon measure theory.
This leads some to that you can select a random real number from a
uniform or pseudouniform distribution over an interval of the reals,
but not an integer. That's because of issues dealing with "zero
probability", or infinitesimal probability.
One consideration of how to sample a real number from the unit interval
is to flip coins, at each step narrowing the resultant interval towards
a binary expansion of the real number.
A consideration with that is that each sample from {0,1} is as well the
beginning of the sample of another real number, and that any other
permutation of the sequence is another sample. Then, if there's an
iterative process, a consideration is whether an infinite number of
ones and zeros is reached at the same time, where the average value is
1/2 as the average of 1/3 and 2/3, or an infinite number of zeros is
reached before infinitely many ones, leading to an average value of
zero and many rationals, or an infinite number of ones and finitely
many zeros, with an average value of one and many rationals. The
integral of a function symmetric about the origin is zero.
That is with the notion that if you have any infinite sequence of
zeros, however you reorder the sequence it represents zero, if you have
a sequence with infinitely many ones and zero you can generate most
irrationals, and half of the possible sequences.
That consideration of sampling real numbers is digression from the
point about measure theory and probability: that the utility is
primarily about the cardinality of the continuum and continua, and thus
the basically geometric nature of the continuum instead of its
cardinality, and that it could be explained that way.
So, aside from that digression, if I want to support measure
theoretical results or provide alternate mechanisms for correct results
using my little theory where infinite sets are equivalent, then it
would lead to some retrofitted underpinnings of measure theory as
necessary, so I wonder: is there any use in meaure of transfinite
cardinality besides the cardinality of the continuum? Second: does
probability theory use transfinite cardinals besides using measure
theory?
Consider "complex measure theory".
Regards,
Ross Finlayson
Ross A. Finlayson
meant normal or usual.
Because there exists a well-ordering of the reals, I wonder what it is.
I already have the notion that the unit interval is a contiguous
sequence of points. With that as an assumption, any subset of those
points is totally ordered and one of those points is the least element.
That kind of consideration doesn't seem to be missing much from the
properties of a well-ordering of the reals.
As a contiguous sequence of points, the least element of the open unit
interval is: not zero, not rational, not algebraic, a real number with
its value hidden, iota, and the next point after iota is not iota and
otherwise similar/different.
It doesn't give a pair of finite integers a, b and say "a/b is the
least positive real number." It just gives a real iota and says "iota
is the least positive real number."
That's into the consideration of the definitions of the line and the
points that comprise it. That's about defining the reals not just as a
field, about multiplication and the inverses or reciprocals, but
simultaneously as an uninterlocked ring.
Anyways, I would enjoy seeing an example of a well-ordering of the
reals of the unit interval, because I think via Cantor/Megill it would
be the normal ordering. I know that's not standard, I know that's not
generally accepted, I simply offer what I hope to be an adequate
rationalization and methods.
Do you have a math degree? I don't. I think they're great. Here on
sci.math and sci.logic there are lots of holders of mathematics
degrees. Cox, I wonder if first-order logic resolves to tautology and
second-order logic is representable in first-order logic, and promote
that it is.
Regards,
Ross Finlayson
George Cox
>
> I'm looking for any well-ordering of the reals. When I wrote "main" I
> meant normal or usual.
>
> Because there exists a well-ordering of the reals, I wonder what it is.
>
> I already have the notion that the unit interval is a contiguous
> sequence of points.
If by sequence of points you mean that there is a first, a second, a
third, and so on until they are all exhausted, then you're wrong. In any
non degenerate interval, between any two numbers there is an infinity of
others, and hence no "first, second,...".
Timothy Little
> if you have a sequence with infinitely many ones and zero you can
> generate most irrationals, and half of the possible sequences.
Actually, you can get all real numbers if you're have a wide enough
deefinition of "reorder".
For example, define a "reordering" to be an injective map f:N->N where
f(n) is the position in the old sequence from which you get the n'th
element in the new. For any finite set of labels, such a "reordering"
is bijective. For infinite sets of labels, it need not be.
> That consideration of sampling real numbers is digression from the
> point about measure theory and probability: that the utility is
> primarily about the cardinality of the continuum and continua, and thus
> the basically geometric nature of the continuum instead of its
> cardinality, and that it could be explained that way.
Not at all. Many of the functions I talk about aren't even defined
over the reals, let alone dependent upon their geometric structure.
> So, aside from that digression, if I want to support measure
> theoretical results or provide alternate mechanisms for correct results
> using my little theory where infinite sets are equivalent, then it
> would lead to some retrofitted underpinnings of measure theory as
> necessary
Definitely. All of analysis to begin with.
> I wonder: is there any use in meaure of transfinite cardinality
> besides the cardinality of the continuum?
Cardinality of the powerset of the continuum is certainly used; I
don't personally recall encountering explicit use of cardinalities
larger than that.
> Second: does probability theory use transfinite cardinals besides
> using measure theory?
As I was taught it, probability theory *is* measure theory for a
particular class of measures.
- Tim
Timothy Little
> Because there exists a well-ordering of the reals, I wonder what it
> is.
There exists a well-ordering of the reals *in ZFC*. I highlight the
qualification, because in other set theories there need not be.
> I already have the notion that the unit interval is a contiguous
> sequence of points. With that as an assumption, any subset of those
> points is totally ordered and one of those points is the least
> element. That kind of consideration doesn't seem to be missing much
> from the properties of a well-ordering of the reals.
Only the biggest property that I've mentioned about a dozen times now:
for a well-ordering, *EVERY SUBSET* has a least element. Not just
intervals.
As I said before, it is easy to define a set, with a total order, that
includes the standard reals with their usual ordering, for which every
interval has a least (and greatest) element.
But it won't be a well-ordering.
