Uncountability of the Rationals?
Tony Orlow
Looking at some of Herc's recent threads about Cantor's diagonal proof
of the uncountability of the reals, a question occurs to me which I
actually don't know how a subscriber to transfinite set theory would
answer. I'm curious.
In Cantor's list of reals, for every digit added, the list doubles in
length. In order to follow his logic, we need not consider any real
numbers which are not rational. In any such list of rationals, it is
always true that we can concoct another rational which is not on the
list. Irrational numbers need not even be considered. So, are the
rationals not to be considered uncountable, by Cantor's own logic?
BTW, I understand that the rationals are considered countable because
of a rather artificial bijection with N, but if Cantor's argument
really has anything to do with the reals, as opposed to the powerset
(which is what it's really all about), then why is it entirely
unnecessary to even consider irrational reals in his construction?
Thanks and Smiles,
Tony
Virgil
<b65b2a33-7335-410a...@q13g2000vbm.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> Hi All -
>
> Looking at some of Herc's recent threads about Cantor's diagonal proof
> of the uncountability of the reals, a question occurs to me which I
> actually don't know how a subscriber to transfinite set theory would
> answer. I'm curious.
>
> In Cantor's list of reals, for every digit added, the list doubles in
> length. In order to follow his logic, we need not consider any real
> numbers which are not rational. In any such list of rationals, it is
> always true that we can concoct another rational which is not on the
> list. Irrational numbers need not even be considered. So, are the
> rationals not to be considered uncountable, by Cantor's own logic?
>
> BTW, I understand that the rationals are considered countable because
> of a rather artificial bijection with N
Countability of any set is established by ANY bijection with N, no
matter how "artificial" it may seem to Tony Orlow, or anyone else.
> but if Cantor's argument
> really has anything to do with the reals, as opposed to the powerset
> (which is what it's really all about), then why is it entirely
> unnecessary to even consider irrational reals in his construction?
Since Cantor has provided at least two proofs based on two entirely
different lines of argument, TO should make clear what proof he is
talking about, particularly as both proofs require consideration of
irrationals so that TO's comment is nonsense with respect to either.
Brian Chandler
> Hi All -
Hello Tony!
> Looking at some of Herc's recent threads about Cantor's diagonal proof
> of the uncountability of the reals, a question occurs to me which I
> actually don't know how a subscriber to transfinite set theory would
> answer. I'm curious.
There are no "subscribers" to set theory (unless this has something
curious to do with using a "full-price" newsreader), only those who
understand it (as distinct from those who don't, of course).
> In Cantor's list of reals, for every digit added, the list doubles in
> length.
It does? Reference please to a textbooks talking about "Cantor's list
of reals", with digits being progressively added. (For extra marks,
you could just explain why you think there's a relation between
"adding a digit" and doubling the number of somethings. This sounds
much more to me like a list of terminating binary fractions of length
not more than n, for some n -- not related in any obvious way to a
"list of reals".)
Brian Chandler
Martin M. + I am right:if I am not tell me:or I am right
Note if NP ⊆ P/poly then SAT has a polynomial-size circuit family.
Namely, for some c > 0 there is a sequence of circuits W1,W2,W3,...,
where Wn is a circuit of size nk deciding satisfiability of all
boolean formulae whose encoding size is ≤ n.
Of course, NP ⊆ P/poly merely implies the existence of such a family;
there may be no easy way to construct the circuits.
Lemma (Self-reducibility of SAT)
There is a polynomial-time computable function h such that if {Wn}n≥1
is a circuit family that solves SAT, then for all boolean formulae ϕ:ϕ
∈ SAT iff h (ϕ, W|ϕ|) is a satisfying assignment for ϕ. (1)
Proof: The main idea is to use the circuit to generate a satisfying
assignment as follows.
Ask the circuit if ϕ is satisfiable.
If so, ask it if ϕ(x1 = T) (i.e., ϕ with the first variable assigned
True) is satisfiable.
The circuit’s answer allows us to reduce the size of the formula.
If the circuit says no, we can conclude ϕ(x1 = F) is true, and have
thus reduced the number of variables by 1.
If the circuit says yes, we can substitute x1 = T and again reduced
the number of variables by 1.
Continuing this way, we can generate a satisfying assignment.
This proves the Lemma. (Aside: The formal name for the above property
of SAT is downward self-reducibility.
All NP-complete languages have this property.)QED
Now we prove the Theorem.
We show NP ⊆ P/poly implies Πp2 ⊆ Σp2. Let L ∈ Πp2. Then there is a
language L1 ∈ NP and c > 0 such that L = {x ∈ {0, 1}∗: ∀ y, |y|≤|x|c,
(x, y) ∈ L1}. (2)
Since L1 ∈ NP, there is a polynomial-time reduction, say g, from L1 to
SAT.
Thus ∀ z ∈ {0,1}∗: z ∈ L1 iff g(z) ∈ SAT.
Suppose further that d > 0 is such that |g(z)|≤|z|d.
Now we can rewrite (2) asL = {x ∈ {0,1}∗: ∀ y,|y|≤|x|c1, g(x, y) ∈
SAT}.
Note that |g(x, y)| ≤ (|x| + |y|)d.
Let us simplify this as |x|cd.
Thus if Wn≥1 is a nk-sized circuit family for SAT, then by (1) we have
L = {x : ∀ y,|y|≤|x|c1, h(g(x, y), W|x|cd ) is a satisfying assignment
for g(x, y)}.
Now we are almost done.
Even though there may be no way for us to construct the circuit for
SAT, we can just try to “guess” it.
Namely, an input x is in L iff ∃W, a circuit with |x|cd inputs and
size |x|cdk such that ∀ y, |y|≤|x|c1, h(g(x, y), W) is a satisfying
assignment for g(x, y).
Since h is computable in polynomial time, we have thus shown L ∈ Σp2.
QED1
On Jun 2, 12:27 pm, Brian Chandler <imaginator...@despammed.com>
wrote:
Mike Terry
news:b65b2a33-7335-410a...@q13g2000vbm.googlegroups.com...
> Hi All -
>
> Looking at some of Herc's recent threads about Cantor's diagonal proof
> of the uncountability of the reals, a question occurs to me which I
> actually don't know how a subscriber to transfinite set theory would
> answer. I'm curious.
>
> In Cantor's list of reals, for every digit added, the list doubles in
> length. In order to follow his logic, we need not consider any real
> numbers which are not rational. In any such list of rationals, it is
> always true that we can concoct another rational which is not on the
> list. Irrational numbers need not even be considered. So, are the
> rationals not to be considered uncountable, by Cantor's own logic?
I don't understand the "list doubles in length" bit. However, I think maybe
I can see what you're trying to do. You want to:
1) treat rationals as real numbers, with a decimal expansion
2) given a listing of all the rational numbers, apply Cantor's
argument regarding the anti-diagonal to construct a new number
not on the list
3) conclude that this means that the rationals would not be countable?
Is this what you're doing? If so, don't bother explaining about the list
doubling stuff, as it doesn't matter - there's an easy explanation for your
problem! The explanation is that steps (1) and (2) are fine. However the
"antidiagonal number" generated in (2) will not be a rational number, so the
conclusion (3) breaks down. Note the difference with Reals instead of
Rationals: when applied to Reals, the antidiagonal number is a *Real* number
not in the list...
Regards,
Mike.
christian.bau
> BTW, I understand that the rationals are considered countable because
> of a rather artificial bijection with N, but if Cantor's argument
> really has anything to do with the reals, as opposed to the powerset
> (which is what it's really all about), then why is it entirely
> unnecessary to even consider irrational reals in his construction?
You call this a "rather artificial bijection", but here is what being
"countable" really means: A set is countable if you can choose a first
element, a second element, a third element, and so on, in such a way
that every single element will be chosen eventually. In other words,
you can systematically count through all the elements.
So what is "artificial" about the bijection between rationals and
natural numbers? If you tried to count the integers (positive and
negative) by starting 1, 2, 3, 4 and so on and then follow with the
negative numbers and zero last, then obviously we will _not_ manage to
eventually reach every integer, because we never finish with the
positive ones. But that is just because we took a stupid ordering: If
you count 0, +1, -1, +2, -2, +3, -3, +4, -4 and so on, then _every_
integer will turn up eventually. Same with the rationals. Start with
zero. Then all fractions where numerator + denominator = 2, followed
by the negative ones, then those where numerator + denominator = 3,
the negatives, and so on. There is nothing "artificial" about this,
just a straightforward way to achieve what you want. And every
fraction will eventually turn up.
Cantor's argument breaks down for rational numbers: Assume we have a
list of all rational numbers. Now we create a rational number that is
not on the list by taking the first decimal of the first number on the
list and adding 1, the second decimal of the second number on the list
and adding one, then the third decimal of the third number adding one
and so on. The resulting number is not in the list. So where does the
argument break down? Very simple: The resulting number is not
rational.
Transfer Principle
> > of the uncountability of the reals, a question occurs to me which I
> > actually don't know how a subscriber to transfinite set theory would
> > answer. I'm curious.
> There are no "subscribers" to set theory (unless this has something
> curious to do with using a "full-price" newsreader)
This is a thinly veiled reference to me, and my argument
that those who post via free web-based news access have
a lower reputation than those who post via traditional
NNTP newsreaders to sci.math. (If Chandler doesn't
believe me, he can just look at the poster Spotter, all
of whose posts are one-liners disparaging those who
post via Google or Mathforum or make even the slightest
spelling error. At least one of his posts is racist. In
no case does Spotter ever address the argument per se.)
Since I'm here in a thread that attacks Cantor anyway,
I might as well address the subject at hand.
> only those who
> understand it (as distinct from those who don't, of course).
I agree with TO here, except that instead of "subscriber,"
I prefer to use a word like "adherent" (so as to avoid the
money issue). So I refer to the adherents of ZFC. Some
posters may have different reasons for not being an
adherent of ZFC -- perhaps because they're constructivists
who oppose AC, perhaps they are finitists who oppose the
Axiom of Infinity, and so on.
The word "adherent" often describes those who have a
particular religion. Notice that Herc, whose posts TO has
read before starting this thread, even refers to standard
set theory based on Cantor as being a "religion." And I
somewhat agree. Adherents of ZFC believe that its axioms
are true, just as adherents of a religion believe that its
unprovable claims are true.
I have nothing against ZFC or religion per se -- just that
adhering to either is somewhat similar.
> > In Cantor's list of reals, for every digit added, the list doubles in
> > length.
> It does? Reference please to a textbooks talking about "Cantor's list
> of reals", with digits being progressively added. (For extra marks,
> you could just explain why you think there's a relation between
> "adding a digit" and doubling the number of somethings. This sounds
> much more to me like a list of terminating binary fractions of length
> not more than n, for some n -- not related in any obvious way to a
> "list of reals".)
TO admits that he's read Herc, and so it's not surprising
that he makes a Herc-like argument here. Herc and TO ask
if Cantor proves that R is uncountable, why doesn't the
same proof show N or Q, respectively, to be uncountable?
As I told Cooper in the other thread, because of the
defintion of countable and the way that quantifiers work,
proving a set to be countable only requires that _one_
list of its elements be complete, while showing that it's
uncountable requires that _all_ (not just _one_) lists of
its elements be incomplete.
But unlike Herc, we _know_ that TO is interested in his
own theory (thus putting him in Case 1 of my list of
possible cases), since he's gone as far as to give the
proposed theory a name, the T-riffics. In the T-riffics,
the set N of natural numbers is said to have a cardinality
(or "bigulosity") of Big'Un. In the T-riffics, sets have
distinct bigulosity from their proper subsets, and thus Q
must have a bigulosity strictly more than Big'Un. (I
forget what TO claims to be the bigulosities of Q or R.)
So unlike Cooper, at least we know what TO is attempting
to accomplish here.
adamk
Fuck you, worthless piece of shit. Go swallow
boiling-hot oil, and go dump your trash somewhere
else.
Fuck Off, now.
Loser.
Tony Orlow
Hi Transfer et al -
Thanks for (most of) the responses. I get what I missed, but probably
have to play with it a little to be fully convinced intuitively that
the diagonal of a list of all rationals must produce an irrational
real. That would be a good response. I am by no means trying to equate
|N| with |R| or any such thing, nor |Q| with |R|. I was just wondering
off the cuff how the logic broke there and it makes sense.
BTW, Transfer, your rendition of Bigulosity Theory is a little off,
but no worries. Hopefully I'll have at least a preliminary version
ready soon for release, review, and excoriation. Thanks to Mike and
Christian, too. Adam, I think you know where you can stick your
comment. Virgil and Brian, nice to see you still holding down the
fort. Martin, you left your medication in the CD ROM drive again....
Have a nice day!