> Anyways, I would enjoy seeing an example of a well-ordering of the
> reals of the unit interval, because I think via Cantor/Megill it
> would be the normal ordering. I know that's not standard, I know
> that's not generally accepted, I simply offer what I hope to be an
> adequate rationalization and methods.
They are not adequate, because you keep missing what a well-ordering
*is*.
You also miss the point that ZFC guarantees a well-ordering, but in
ZFC the usual ordering is not a well-ordering (and wishing it were so
does not make it so).
You seem to want to jump to a different system from ZFC, yet once you
do that you no longer have the guarantee that a well-ordering exists.
As an example, "the reals do not have a well-ordering" is consistent
with ZF (without C).
- Tim
Ross A. Finlayson
One point of clarification is "well-ordered" vs. "well-orderable",
where "well-ordered" means "well-ordered by the normal ordering" and
"well-orderable" means "well-ordered by some ordering that is not
necessarily the normal ordering." That is to say, I'm under the
impression that "well-orderable" is to be used instead of
"well-ordered" that is not qualified "by the ordering <[", when the
normal total ordering is not deemed a well-ordering. Is that
appropriate?
Then I wonder if Caicedo or the abstract writer uses "well-ordered" and
"well-orderable" interchangeably, instead of as in this context where
"well-ordered" by itself means "by the normal ordering."
In strict usage does "well-ordered" as applied to a set mean "by its
normal ordering"? In particular what does it mean in the context of
Andres Caicedo's "projective reals"? I wrote to ask him.
Another thing I hope you can summarize is the concept of "natural
well-ordering." In a very casual research Feferman describes the
concept of "natural well-ordering" as a problem area, where "problem
area" means an area where there is room for research and discovery but
there is a need for some direction.
I'm hoping you would discuss briefly the "natural well-ordering", and
the historical and logical context of the "natural well-ordering."
About V = L, what do you see as the justification for higher order
logic, or, for what reasons are higher order logics necessary? Is the
universe of all higher order logics not a universe? That is probably
not strict usage, can you tell me why everything is first order, or
not?
Thank you. Have a nice day. Regards,
Ross Finlayson
Ross A. Finlayson
About the infinite binary sequences, there are definitely
considerations to be made about the density of the elements, and
restrictions on the sequence element interchange in permutation.
There's only one sequence of binary digits that represents the number
zero.
I'm hoping you would briefly summarize, in an obvious way, the
functions you mention not defined over the reals, and the utility of
transfinite cardinals. For many, the lack of applications of
transfinite cardinals for solving real-world problems is the (a)
primary source of apathy about them. I have just been very casually
examining the measure theoretical foundations and have not seen
non-geometric examples.
I'm not a geometer, but I notice that geometry is very useful in
solving real-world problems.
You say all of analysis is under measure theory, or that it's
expressible in terms of measure theory. Where analysis is the integral
calculus, is not that all about functions on the real (and complex,
hypercomplex) numbers? A lot of people do integral calculus without
regard of measure theory. I don't think of functional domains in the
integral calculus without geometric structure.
Could you provide an example of the use of the powerset of the
continuum? Do you have an example of an analytical result over a
non-empty domain of measure zero with a geometric analog? That is to
say, is there a simple example of a set with measure zero leading to a
result that is any different than that for an empty set, and derivable
by other means, or are all sets of measure zero the empty set?
Searching for information about "zero probability", "infinitesimal
probabilities", "nonstandard measure theory", "nonstandard
probability", leads to some discussion of intuitive and also
counterintuitive results in measure theory. When I ask about "complex
measure theory" I'm asking about "multivariate" and "multidimensional
measure theory." I hear buzz about these "probability density
functions."
There are everywhere reals between zero and one. They're not "only"
real numbers, but they're all real numbers.
Hey thanks, that helps me understand. Regards,
Ross Finlayson
Timothy Little
> About the infinite binary sequences, there are definitely
> considerations to be made about the density of the elements, and
> restrictions on the sequence element interchange in permutation.
Sure, I was just pointing out one way in which saying "all infinite
sets are equivalent" can end up with very counterintuitive results.
> I'm hoping you would briefly summarize, in an obvious way, the
> functions you mention not defined over the reals
A single example of many: in one application I was dealing with
measures of sets of functions from certain grammars to natural
numbers. Some of the grammars gave rise to uncountable sets of
functions, some gave rise to countable.
Of course, in a system where "infinite sets are equivalent" it would
have been pointless, because the infinite sets I was dealing with
would all have been equivalent.
> I have just been very casually examining the measure theoretical
> foundations and have not seen non-geometric examples.
Measure theory may have started with geometric motivation (I'm not
sure of its history), but if so then it has certainly grown far beyond
that.
> You say all of analysis is under measure theory, or that it's
> expressible in terms of measure theory.
Not exactly; I just said that if you're going to retrofit measure
theory to fit an "all infinite sets are equivalent" axiom, you'll need
to retrofit all of analysis just to begin with.
> A lot of people do integral calculus without regard of measure
> theory.
A lot of people balance their books without regard of the abelian
group theory underlying addition of integers, either. However, if
addition were no longer commutative, associative, or had inverses then
the calculations wouldn't balance because they use them *implicity*.
- Tim
ken quirici
Hi Ross,
Thanks for those as well. Sorry for the long time to reply.
Ken
Ross A. Finlayson
many results, one is this paper from the Bulletin of Symbolic Logic, H.
Enderton et al. eds., Erik Palmgren's "Developments in Constructive
Nonstandard Analysis."