Tony
Jesse F. Hughes
> On Jun 2, 12:27 pm, Brian Chandler <imaginator...@despammed.com>
> wrote:
>> > Looking at some of Herc's recent threads about Cantor's diagonal proof
>> > of the uncountability of the reals, a question occurs to me which I
>> > actually don't know how a subscriber to transfinite set theory would
>> > answer. I'm curious.
>> There are no "subscribers" to set theory (unless this has something
>> curious to do with using a "full-price" newsreader)
>
> This is a thinly veiled reference to me, and my argument
> that those who post via free web-based news access have
> a lower reputation than those who post via traditional
> NNTP newsreaders to sci.math. (If Chandler doesn't
> believe me, he can just look at the poster Spotter, all
> of whose posts are one-liners disparaging those who
> post via Google or Mathforum or make even the slightest
> spelling error. At least one of his posts is racist. In
> no case does Spotter ever address the argument per se.)
You care about what Spotter thinks?
Spotter is an asshole and a troll. I can't imagine why you think his
posts indicate anything about the "reputations" of Google users.
> Since I'm here in a thread that attacks Cantor anyway,
> I might as well address the subject at hand.
>
>> only those who
>> understand it (as distinct from those who don't, of course).
>
> I agree with TO here, except that instead of "subscriber,"
> I prefer to use a word like "adherent" (so as to avoid the
> money issue). So I refer to the adherents of ZFC.
You are honestly an odd person.
In any case, what does "adherent" mean? What distinguishes an
adherent from someone else acquainted with a theory? Brian's comment
is spot on once again.
There are no ["adherents"] to set theory ...
only those who understand it (as distinct from those who don't, of
course).
> Some posters may have different reasons for not being an adherent of
> ZFC -- perhaps because they're constructivists who oppose AC,
> perhaps they are finitists who oppose the Axiom of Infinity, and so
> on.
>
> The word "adherent" often describes those who have a
> particular religion. Notice that Herc, whose posts TO has
> read before starting this thread, even refers to standard
> set theory based on Cantor as being a "religion." And I
> somewhat agree. Adherents of ZFC believe that its axioms
> are true, just as adherents of a religion believe that its
> unprovable claims are true.
Nonsense!
I honestly have no idea what it means to say that the axioms of ZFC
are true. I understand what it means in certain contexts, but not
that they are true simpliciter. Does this mean I'm not an adherent?
It's important that we get this clear. I want to know what labels
apply to me. (It's for bumper stickers.)
--
Jesse F. Hughes
Playin' dismal hollers for abysmal dollars,
Those were the days, best I can recall.
-- Austin Lounge Lizards, "Rocky Byways"
christian.bau
> Thanks for (most of) the responses. I get what I missed, but probably
> have to play with it a little to be fully convinced intuitively that
> the diagonal of a list of all rationals must produce an irrational
> real. That would be a good response.
That's quite easy if you go the other way round. Instead of proving
that applying the Cantor process to a complete list of all rational
numbers will necessarily produce a real number, you can show: If the
Cantor process applied to an infinite list of numbers produces a
rational result, then there is a rational number which is not in the
list.
So take a list of numbers (we don't even care if they are rational).
Construct a real number x by taking the first digit of the first
number in the list, the second decimal digit of the first number and
so on, and increasing each digit by 1 (changing a digit 9 to a 0).
Assume the result x is a rational number. Now construct a number y by
subtracting 2 from each digit of x, replacing 0 with 8 and 1 with 9. A
number is rational if and only if its decimal representation is
periodic. x is periodic, therefore y is periodic, therefore y is
rational. But y cannot be in the list of numbers, because if it was
element number n, then the n-th decimal digit of x would be 1 higher
than the n-th decimal digit of y, but it is 2 higher.
So: The Cantor process applied to _any_ infinite list of real numbers
produces a result x which is either rational or irrational. If x is
rational then there is a different rational number y which is not in
the list. So if the list contains all rational numbers (plus possibly
a finite or infinite number of irrational numbers), then the result of
the Cantor process must be irrational.
Tim Little
> In Cantor's list of reals, for every digit added, the list doubles in
> length.
In Cantor's list of reals, there are no digits "added", nor changes in
length of the list. The list is presumed to be given, and regardless
of what it contains, is shown to be necessarily incomplete.
> In order to follow his logic, we need not consider any real
> numbers which are not rational. In any such list of rationals, it is
> always true that we can concoct another rational which is not on the
> list.
No, it is not always true. It is not difficult to demonstrate a list
of rationals for which the antidiagonal is irrational.
- Tim
Brian Chandler
> On Jun 2, 12:27 pm, Brian Chandler <imaginator...@despammed.com>
> wrote:
> > > Looking at some of Herc's recent threads about Cantor's diagonal proof
> > > of the uncountability of the reals, a question occurs to me which I
> > > actually don't know how a subscriber to transfinite set theory would
> > > answer. I'm curious.
> > There are no "subscribers" to set theory (unless this has something
> > curious to do with using a "full-price" newsreader)
>
> This is a thinly veiled reference to me,
I'd have called it a "weak joke", rather than a "thinly-veiled
reference", but never mind. FWIW, you might notice I'm posting this
from Google, so you and I share the same low esteem rating, it seems.
> > only those who
> > understand it (as distinct from those who don't, of course).
>
> I agree with TO here, except that instead of "subscriber,"
> I prefer to use a word like "adherent" ...
>
> The word "adherent" often describes those who have a
> particular religion. ... Adherents of ZFC believe that its axioms
> are true, just as adherents of a religion believe that its
> unprovable claims are true.
Hmm. But non-adherents of religion (and possible adherents, too, with
respect to all but the one they adhere to*) can generally say just
where they disagree with the axioms of the religion. "Non-adherents"
to ZFC typically flunk out when challenged to exactly what they
disagree with.
* I'm reminded of the review of Martin Gardner's seminal anti-woo
bookl published around 1950. The second edition is generally preferred
because it includes samples of the hate mail he received after the
first edition, from people who generally loved all chapters but
exactly one of the book. **
** Sorry, I chickened out, because I just couldn't choose between "all
but exactly one chapter" and "all but exactly one chapters" ...
> I have nothing against ZFC or religion per se -- just that
> adhering to either is somewhat similar.
Ah, now I'm stuck.
Brian Chandler
Rob Johnson
In article <f1b4a519-f904-43ca...@q8g2000vbm.googlegroups.com>,
"christian.bau" <christ...@cbau.wanadoo.co.uk> wrote:
>Cantor's argument breaks down for rational numbers: Assume we have a
>list of all rational numbers. Now we create a rational number that is
>not on the list by taking the first decimal of the first number on the
>list and adding 1, the second decimal of the second number on the list
>and adding one, then the third decimal of the third number adding one
>and so on. The resulting number is not in the list. So where does the
>argument break down? Very simple: The resulting number is not
>rational.
In article <d95e1d01-9992-4e8d...@z33g2000vbb.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
>Thanks for (most of) the responses. I get what I missed, but probably
>have to play with it a little to be fully convinced intuitively that
>the diagonal of a list of all rationals must produce an irrational
>real. That would be a good response.
I think that you are missing the point. In the Cantor diagonal proof,
the number constructed from the diagonal is easily a real number.
However, when we try the same argument with the rationals, the proof
breaks down because we cannot show that the number constructed from
the diagonal is a rational. We don't need to prove it's irrational.
It's up to the proof writer to show that the constructed number is
rational; if they can't, the proof fails.
It is provable that the diagonal element in the argument with the
rationals is irrational. However, the only relevance of this fact
to the discussion above is to assure us that no one can show that
the diagonal element is rational.
christian.bau gives an outline of an enumeration of the rationals:
In article <f1b4a519-f904-43ca...@q8g2000vbm.googlegroups.com>,
"christian.bau" <christ...@cbau.wanadoo.co.uk> wrote:
> Start with
>zero. Then all fractions where numerator + denominator = 2, followed
>by the negative ones, then those where numerator + denominator = 3,
>the negatives, and so on. There is nothing "artificial" about this,
>just a straightforward way to achieve what you want. And every
>fraction will eventually turn up.
Since the rationals are countable, if all of them are in a list,
the diagonal element from that list must be irrational.
Rob Johnson <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
Tony Orlow
> Transfer Principle wrote:
> > On Jun 2, 12:27 pm, Brian Chandler <imaginator...@despammed.com>
> > wrote:
> > > > Looking at some of Herc's recent threads about Cantor's diagonal proof
> > > > of the uncountability of the reals, a question occurs to me which I
> > > > actually don't know how a subscriber to transfinite set theory would
> > > > answer. I'm curious.
> > > There are no "subscribers" to set theory (unless this has something
> > > curious to do with using a "full-price" newsreader)
>
> > This is a thinly veiled reference to me,
>
> I'd have called it a "weak joke", rather than a "thinly-veiled
> reference", but never mind. FWIW, you might notice I'm posting this
> from Google, so you and I share the same low esteem rating, it seems.
I used to use a real news reader at Cornell, but now I have
Roadrunner, and they conveniently discontinued newsgroup support right
before I joined. Hrummmmph!! Not my fault.
>
> > > only those who
> > > understand it (as distinct from those who don't, of course).
>
> > I agree with TO here, except that instead of "subscriber,"
> > I prefer to use a word like "adherent" ...
>
> > The word "adherent" often describes those who have a
> > particular religion. ... Adherents of ZFC believe that its axioms
> > are true, just as adherents of a religion believe that its
> > unprovable claims are true.
>
> Hmm. But non-adherents of religion (and possible adherents, too, with
> respect to all but the one they adhere to*) can generally say just
> where they disagree with the axioms of the religion. "Non-adherents"
> to ZFC typically flunk out when challenged to exactly what they
> disagree with.
Actually, Brian, that statement is pure bunk, no offense intended.
Most adherents of a given religion simply believe that theirs is the
right belief because their parents or some authority figure told them
that some prophet or book says so, and rarely do they really
understand the tenets of their own religion, much less those in
competition. Can you say, for instance, why you disagree with Islam or
Tibetan Buddhism? If not, do you then accept what they believe?
In the same vein, most objectors to transfinitology have a hard time
identifying where exactly they disagree with the construction of the
theory, but have at least a few examples of conclusions drawn from it
which are completely incorrect when approached through other avenues.
This is partly because mathematicians misrepresent the theory,
claiming that every conclusion they draw "follows logicallly from the
axioms." This claim is simply not true, as cardinality is not
mentioned in the axioms, much less anything about omega or the alephs,
except for a declaration that something homomorphic to the naturals
exists, which is ultimately not a set, but a sequence. By demanding to
know with which axioms a non-transfinitologist disagrees,
mathematicians obscure the question. After some years of this
discussion have arisen some obvious culprits: the von Neumann ordinals
as some complete set, and the simplistic assumption of equivalence of
"size" based on bijection alone. If bijection determines equal
cardinality that's fine, but to equate cardinality with set size
simply does not work for infinite sets to the satisfaction of most
people's intuitions. To claim that they are being illogical by
objecting to the theory without identifying an axiom at fault is
disingenuous, since the axioms are clearly not the problem.
>
> * I'm reminded of the review of Martin Gardner's seminal anti-woo
> bookl published around 1950. The second edition is generally preferred
> because it includes samples of the hate mail he received after the
> first edition, from people who generally loved all chapters but
> exactly one of the book. **
>
> ** Sorry, I chickened out, because I just couldn't choose between "all
> but exactly one chapter" and "all but exactly one chapters" ...
The physics isn't wrong. The bomb is. The axioms aren't wrong (though
some could be better stated). The von Neumann limit ordinals are
simply bunk. Everything Jesus said was true and wise, but the Trinity
and the Immaculate Conception are Christianity's von Neumann limit
ordinals, thrown in later for Constantine's political goal of state
religion incorporating pagan beliefs and christian.
>
> > I have nothing against ZFC or religion per se -- just that
> > adhering to either is somewhat similar.
>
> Ah, now I'm stuck.
What's brown and sticky?
>
> Brian Chandler
Peace,
Tony
Ludovicus
>.. in any such list of rationals, it is
> always true that we can concoct another rational which is not on the
> list. Irrational numbers need not even be considered. So, are the
> rationals not to be considered uncountable, by Cantor's own logic?
> Tony
That is false.
There is not rational that is not in the list.
Given any rational n/d, its place N in Cantor's list is:
Be S =n + d
If S is even, N = (S-1)(S-2) + d
If S is odd , N = (S-1)(S-2) + n
Ludovicus
kunzmilan
there are more rational numbers than there are the natural numbers.
The rational numbers are defined as the natural number i divided by
the natural number j. Thus, rational numbers can be greater than 1,
too.