Palmgren says Schmieden and Laugwitz might have been already
considering the nonstandard reals as a partially ordered ring, with a
"rather restricted transfer principle", before Robinson and his
hyperreals a decade or so. Then he goes on to say he's a fan of Bishop
and Cheng. Ready results lead to Peter Zahn and Palmgren as
contemporary nonstandard constructivists, among the many nonstandard
constructivists.
http://mathworld.wolfram.com/TransferPrinciple.html
http://www.mathematik.uni-muenchen.de/~antipode/abstracts.html
There is lots of stuff about nonstandard analysis and infinitesimals,
greater than zero and thus between zero and one. Compare to "Continuum
Hypothesis". There's a lot in real analysis.
I guess the idea here is that measure theory doesn't differentiate
between empty/zero and countable/non-zero, calling them both zero.
Infinite sets are equivalent.
Regards,
Ross Finlayson
Ross A. Finlayson
George Cox
>
> Infinite sets are equivalent.
>
What do you mean by that? That all infinite sets are equipollent or of
equal cardinality?
H. Enderton
When Caicedo said that there was a model of ZF in which
"the reals are projectively well-ordered" he meant that
in the model there was some projective relation that well ordered
the real numbers. Of course, this relation was not the usual
ordering on the reals, which we all know is not a well-ordering.
>Another thing I hope you can summarize is the concept of "natural
>well-ordering."
Don't get your hopes up too high -- "natural" is a slippery term.
But the word does suggest that the ordering should be definable
or describable in some way that does not appeal to some mere statement
of existence.
--Herb Enderton
Ross A. Finlayson
Hi,
It's meaningless.
Yes: equivalent, equipotent, equipollent, and so on, except equipollent
may have a different meaning than that there exists a bijection between
the two infinite sets.
Here's my understanding of some arguments to the contrary: the
antidiagonal argument as applied to reals, the antidiagonal argument as
applied to the coded powerset of the naturals, Cantor/Megill (Cantor's
first proof), and the powerset result, or "Dave's".
So, if you say that the reals are uncountable or the powerset of the
naturals is uncountable then those are the arguments you use to support
that claim. Do you know other arguments to that effect? I'd be
interested to hear about them.
For me to claim otherwise I use induction in the sense that for an
infinite set it is possible to remove elements one at a time and there
are always infinitely many more, as it is infinite. Then, for the
mapping of naturals and reals I described the Equivalency Function,
which is subject to many of the same concerns as examinations of the
reals' normal ordering, and for the coded antidiagonal and powerset
result the dual representation of zero and infinity, and an
indeterminacy of which it is.
The set of all sets would be its own powerset, the order type of all
ordinals would be an ordinal, the set of all sets not containing
themselves would not be a set, I try to resolve those at one stroke.
This is where an ordinal is a set and the powerset is order type is
successor. There is still the problem the Ord would be an ordinal, and
that leads to the notion of a confirmator successor that removes the
indeterminacy.
That's visualizable as the integers being a circle, a gear, that meshes
with the identical gear, with regards to the space-time wheel, as a
platonist. Where it would seem to take two ticks, by then then it had
already gone over one tick, in terms of a clockworks.
That's again digression from the subject at hand, well-orderings of the
reals and sets dense in the reals. While that is so, they're related
topics: foundations and application of a sort.
Consider Cantor's first proof with regards to a well-ordering of the
reals. You ponder that and consider that the well-ordering would be
indexed by more than infinitely many numbers. Then, consider why you
could construct it thus that the convergent sequences, monotonically
increasing a and montonically decreasing b, indexed by more than
infinitely many numbers, converge. Why would not the well-ordering
have to be in a form that disallows a contradiction?
I guess the idea here is to supplement the definition of the real
numbers with a coincident structure that is useful for their definite
well-ordering: the consideration of the line as a sequence of points:
points on a line.
Then, with regards to a transfer principle, the elements of the
coincident structure are real numbers, and for a finite interval
well-ordered.
Timothy Little
> except equipollent may have a different meaning than that there
> exists a bijection between the two infinite sets.
What different meaning? The only two meanings I've seen used are:
1) of sentences, that one may be derived from another by a finite
sequence of application of derivation rules;
2) of sets, that there exists a bijection between them.
Do you have a private second meaning as applied to sets, or is there
some other standard meaning of which I'm not aware?
> Consider Cantor's first proof with regards to a well-ordering of the
> reals. You ponder that and consider that the well-ordering would be
> indexed by more than infinitely many numbers.
More than aleph_0, to be more precise.
> Then, consider why you could construct it thus that the convergent
> sequences, monotonically increasing a and montonically decreasing b,
> indexed by more than infinitely many numbers, converge. Why would
> not the well-ordering have to be in a form that disallows a
> contradiction?
This is gibberish. Are you attempting to define a convergent sequence
over some unspecified uncountable ordinal?
> I guess the idea here [...]
I guess there's no idea there. Do you know the definition of a
well-ordering? Do you have any idea of how to check whether an
ordering is a well-ordering? You've shown no sign of either. Learn
what a well-ordering is before making statements about them, and you
might start to be able to form ideas about them worth exploring.
> Infinite sets are equivalent.
You still haven't said in which sense they're equivalent in your
private "set theory". "Equivalent" is a very vague term that really
only makes sense in a context. You've removed all context, and hence
you're not making any sense.
- Tim
Ross A. Finlayson
Hi Tim,
I think that's not fair, besides that it's unfactual. We have some
level of agreement of the terms discussed. Did you have something to
say that's not hot air and handwaving? I'm concerned to be asking
that, because I don't care to devolve into pointless ad hominen attacks
that only serve to degrade each others' self-esteem and easily swayed
opinion. Your untoward response that denies the evidence of this
immediate context is an insult to other participants and drastically
cheapens your arguments. Don't you think anyone else can read, and
does? That's a pointed question.
We have been discussing, and perhaps continue to discuss,
well-orderings of the real numbers. I'm not mollified by your
suggestion, rather irked, learning is an iterative process. If you had
heard of "natural well-orderings" before reading about it in this
thread, you're probably an academic logician. You probably hadn't,
that is not a problem. It's a discussion of mathematical logic by
participants with years of study in the field. Please respect the
outstanding academic credentials and analytic ability of contributors
to this discussion, including yourself.
You claim some familiarity with "Cantor's first proof" and
"uncountability of the reals", or rather, I do. The notion is that if
you construct from a sequence of reals two other sequences: an
increasing and decreasing, where elements are added to first a
monotonically increasing sequence, that means for each element the next
element of the sequence is greater than that previous. As elements in
a well-ordering are encountered, where the well-ordering implies the
existence of a choice function to return the least element of each
subset by some total ordering, then if the element is in the interval
between the last elements of the "increasing and "decreasing"
sequences, that it is added to one or the other. Now, "Cantor's first"
has it that those two sequences demonstrate a real number not existing
in the original sequence of real numbers, for the reals' completion.
Consider that with regards to the transfer principle and nonstandard
models of the integers as hyperintegers. That's about "Cantor's first"
applying to either: both of the integers and hyperintegers as indices;
or neither. The transfer principle implies that the standard and
nonstandard result are the same, or rather that proof in one is proof
in the other.
To reiterate a point of my previous e-mail here: an alternate (to that
solely based on the field structure or rational approximation),
coincident structure of the real numbers is considered, with regards to
the transfer principle, to, yes, formalize a description of the real
numbers as contiguous points on the line and provide an explicit
example of a well-ordering of the reals. This is preferably done with
duly diligent reference to the historical context.
Are the context of this discussion and that statement clear enough for
you to make sense of them? I hope they are.
The context is the complement in the universal set.
This is unfortunately not a compliment. Ha ha ha ha!
Regards,
Ross Finlayson
Timothy Little
> I think that's not fair, besides that it's unfactual. We have some
> level of agreement of the terms discussed.
The reason I posted as I did, was because it seems we don't. In
regard to sets, I have only ever seen one definition of "equipollent":
that a bijection exists between the sets. You're saying there's
another, and implicitly using it (whatever it is) in what you mean by
"infinite sets are equivalent".
That's the missing context, without which your assertion makes no
sense.
> If you had heard of "natural well-orderings" before reading about it
> in this thread, you're probably an academic logician.
I had heard of natural well-orderings, but it's rather the basic
concept of a well-ordering itself that I'm concerned with.
You keep appealing to ZFC to assert that the reals have a
well-ordering. The normal ordering is not a well-ordering, and your
statements to the contrary waffle around "iota" and "contiguous
points". Neither of those have much to do with a well-ordering, hence
I really do seriously wonder whether you know what a well-ordering is.
You haven't provided any evidence that you do, but have provided a lot
of evidence that suggests you don't.
Every time I mention the definition of a well-ordering, and show a
counterexample to the definition for the normal ordering, you ignore
it or treat it as irrelevant.
As I see it, the situation is simple:
1) If you want a model of reals that even vaguely follows the
geometric one, you will need to include the rationals with their
normal order, since every rational can be geometrically constructed.
2) If an order on a set A is not a well-order, then any extension of
that order is not a well-order. This follows trivially from the
definition.
3) The rationals are not well-ordered by their normal order. This is
easily shown by {1/n:n in N} and the definition of a well-order.
4) By (1), (2), and (3), no such model of the reals will ever be
well-ordered by its normal order.
Vague generalities about "contiguous points" and "iota" don't mean a
thing, because they have nothing to do with whether an order is a
well-order or not.
> To reiterate a point of my previous e-mail here: an alternate (to
> that solely based on the field structure or rational approximation),
> coincident structure of the real numbers is considered, with regards
> to the transfer principle, to, yes, formalize a description of the
> real numbers as contiguous points on the line and provide an
> explicit example of a well-ordering of the reals.
If an explicit well-ordering and "contiguous" is enough for you, why
not just use lexicographically ordered R'xZ where R' is the set of
computable reals? This set isn't hard to explicitly well-order.
Every point has a successor and predecessor which seems to be what you
mean by "contiguous". If you call "iota" (0,1), then the first few
elements after 0 are iota, 2iota, ... .
However, the usual ordering on this set won't be a well-ordering. Nor
will it be in any model that includes the rationals.
- Tim
Ross A. Finlayson
The computable reals are those with some "finite expression of their
value". Why are not "otherwise uncomputable real the first",
"otherwise uncomputable real the second", ... finite expressions of
those values? Consider as well a proof of the existence of an
uncomputable real, if the proof was finite would not that value have
just been summarily given, thus that it could not be? Does that agree
with that the universe is the computable universe?
Then, you consider to order the computable reals, say from the unit
interval, by their lexicographic ordering? The "computable" reals are
dense in the reals. What is their lexicographic ordering? Is it to
eachs' Godel number? As I stated here, I'm not very familiar with the
definition of computable besides as "with some finite definition."
You're telling me the computable reals are countable. I want an
example of a well-ordering of _all_ the real numbers.
About "equipollent", I think I've seen it used differently in set
theory. I just used it as an example of a synonym for "equivalent", in
that list of other equivalent terms, that's not a big deal.
That's to address part of what you noted, but I'm a little more
interested in that which you did not: about the transfer principle and
nonstandard integers, with the notion of mapping the hyperintegers and
reals. That's because due to the completion of the reals, that
"Cantor's first" would apply to mapping the hyperintegers onto the
reals.
The notion of iota is similar to that of constructing a finite
definition for some real number between zero and one. Do you want to
compute iota to any finite precision: it's point zero, zero, zero, ...
(n many times), one. Is it thus computable in that way? It would be
different than zero at any finite precision, and it's between -1/n and
1/n. There are obviously ways to construct in a similar manner values
between zero and that.
http://planetmath.org/encyclopedia/ComputableNumber.html
I'm not so interested in the computable reals, I'm interested in a
well-ordering of the reals, the set of all real numbers, or just to
simplify notions, the unit interval of reals.