When we form the matrix R of rational numbers, all rationals lesser
than 1 are bellow its diagonal, all rationals greater than 1 are ower
its diagonal. In the first row are all natural numbers defined as n/1.
All rationals greater than 1, as 3/2, 4/3, are in the triangle over
the diagonal.
All rationals lesser than 1, as 2/3, 3/4, are in the triangle under
the diagonal.
Since in both triangles are more different elements than in the first
row of the matrix R, there are more rational numbers than in the set
of the natural numbers and the rational numbers are uncoutable.
kunzmilan
Rick Decker
> On 2 čvn, 20:56, Tony Orlow<t...@lightlink.com> wrote:
>> Hi All -
>>
>> Tony
> The uncountability of the reals is simply based on the fact, that
> there are more rational numbers than there are the natural numbers.
> The rational numbers are defined as the natural number i divided by
> the natural number j. Thus, rational numbers can be greater than 1,
> too.
> When we form the matrix R of rational numbers, all rationals lesser
> than 1 are bellow its diagonal, all rationals greater than 1 are ower
> its diagonal. In the first row are all natural numbers defined as n/1.
> All rationals greater than 1, as 3/2, 4/3, are in the triangle over
> the diagonal.
> All rationals lesser than 1, as 2/3, 3/4, are in the triangle under
> the diagonal.
> Since in both triangles are more different elements than in the first
> row of the matrix R, there are more rational numbers than in the set
> of the natural numbers and the rational numbers are uncoutable.
No. While there are "more" rationals than natural numbers, in the
sense that the naturals can be viewed as a proper subset of the
rationals, that does not prove that the two sets are equinumerous.
In fact, there is a rather trivial bijection between the two sets.
Regards,
Rick
Tony Orlow
> On 2 čvn, 20:56, Tony Orlow <t...@lightlink.com> wrote:
>
>
>
> > Hi All -
>
> > Looking at some of Herc's recent threads about Cantor's diagonal proof
> > of the uncountability of the reals, a question occurs to me which I
> > actually don't know how a subscriber to transfinite set theory would
> > answer. I'm curious.
>
> > In Cantor's list of reals, for every digit added, the list doubles in
> > length. In order to follow his logic, we need not consider any real
> > numbers which are not rational. In any such list of rationals, it is
> > always true that we can concoct another rational which is not on the
> > list. Irrational numbers need not even be considered. So, are the
> > rationals not to be considered uncountable, by Cantor's own logic?
>
> > BTW, I understand that the rationals are considered countable because
> > of a rather artificial bijection with N, but if Cantor's argument
> > really has anything to do with the reals, as opposed to the powerset
> > (which is what it's really all about), then why is it entirely
> > unnecessary to even consider irrational reals in his construction?
>
> > Thanks and Smiles,
>
> > Tony
>
> The uncountability of the reals is simply based on the fact, that
> there are more rational numbers than there are the natural numbers.
The rationals have the same cardinality as the naturals, though there
are surely more rationals per unit interval of the real line than
there are naturals.
> The rational numbers are defined as the natural number i divided by
> the natural number j. Thus, rational numbers can be greater than 1,
> too.
One might think there were something like aleph_0^2 rationals, but
that's not standard theory.
> When we form the matrix R of rational numbers, all rationals lesser
> than 1 are bellow its diagonal, all rationals greater than 1 are ower
> its diagonal. In the first row are all natural numbers defined as n/1.
> All rationals greater than 1, as 3/2, 4/3, are in the triangle over
> the diagonal.
> All rationals lesser than 1, as 2/3, 3/4, are in the triangle under
> the diagonal.
> Since in both triangles are more different elements than in the first
> row of the matrix R, there are more rational numbers than in the set
> of the natural numbers and the rational numbers are uncoutable.
That's not a bad alternative argument. Of course, they still have the
same cardinality, but the axioms never say anything about cardinality
to begin with. Nice comment. You get five stars!
> kunzmilan- Hide quoted text -
>
> - Show quoted text -
Regards,
Tony
Virgil
<1bb4e64e-9dd5-45a2...@c7g2000vbc.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> In the same vein, most objectors to transfinitology have a hard time
> identifying where exactly they disagree with the construction of the
> theory, but have at least a few examples of conclusions drawn from it
> which are completely incorrect when approached through other avenues.
> This is partly because mathematicians misrepresent the theory,
> claiming that every conclusion they draw "follows logicallly from the
> axioms." This claim is simply not true, as cardinality is not
> mentioned in the axioms, much less anything about omega or the alephs,
> except for a declaration that something homomorphic to the naturals
> exists, which is ultimately not a set, but a sequence.
The issue is whether cardinality CAN BE defined within an axiom system,
not whether it is explicitly mentioned in some axiom(s) of that system.
And it can be.
How does TO define 'sequence' without reference to something like the
SET of naturals as indices?
K_h
On Jun 3, 4:07 am, Brian Chandler
<imaginator...@despammed.com> wrote:
> Transfer Principle wrote:
> > On Jun 2, 12:27 pm, Brian Chandler
> > <imaginator...@despammed.com>
> > wrote:
> discussion have arisen some obvious culprits: the von
> Neumann ordinals as some complete set, and the
> simplistic assumption of equivalence of "size" based on
> bijection alone. If bijection determines equal cardinality
> that's fine, but to equate cardinality with set size
> simply
> does not work for infinite sets to the satisfaction of
> most
Tony, first, the von-Neuman ordinals do not form a set,
rather they form a proper class. Second, cardinal size is
only one measure of size. There are other measures, for
example ordinal size. The ordinal w2 has an ordinal size
greater than w but they both have the same cardinal size.
The same is true of w+1 and w: w<w+1 ordinally but
|w|=|w+1| cardinally.
To be clear on your issue about Cantor's diagonal argument,
there is nothing wrong with Cantor's argument. If there is
a list containing all rationals then its `anti-diagonal'
numbers will not be rational. The proof is easy: use one of
the common `zig-zag' bijections between the naturals and the
rationals so that every ratio p/q appears in the list once
and only once with p and q containing no common factors
except 1. (Note negative rationals are in the list and this
is not a problem: there will be two anti-diagonals because
of opposite signs). Create an `anti-diagonal' number for
the list and then note that this number is nowhere in the
list and so it must be different from each p/q. Therefore,
it cannot be written as the ratio of two integers p/q.
_
porky_...@my-deja.com
So we had Herc claiming that reals are countable. now we have
kunzmilan claiming the rationals are uncountable. The plot thickens.