A well-ordering is a relation or comparator on the elements of a set
thus that for any non-empty subset of the set, including the set, there
is a least element by the comparator among all those elements, that is
satisfied being on the "left hand side" compared to any other element
of that subset.
Is that not a correct definition?
Then, there is the notion that the reals are a contiguous sequence of
points. "Contiguous" is the discrete analog of "continuous". The
points that comprise the reals satisfy other things than merely being
constituent parts of some line. For example, the integers each refer
to a point on the line. You can take line segments of various lengths
and join them as a larger line segment and the length exactly matches
that of the sum of the numbers that are their lengths.
As the infinitesimals are considered, and those as fluents to their
infinitesimal fluxions, and those to theirs, in the Newtonian
definition of infinitesimal, it makes some sense to stop at some point
and say "there are smaller infinitesimals, this is the smallest working
infinitesimal", where that is satisfied by the fluxion of the unit.
With only rational numbers, there is no immediate need for
infinitesimals. With the real numbers, though, they're complete,
towards the notion that there has to exist infinitesimals (0 < x < 1/n)
within the real numbers else there could be no real numbers greater
than zero.
So the infinitesimal becomes amorphous, if it's a real number x greater
than zero then as the reals are a field then there exists x/2. If x
was supposed to be some smallest infinitesimal, then x/2 is a
counterexample unless x=0 or x/2 = x, which is certainly not so for
some non-infinitesimal and non-zero x. That's similar to infinity or
"oo", and how oo + 1 is no different than infinity, that iota/2 is
undefined or no different than iota. There are only and everywhere
real numbers between zero and one.
While that is the case, infinitesimals are slippery, there is the
motivation to start somewhere. One way to see that is to halve,
quarter, then make into eighths, etcetera, the unit interval ad
infinitum. That leads into stuff like the unit impulse function, "not
a real function." There is a lot of discussion of "infinitesimal
measure."
So, an expanded definition for the real numbers is considered to be as
a contiguous sequence of totally ordered points, with the caveat that
except for a possibly definite value they're not "computable" any other
way. Where the unit interval of reals is composed of those points, by
that ordering there is a least for each non-empty subset. There is at
least one real in a non-empty subset of the reals. (That's humorous.)
Where the reals are this contiguous sequence of atomic iota-values,
that's a justification for calling the normal ordering a well-ordering.
Where the infinitesimals iota-values have their own deficiencies in
usage as real numbers, that's made a separable concern. I'm not
ignoring or calling irrelevant those points of yours, but rather
describing a synthetic definition that allows both of these
perspectives of the real numbers non-conflictingly with a
discretization of the continuum's parts.
In some casual research of what others have recorded of their notion of
these things, I learn about Schmieden and Laugwitz in the 20'th century
talking about the nonstandard reals as a ring some time before A.
Robinso(h)n's hyperreals. That's only mentioned briefly in passing,
but it makes sense that something fundamental in this way about the
real numbers would have been at least considered and noted in the past
hundred years, besides the earlier considerations. There are quite a
few "nonstandard" considerations of the real numbers and continuum, and
they're there for reason. I'm not aware of any existing system that
meets my requirements in this matter.
In ZFC there is a well-ordering of the unit interval of reals.
Indexing that well-ordering of the complete unit interval of reals by
the hyperintegers, why does not "Cantor's first" apply with regards to
the transfer principle? Where it doesn't, is it for negation of a: the
transfer principle, b: the completeness of the reals, or c: the
uncountability of the reals, or d: other, or e: normal ordering as not
well-ordering, and why?
There's lots of discussion of iota. It is what it is. It holds many
of the variabilities of the infinite: it's the infinitely small for
the infinitely large. I hope I'm being fair, in sincerely presenting
these notions of mathematical logic with acceptance and understanding
of common terms and adequate reference to the larger context so that
the words have meaning. There is mathematical utility in iota.
Also: infinite sets are equivalent.
Regards,
Ross Finlayson
Timothy Little
> The computable reals are those with some "finite expression of their
> value".
No, they are not. Chaitin's Omega has a finite expression of its
value, but it is not computable.
A real number is computable iff there exists a finite Turing machine
for which you can feed in a rational, and it tells you whether the
rational is less than, equal or, or greater than the real number.
Equivalently, for any numeric base B there is a Turing machine for
which you can feed in a natural number N, and it will tell you the
N'th digit of that number in that base.
> Why are not "otherwise uncomputable real the first", "otherwise
> uncomputable real the second", ... finite expressions of those
> values?
Because you don't know what a computable real number is.
> Then, you consider to order the computable reals, say from the unit
> interval, by their lexicographic ordering?
Sure.
> The "computable" reals are dense in the reals. What is their
> lexicographic ordering? Is it to eachs' Godel number?
Pretty close -- it can be considered a Godel numbering of the Turing
machine description, and it's a well-ordering.
> As I stated here, I'm not very familiar with the definition of
> computable besides as "with some finite definition." You're telling
> me the computable reals are countable. I want an example of a
> well-ordering of _all_ the real numbers.
An interesting goal, and I wish you luck. Demanding that the
well-ordering also be the normal ordering is contradictory. You just
can't see it because you don't know, or can't use, the definition of a
well-ordering.
> About "equipollent", I think I've seen it used differently in set
> theory. I just used it as an example of a synonym for "equivalent",
> in that list of other equivalent terms, that's not a big deal.
It would be a big deal if it was part of the meaning of "infinite sets
are equivalent", when you haven't otherwise defined "equivalent". Is
it part of the meaning?
> That's because due to the completion of the reals, that "Cantor's
> first" would apply to mapping the hyperintegers onto the reals.
The transfer principle applies only to certain types of sentences, and
it's not immediately obvious that it applies to Cantor's first proof.
It certainly does not apply to the statement of the theorem directly.
Besides which, even if it were valid, it would not apply to mapping
the hyperintegers onto the reals. It would apply to mapping the
hyperintegers onto the hyperreals.
Thirdly, what does this have to do with constructing a well-ordering
of the reals? The positive hyperintegers aren't well-ordered.
> The notion of iota is similar to that of constructing a finite
> definition for some real number between zero and one. Do you want
> to compute iota to any finite precision: it's point zero, zero,
> zero, ... (n many times), one. Is it thus computable in that way?
In the definition of computability, the program is fixed, you provide
it with variable n, and it returns the n'th digit (in base 10, say).
Which digits are nonzero for iota?
Iota is not at all similar to that of constructing a finite definition
for some real number.
> A well-ordering is a relation or comparator on the elements of a set
> thus that for any non-empty subset of the set, including the set, there
> is a least element by the comparator among all those elements, that is
> satisfied being on the "left hand side" compared to any other element
> of that subset.
>
> Is that not a correct definition?
It is; now all you have to do is remember it and be able to use it.
> Then, there is the notion that the reals are a contiguous sequence of
> points. "Contiguous" is the discrete analog of "continuous".
Contiguous normally means "adjacent" or "touching". That seems
reasonable as a property of pseudo-reals that you're defining
yourself, but it's not a property of the real reals.
> With only rational numbers, there is no immediate need for
> infinitesimals. With the real numbers, though, they're complete,
> towards the notion that there has to exist infinitesimals (0 < x < 1/n)
> within the real numbers else there could be no real numbers greater
> than zero.
The reals are defined without infinitesimals. Hence your assertion is
false. If you introduce infinitesimals it's because you brought them
in yourself as an addition assertion, not because they're logically
necessary.
> That's similar to infinity or "oo", and how oo + 1 is no different
> than infinity, that iota/2 is undefined or no different than iota.
Sure, as I said I have no problem with infinitesimals. They're not in
the reals, but there's no law against creating larger sets that
contain the reals.
> So, an expanded definition for the real numbers is considered to be as
> a contiguous sequence of totally ordered points, with the caveat that
> except for a possibly definite value they're not "computable" any other
> way. Where the unit interval of reals is composed of those points, by
> that ordering there is a least for each non-empty subset.
Prove it.
> Where the reals are this contiguous sequence of atomic iota-values,
> that's a justification for calling the normal ordering a well-ordering.
There is no such justification. Look up, to where you gave the
definition of a well-ordering. *EVERY* nonempty subset must have a
least element. In particular, {1/n:n in N} must have a smallest
element. N has no largest element, so this set has no smallest
element. Hence the ordering is not a well-ordering.
Iota isn't in the set {1/n: n in N}, so it certainly isn't the
smallest element of it. It may be the least element of (0,1] in your
extension of the reals, but that's just one subset. A well-ordering
requires the property for every subset. Did you see the word "every"?
Once again, you demonstrate your inability to use the definition.
> Where the infinitesimals iota-values have their own deficiencies in
> usage as real numbers, that's made a separable concern. I'm not
> ignoring or calling irrelevant those points of yours,
Actually, you are. You're ignoring the definition of a well-ordering.
Quoting it isn't the same thing as using it, and you're ignoring the
examples that show why the usual ordering can't be a well-ordering.
Then you go on to blather about iota, contiguous points,
infinitesimals, hyperreals, "infinite sets are equivalent", and make
unsupported assertions that are obviously false.
> I'm not aware of any existing system that meets my requirements in
> this matter.
That's because your requirements are contradictory, obviously so once
you actually use their definitions.
> In ZFC there is a well-ordering of the unit interval of reals.
There is.
> Indexing that well-ordering of the complete unit interval of reals
> by the hyperintegers, why does not "Cantor's first" apply with
> regards to the transfer principle?
1) The hyperintegers are not well-ordered.
2) Cantor's First proof uses properties of the reals that are not true
of hyperreals. This does not counter the Transfer Principle because
3) The transfer principle maps certain first-order statements
quantified over reals with corresponding statements about hyperreals.
Cantor's first proof is not such a statement.
4) The transfer principle has nothing to say about statements that mix
hyperintegers and reals.
5) There may not even be a surjection from the hyperintegers to the
reals. I think this depends upon the continuum hypothesis, which is
undecidable within ZFC.
> Where it doesn't, is it for negation of a: the transfer principle,
> b: the completeness of the reals, or c: the uncountability of the
> reals, or d: other, or e: normal ordering as not well-ordering, and
> why?
Pretty much all of these, even though just one would suffice.
> I hope I'm being fair, in sincerely presenting these notions of
> mathematical logic with acceptance and understanding of common terms
> and adequate reference to the larger context so that the words have
> meaning.
You could start by accepting and understanding the common term
"well-ordering".
> There is mathematical utility in iota.
There may well be, but you haven't shown any yet.
> Also: infinite sets are equivalent.
This statement however does not yet have any mathematical utility
whatsoever. Replace "equivalent" with a meaningful mathematical
statement and it might have some utility.
As it stands, "equivalent" just means "there exists an equivalence
class that contains them", which is a completely trivial property.
Which equivalence relation are you talking about?
- Tim
Ross A. Finlayson
of "computable" is somewhat more pathological than that of, say,
"equipollent", please disregard use of that term, re Torkel's "which
definition of computable". In examining the "PlanetMath" entry for
"computable number", you can currently note a minor discrepancy between
the current entry: empty, and the cached entry via Google, with an
"alternate definition" about a recursive function to generate a value
to any given precision, with some notion of a single free variable to
combinatorially enumerate the numbers.
http://www.google.com/search?q=cache:O-pLJ5ALioYJ:planetmath.org/encyclopedia/ComputableNumber.html
As I said, I'm not very familiar with the definition of "computable".