Jesse F. Hughes
> The uncountability of the reals is simply based on the fact, that
> there are more rational numbers than there are the natural numbers.
This is, of course, utterly butt-wrong.
There are not more rational numbers than naturals -- that is, |N|=|Q|.
Even Tony knows that.
--
Jesse F. Hughes
One is not superior merely because one sees the world as odious.
-- Chateaubriand (1768-1848)
Tim Little
> This is partly because mathematicians misrepresent the theory,
> claiming that every conclusion they draw "follows logicallly from
> the axioms."
The conclusions *do* follow logically from the axioms.
> This claim is simply not true, as cardinality is not mentioned in
> the axioms
Calling a given symbol "cardinality" is just a convenient notational
shorthand for a set of logical propositions expressible in terms of
the foundation symbols of the theory. That's what is meant by a
mathematical definition.
> much less anything about omega or the alephs
Actually the symbol omega is frequently expressed directly in the
Axiom of Infinity of ZF: "There exists omega such that ...". In
expressions of the axioms that do not directly use the symbol omega,
it is a notational shorthand for such a set. Likewise the aleph
symbols are notational shorthands for properties expressed in the
language of the theory.
> except for a declaration that something homomorphic to the naturals
> exists, which is ultimately not a set, but a sequence.
In the theory of ZF, it is provable that there exist sets that model
the naturals. One example is the set of von Neumann ordinals.
> By demanding to know with which axioms a non-transfinitologist
> disagrees, mathematicians obscure the question. After some years of
> this discussion have arisen some obvious culprits: the von Neumann
> ordinals as some complete set
The existence of a set satisfying the properties defining the von
Neumann ordinals follows as a purely logical deduction from the
axioms. You appear to be objecting to nothing more than giving that
set a name.
> and the simplistic assumption of equivalence of "size" based on
> bijection alone.
If you don't like the notion of size being determined by cardinality,
feel free to come up with a better formalization that doesn't depend
on mere labelling of elements of the set. Zuhair did try for a while
in this newsgroup, but didn't really succeed.
- Tim
Tim Little
> that's not standard theory.
That's perfectly standard theory. It's just that aleph_0^2 = aleph_0.
- Tim
Tim Little
>> except for a declaration that something homomorphic to the naturals
>> exists, which is ultimately not a set, but a sequence.
>
> In the theory of ZF, it is provable that there exist sets that model
> the naturals. One example is the set of von Neumann ordinals.
My apologies, "the set of *finite* von Neumann ordinals".
> The existence of a set satisfying the properties defining the von
> Neumann ordinals follows as a purely logical deduction from the
> axioms.
Once again, *finite* ordinals. There is no set of all von Neumann
ordinals.
- Tim
Transfer Principle
> kunzmilan claiming the rationals are uncountable. The plot thickens.
At one point, I'd wanted to introduce phrases such as
"substandard theorists" and "superstandard theorists." The
former would refer to posters like Herc/Cooper who believe
in _fewer_ types of infinity than the standard (sub- means
below -- Herc believes that the infinity of the reals and
the infinity of the naturals are the same.) The latter
would refer to posters such as TO and kunzmilan who
believe in _more_ typpes of infinity than the standard
(super- means above -- these two posters believe that the
infinity of rationals and the infinity of the reals are
in fact distinct).
This would have emphasized that there are posters who
oppose ZFC from two different sides -- those who believe
that ZFC has too many set sizes and those who believe that
ZFC doesn't have enough set sizes.
But I've said that I would avoid calling different types
of posters as so-and-so "theorists," and so I will not be
using those labels at all. I only point out what I once
had in mind.
Michael Stemper
>Looking at some of Herc's recent threads about Cantor's diagonal proof
>of the uncountability of the reals, a question occurs to me which I
>actually don't know how a subscriber to transfinite set theory would
>answer. I'm curious.
>
>In Cantor's list of reals, for every digit added, the list doubles in
>length. In order to follow his logic, we need not consider any real
>numbers which are not rational. In any such list of rationals, it is
>always true that we can concoct another rational which is not on the
As others have pointed out, the number that you construct is not
known to be rational.
>BTW, I understand that the rationals are considered countable because
>of a rather artificial bijection with N,
Better to say "the rationals *are* countable because they biject
with N". As a matter of fact, there are many possible bijections
between Q and N. The use of the word "artificial" is kind of silly,
because most of what we talk about in math is "artificial". I'm not
sure how one could define the difference between an "artificial"
function and a "natural" one.
But, it's not only possible to establish bijections between Q and N,
it's even possible to define injections from Q to N that are not onto.
In other words, you can map every rational to a unique natural, while
still having an infinite set of naturals left over.
Express any rational, q, in the form a*(m/n), where a = +/- 1 and m,n
are naturals with gcd(m,n) = 1. Define f(q) to be:
(2^m)*(3^n) if a == +1
(5^m)*(7^n) if a == -1
Do you see that this maps every rational to a different natural? So,
it's an injection from Q to N. Do you also see that no rational gets
mapped to 11 or 17 or 42?
--
Michael F. Stemper
#include <Standard_Disclaimer>
Indians scattered on dawn's highway bleeding;
Ghosts crowd the young child's fragile eggshell mind.
Tony Orlow
> In article
> <1bb4e64e-9dd5-45a2-8c42-9ffa430f8...@c7g2000vbc.googlegroups.com>,
> Tony Orlow <t...@lightlink.com> wrote:
>
> > In the same vein, most objectors to transfinitology have a hard time
> > identifying where exactly they disagree with the construction of the
> > theory, but have at least a few examples of conclusions drawn from it
> > which are completely incorrect when approached through other avenues.
> > This is partly because mathematicians misrepresent the theory,
> > claiming that every conclusion they draw "follows logicallly from the
> > axioms." This claim is simply not true, as cardinality is not
> > mentioned in the axioms, much less anything about omega or the alephs,
> > except for a declaration that something homomorphic to the naturals
> > exists, which is ultimately not a set, but a sequence.
>
> The issue is whether cardinality CAN BE defined within an axiom system,
> not whether it is explicitly mentioned in some axiom(s) of that system.
>
> And it can be.
And yet, it does not logically follow from the axioms. It is simply
consistent in the sense of not contradicting them. Therefore, other
measures of sets can also exist, consistent with the axioms, but
inconsistent with cardinality.
>
> How does TO define 'sequence' without reference to something like the
> SET of naturals as indices?