Is Chaitin's constant useful for anything? I'd like to know an
application of it.
With computable versus non-computable numbers there seems to be a
similar inability as Dedekind/Cauchy to describe each real number.
I figured you would note the "apples and oranges" of reals vis-a-vis
hyperreals, but I think it's possible that the transfer principle still
applies because the reals are _complete_. I'm glad you did. I don't
grasp why the positive hyperintegers are not well-ordered, as the
positive integers are, and those are the ones under consideration. As
well, with regards to indexing the elements of the well-ordering of the
reals by the ordinals, it still seems unclear why "Cantor's first"
wouldn't apply, due to the completeness of the reals. Does Cantor's
first not apply, or do two successive elements of the well-ordering
contain no real between them in the normal ordering of the reals?
How does any well-ordering of the real numbers avoid the contradictory
consequences of Cantor's first?
About a working (agreed upon) definition of "well-ordering", it is
obvious that that is one currently available to this discussion and
that is in use for some time. The definition and its usage are quite
clear.
About proving the existence of a least element in each subset of the
reals, that's part of the rationale of addending to the definition of
real numbers iota-value or atomic infinitesimals towards what you have
labelled the "Finlayson numbers" or "pseudo-reals", which I call the
"reals" or real numbers. I derive enjoyment from calling them
"Finlayson numbers" to the extent that I think they're useful, is that
selfish or wrong? I'm reminded of Glass' pronouncements in joviality.
I just call them the reals or real numbers with atomic iota-values,
perhaps R-bar or R-umlaut. This is where re-reading the entire thread
helps to refresh understanding.
Here's another analogy, it's like a Pachinko machine, a method to
randomly generate a real. That's where there is a triangular (or
lattice) grid of pegs at an incline and a marble is dropped on the top
of the triangle (from the center or evenly distributed side-to-side),
bouncing somewhat randomly at each peg to the left or right, to
eventually fall within one of many separate bins, labelled left to
right by ascending or descending ordinals. The average value of all of
the reals of the unit interval is 1/2, as discussed above about the
fair coin and random reals. That adds a notion of "computability" to
"beads on a string" or points on a line.
About the notion that 1/n has no least element, the idea is that for
some ordinal x that is indefinite and unknown except to be greater than
one and less than for any other value of n that x*iota is the least
element of the range of 1/n. _If_ you accept that iota is the least
value of the interval (0,1) by the normal ordering, _then_
correspondingly every subset of (0,1) has a least element.
About the equivalency of infinite sets, I've argued that infinite sets
are equivalent, where that means there is a bijection between them, for
some years. "Equivalent" is here seeing acceptible usage, and it is in
regular usage in that way, and in the context it is quite clear.
I'm glad that you consider these notions of the extension of the real
numbers with atomic iota-values, indefinite and ubiquitous. While that
is so, the _completeness_ of the reals implies that the elements of
these nonstandard extensions can only be real numbers.
So, what insight you give towards helping to establish the definition
of these "extensions" to the real numbers is appreciated.
Thanks for briefly re-reading the entire thread.
Regards,
Ross Finlayson
Timothy Little
> We're arguing about definitions here. It appears that the definition
> of "computable" is somewhat more pathological than that of, say,
> "equipollent", please disregard use of that term
OK. Disregarded.
> I don't grasp why the positive hyperintegers are not well-ordered,
It is not difficult to construct a set of hyperintegers that has no
least element. For example, pick any infinite hyperinteger m*, and
form the set {m* - n: n in N}. Every element of that set is a
positive hyperinteger, and it has no smallest element.
> as the positive integers are, and those are the ones under
> consideration.
The positive integers are, the positive hyperintegers aren't.
Well-ordering is not a property that the transfer principle carries
across since it quantifies over subsets, not just over numbers.
> As well, with regards to indexing the elements of the well-ordering
> of the reals by the ordinals, it still seems unclear why "Cantor's
> first" wouldn't apply, due to the completeness of the reals.
The problem with extending Cantor's first proof to arbitrary ordinals
is that it is based on ordinary induction rather than transfinite
induction.
Let's look at where the construction fails in a specific example. The
theorem says that given any sequence of reals, and any interval, we
can construct an element not in that sequence.
Let alpha = N union {N}. That is, it is the ordinal set consisting of
all the natural numbers, as well as the set N included as an element
greater than all the rest.
Define a sequence of reals A:alpha -> R by:
A(2n-1) = 1/n
A(2n) = -1/n
A(N) = 0.
Let our initial interval be [-1, 1].
We begin by taking the first two sequence members within the interval,
and forming a new interval: (a_1, b_1) = (-1, 1). Next we find
the first two sequence members inside (a_1, b_1): we find
(a_2, b_2) = (-1/2, 1/2). Continuing on, it is clear that
(a_n, b_n) = (-1/n, 1/n) for all n in N. Now we take the upper and
lower limits: in this case both are 0. A naive extension of Cantor's
proof would conclude that 0 is not a member of the original sequence!
It failed because it couldn't make the jump from N to the larger
ordinal that the sequence was based on. So we cannot apply the
original proof to arbitrary well-orderings.
> About a working (agreed upon) definition of "well-ordering", it is
> obvious that that is one currently available to this discussion and
> that is in use for some time. The definition and its usage are quite
> clear.
Glad you agree. How about you start actually using it?
> I derive enjoyment from calling them "Finlayson numbers" to the
> extent that I think they're useful, is that selfish or wrong?
That sounds fine to me, and makes it clear you're not talking about
the standard definition of the reals.
> About the notion that 1/n has no least element, the idea is that for
> some ordinal x that is indefinite and unknown except to be greater
> than one and less than for any other value of n that x*iota is the
> least element of the range of 1/n.