It is a set wherein every element is either before or after (not
immediately) every other element. That's one possible definition,
though you undoubtedly have some objection.
Peace, Virgil.
Tony
Tony Orlow
Hi Jesse -
Just because I concede that both sets are countably infinite and
therefore of the same cardinality, nevertheless the sparse proper
subset of the rationals called the naturals should not be equated in
size with its dense proper superset.
Tony
Tony Orlow
Hi Transfer -
As always your comments are appreciated. Just so you know, there
already exist labels describing the two types of objectors to
transfinitology: Anti-Cantorians and Post-Cantorians. I think
"substandard" probably gives the wrong connotation, though it doesn't
offend me, since I'm a "superstandard" Post-Cantorian. Heh :)
Have a nice day.
Tony
Tonico
Yes, Virgil will surely object something...I don't know, he may even
ask you to be logical or to base your arguments...that bastard!
So a sequence "is a set wherein every element is either before [sic]
or after (not immediately)[sic] every [sic] other element" ...leaving
aside that wonderful and startling "every" in that ""definition"" ,
would you mind to tell us what do you mean by an element of a general
set "being before or after" some other one?! I mean, this is the kind
of things Virgil, mathematicians and people used to think more or less
logically can object to...
For example, what'd mean to be "before" or "after" an element in, say
the set of all the red apples from California?
Tonio
Tony Orlow
Duh-uh....is there a sequence of apples? Sets are not necessarily
sequences. I might as well bring up monkeys and typewriters in some
dicussion of Shakespeare, and it would be no more of a nonsequitur
than your comment.
>
> Tonio- Hide quoted text -
>
> - Show quoted text -
Look, "Tonico", you can cop whatever attitude you want with me, but in
a conversation in this very newsgroup not long ago it became obvious
that seasoned mathematicians don't even agree on what a sequence is,
some considering it isomorphic to the naturals, and others to the
entire class of von Neumann ordinals. Besides, this is unrelated to
the topic, and merely finger exercise for your pallid digits and
tangled neurons.
Regards,
TOny
Pubkeybreaker
You have been given an explicit injection from Q to N.
If you think it is wrong, explain why.
Otherwise, explain this idiocy in which you believe that there are
more rationals than
integers.
Aatu Koskensilta
> Otherwise, explain this idiocy in which you believe that there are
> more rationals than integers.
It makes perfect sense to say there are more rationals than integers,
for example to mean that there are many reals that are rational but not
integral. Similarly, in logic we might say -- and indeed do say -- that
PA + Con(PA) proves more theorems than PA, in the sense that the set of
theorems of PA + Con(PA) is a proper superset of the set of theorems of
PA. What is difficult is to come up with any notion of size that applies
to sets in general according to which there are more rationals than
integers. This is an interesting problem, and we find in the literature
many not at all nonsensical attempts at an answer that respects the
various intuitions we have, or some people have, or have had, about
size.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Pubkeybreaker
> Pubkeybreaker <pubkeybrea...@aol.com> writes:
> > Otherwise, explain this idiocy in which you believe that there are
> > more rationals than integers.
>
> It makes perfect sense to say there are more rationals than integers,
Another moron weighs in.
So we have two sets, Q and N. Q is bigger than N. (according to
you) But then
we also have a map from Q to N that assigns **each** element in Q to
a unique
element in N , and yet there are still elements of N that are not
assigned!
Hence, N is also bigger than Q.
It makes ZERO sense to say that there are more rationals than
integers.
All countably infinite sets are the same size.
Aatu Koskensilta
> Another moron weighs in.
The supply of morons in this world is inexhaustible.
> It makes ZERO sense to say that there are more rationals than
> integers. All countably infinite sets are the same size.
So naturally you object also to the statement
PA + Con(PA) proves more true statements about naturals than PA.
which we find in the logical literature? There is of course no
suggestion the set of theorems of PA + Con(PA) is uncountable.
MoeBlee
> mathematicians misrepresent the theory,
> claiming that every conclusion they draw "follows logicallly from the
> axioms." This claim is simply not true, as cardinality is not
> mentioned in the axioms, much less anything about omega or the alephs,
> except for a declaration that something homomorphic to the naturals
> exists, which is ultimately not a set, but a sequence.
No, all of those terms are given by DEFINITIONS, which may take the
form of definitional AXIOMS, which provide a conservative extension of
the theory in its primnitive form. This has been explained to you at
least a hundred times already, but you choose just to ignore.
Moreover, we don't just show that there exists a set homomorphic to
the naturals (and their customary ordering) but rather that we define
'w' (omega) to be the least set that has 0 as a member and is closed
under successsorship. Moreover, in set theory, sequences are sets.
> By demanding to
> know with which axioms a non-transfinitologist disagrees,
> mathematicians obscure the question.
WHAT question?
> After some years of this
> discussion have arisen some obvious culprits: the von Neumann ordinals
> as some complete set,
You're confused. Ordinary set theory does not assert that there is a
set of all the von Neumann ordinals. However, in ordinary set theory
it is proven that there exists a set of all the FINITE von Neumann
ordinals. But fine if you reject that there exists a set of all finite
von Neumann ordinals. Then, if you accept that there is an empty set
and the pairing axiom, then you only (if I'm not missing a detail
here) need reject the axiom of infinity to decline that there is a set
that has all the von Neumann ordinals.
> and the simplistic assumption of equivalence of
> "size" based on bijection alone.
I've presented to you at least a hundred times the answer (in
variation) that instead of the word 'size' we could use the word
'zize'. Then your protest disappears. You've never dealt with that
point.
> If bijection determines equal
> cardinality that's fine, but to equate cardinality with set size
> simply does not work for infinite sets to the satisfaction of most
> people's intuitions.
Most people haven't studied the subject. And, again, everywhere in
discussions about set theory you may substitute 'size' with 'zize'. No
substantive matter to the FORMAL theory.
> To claim that they are being illogical by
> objecting to the theory without identifying an axiom at fault is
> disingenuous, since the axioms are clearly not the problem.
The problem is that you demand that the word 'size' be used in a way
that does not conflict with your ordinary notions about size. But set
theory does not represent that such words are used in a way that
conforms in every sense to such everyday notions.
> The von Neumann limit ordinals are
> simply bunk.
Fine, then probably you want to throw out the axiom of infinity.
> Everything Jesus said was true and wise, but the Trinity
> and the Immaculate Conception are Christianity's von Neumann limit
> ordinals, thrown in later for Constantine's political goal of state
> religion incorporating pagan beliefs and christian.