If x*iota is in the set, then there exists n in N such that
x*iota = 1/n, because that's what being in {1/n:n in N} *means*.
Then 1/(n+1) is a smaller element in the set, contradicting your
assertion that x*iota is the smallest. If you can't grasp this simple
point, there really is no point in discussing this any further with
you.
At first I thought you might have a problem with what a well-ordering
means. Now I think you don't even know what it means for an element
to be in a set, even an explicitly enumerated set.
> About the equivalency of infinite sets, I've argued that infinite
> sets are equivalent, where that means there is a bijection between
> them, for some years.
OK, so "infinite sets are equipollent" or "there is only one infinite
cardinal" would be a more precise and mathematically useful way to put
it.
ZF + "Infinite sets are equivalent" = contradiction
... but that's not fatal. It just means you're talking about
different sets than most other people do. It also means that you need
to come up with some consistent axioms for your personal set theory
before it's worth using. Then you need to tune your axioms so they
are capable of supporting something like numbers without collapsing
into universal provability, then come up with something like the
reals, come up with a well-ordering in your system, and then prove a
correspondence between the reals in your system and the reals in ZFC,
and that well-orderings in your system correspond to well-orderings in
ZFC.
> I'm glad that you consider these notions of the extension of the real
> numbers with atomic iota-values, indefinite and ubiquitous. While that
> is so, the _completeness_ of the reals implies that the elements of
> these nonstandard extensions can only be real numbers.
Completeness is a specific topological property. It doesn't mean that
the reals have everything conceivable in them. The standard reals
don't have infinitesimals.
(I add completeness to the list of terms that RF doesn't understand)
- Tim
Dave Seaman
> Ross A. Finlayson wrote:
>> I don't grasp why the positive hyperintegers are not well-ordered,
> It is not difficult to construct a set of hyperintegers that has no
> least element. For example, pick any infinite hyperinteger m*, and
> form the set {m* - n: n in N}. Every element of that set is a
> positive hyperinteger, and it has no smallest element.
That is an external set, not an internal set. Isn't it true that every
internal set of positive hyperintegers has a least element?
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
Timothy Little
> About Cantor's first and the ordinal index of the well-ordering, if
> there only needs be one infinite ordinal, N, appended to the natural
> index of a soi-disant well-ordering, then 1: that's not an uncountable
> set of ordinals, and 2: it would imply in the well-ordering two points
> of the reals between which there are not other reals.
What has this to do with anything? You asked why Cantor's first
didn't apply for other ordinals, I gave an example. It didn't have to
be an uncountable example. No countable ordinal sequence can
enumerate the reals, but the method in Cantor's first proof is
insufficiently powerful to demonstrate it. That's why he came up with
a better proof and hardly anyone bothers to remember his first.
> With only the knowledge that these points comprise the line and that
> the line or rather segment or ray represents the continuum (of real
> numbers), then one of those points is the least, by the usual
> ordering of points on a line, in any non-empty collection, or set,
> of those points.
Once again, you fail to understand "any non-empty set". It does not
just mean "any interval" or "any contiguous set" or whatever other
derangement you can come up with. It includes infinite sequences of
isolated points approaching a given point which isn't in the set.
We're done here. You can't grasp it.
- Tim
Timothy Little
> That is an external set, not an internal set.
Yes, it is. So is Ross' putative sequence of reals indexed by the
hyperintegers, so we're clearly not talking just about internal sets.
- Tim
Ross A. Finlayson
The reals are complete. Thus, the sequences a and b that converge
toward each other must at some point either 1: come to a conclusion
where the last element of a and the last element of b have no
intervening elements in the normal ordering of the real numbers, or 2:
be infinite, continue on forever, and the well-ordering never contains
each element of the reals. So, the sequences terminate or don't.
According to some, the hyperintegers are the same thing as the
integers, and the hyperreals the same thing as the reals. Some have
that sum 1/2^n = 2. That's not generally the understanding, not
standard.
If we discuss some extended definition of hyperintegers for the nested
intervals, then perhaps it might be considered for the antidiagonal
argument about a well-ordering of the reals. A problem with that is
that for an expansion that can represent a number, if there ever were
an infinite expansion, then that sequence could be considered for the
positive hyper-integers in this case. For each real number by the
well-ordering's index, assign an element of the punitive antidiagonal's
expansion at that index according to your rule. Is that thus an
element of the reals, different from each element indexed by the
well-ordering in the sequence at that index, and not dually
represented?
That might be easily countered by saying that there's no notion of an
expansion or sequence indexed by the positive hyperintegers or infinite
ordinals, except the positive hyperintegers are, because they're just
the positive integers. The reals are complete.
I guess the problem here is that I'm discussing a well-ordering of the
reals as if it were a "list" of the reals, where the status quo says "a
well-ordering exists", yet "no list exists". Another problem is the
lack of any example of a well-ordering.
This is where I think a list of the reals from the unit interval, and a
well-ordering, is something along the lines of: 0, iota, 2iota, ...,
for various reasons as discussed here and elsewhere, for example
Cantor's first.
Perhaps instead you mean a bijection from the reals to the powerset of
naturals and Cantor-Bernstein. Some people have the cardinality of the
continuum being greater than aleph_1.
The set of all sets would be its own powerset. Indeed, in some
theories there is a set of all sets or "universal set", not subject to
the powerset result.
Where infinite sets are equivalent, those are not problems. Infinite
Piotr Sawuk
Timothy Little <tim-via...@little-possums.net> writes:
> Ross A. Finlayson wrote:
> As an example, "the reals do not have a well-ordering" is consistent
> with ZF (without C).
yes, I remember, it was ZF with non C where "all infinities are the
same" did make at least some sense...
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P