So who is the Jesus of set theory before statist limit ordinals
corrupted the whole thing?
MoeBlee
MoeBlee
> other
> measures of sets can also exist, consistent with the axioms, but
> inconsistent with cardinality.
It is not ordinarily disputed that definitions may be given
stipulatively. So what?
> > How does TO define 'sequence' without reference to something like the
> > SET of naturals as indices?
>
> It is a set wherein every element is either before or after (not
> immediately) every other element. That's one possible definition,
Yes, the objection is that it's more general than the ordinary
definition, thus loses the special sense obtained from the ordinary
definition. But again, if you don't like the ordinary definition of
'sequence', then just imagine everywhere 'sequence' appears in set
theory discussion that the word is 'zequence'.
MoeBlee
MoeBlee
> seasoned mathematicians don't even agree on what a sequence is,
> some considering it isomorphic to the naturals, and others to the
> entire class of von Neumann ordinals.
No, you're confused, in Z set theory we don't adopt any such
difference. We don't speak of isomorphism with the class of ordinal
numbers. In Z set theory there is no such thing. In such context, it
is only in an informal sense that we may speak of a sequence on the
entire class of ordinal numbers.
Rather, we take a sequence to be a function on AN ordinal number (not
the entire class).
In any given context, we simply make clear what sense we mean:
A sequence as a function whose domain is a natural number.
A sequence as a function whose domain is either a natural number or w.
A sequence as a function whose domain is an ordinal number.
A sequence as a function whose domain is some well ordered set.
And, in an INformal sense, the ordinal sequence or a sequence (such
the levels in the universe V) on the ordinals.
That different texts use different defintions does not signal that
there is confusion over the matter, but rather only that in a given
context we need to make clear which definition is being used.
MoeBlee
Tony Orlow
>
> - Show quoted text -
Oy Vey!
I have stated clearly that I understand there is a bijection between
the countable infinities of N and Q. Are you daft? Why correct
something which isn't wrong? Sure, they have the same cardinality. So,
what?
You cannot disagree with the fact that the rationals are a dense set
whereas the naturals are sparse, with a countably infinite number of
rationals lying quantitatively between any two given naturals, can
you? Do you not see that in some sense there appear to be more
rationals than naturals? Are you incapable of considering any other
system of set measurement besides what was drilled into you in class?
It's really sad when those who claim to be so smart have sacrificed
their creativity and imagination to reach such an "esteemed position".
Perhaps you should partake in some lamb's bread and then reconsider
the issue.
Peace,
Tony
Aatu Koskensilta
> It's really sad when those who claim to be so smart have sacrificed
> their creativity and imagination to reach such an "esteemed position".
> Perhaps you should partake in some lamb's bread and then reconsider
> the issue.
I think you should reconsider your choice in rhetoric gambits.
Tony Orlow
> Pubkeybreaker <pubkeybrea...@aol.com> writes:
> > Another moron weighs in.
>
> The supply of morons in this world is inexhaustible.
Hi Aatu -
Thankfully you are not among their ranks. Your input is quite
reasonable, and it would be interesting to see some other approaches
to quantifying the relationship between the two sets. As I said in one
post (which received a response much less sophisticated and more
robotic than any of yours have ever been) that one might imagine there
were aleph_0^2 rationals, but if one wants to account for and
eliminate any duplicate quantities expressed in different terms within
that matrix, the problem becomes a little more complicated. I started
thing about that a bit the other day again, and think there might be a
pretty reasonable solution, but haven't quite discovered it yet. In
any case, it's always nice to hear your input.
>
> > It makes ZERO sense to say that there are more rationals than
> > integers. All countably infinite sets are the same size.
>
> So naturally you object also to the statement
>
> PA + Con(PA) proves more true statements about naturals than PA.
>
> which we find in the logical literature? There is of course no
> suggestion the set of theorems of PA + Con(PA) is uncountable.
This also sounds very fascinating, and I'd like to know more. Indeed,
you are one of the very least moronic contributors here. :)
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Thanks, Aatu.
Peace,
Tony
David R Tribble
> In Cantor's list of reals, for every digit added, the list doubles in
> length.
I assume you mean a list of finite-length binary fractions
representing reals in [0,1).
> In order to follow his logic, we need not consider any real
> numbers which are not rational. In any such list of rationals, it is
> always true that we can concoct another rational which is not on the
> list.
Oh come on, Tony. You know this isn't right.
A list of finite-length rationals is either a finite list itself,
being a list of reals of n bits or less and therefore containing
no more than 2^n reals in total; or it's an infinite list of
finte-length reals (i.e., all of them are binary fraction that
end with a trailing list of repeating 000... digits).
In the first case, it's not Cantor's list of reals, so it's moot.
In the second case, the real that's not in the list is a
sequence of binary digits that differs by at least one digit
from every other (rational) real in the list. This diagonal
number, by the way, is most likely not rational (unless it
just so happens to end in a repeating binary digit sequence).
> Irrational numbers need not even be considered. So, are the
> rationals not to be considered uncountable, by Cantor's own logic?
Well, first you have to have the correct description of Cantor's
diagonal argument. You don't.
> BTW, I understand that the rationals are considered countable because
> of a rather artificial bijection with N,
Or any other bijection with N. There are an infinite number of
them to choose from. I personally like the one I independently
re-discovered a few years ago (as a response to one of your posts,
in fact, if I recall correctly), which is at:
http://david.tribble.com/text/sequence1.html
> but if Cantor's argument
> really has anything to do with the reals, as opposed to the powerset
> (which is what it's really all about), then why is it entirely
> unnecessary to even consider irrational reals in his construction?
Again, you have to have the right description of the list.
Once you do, you can see that your conclusion is not warranted.
-drt
MoeBlee
> On Jun 3, 7:34 am, Tony Orlow <t...@lightlink.com> wrote:
> > and the simplistic assumption of equivalence of
> > "size" based on bijection alone.
>
> I've presented to you at least a hundred times the answer (in
> variation) that instead of the word 'size' we could use the word
> 'zize'. Then your protest disappears. You've never dealt with that
> point.
P.S. I don't claim that such a "formalistic" rebuttal to you settles
the question. That is to say, I don't begrudge that there may be a
legitimate philosophic dispute over whether set theoretic cardinality
does or does not capture our intuitive notion of size in regards
infinite sets. (That set theoretic cardinality does capture our notion
of size in finite cases I take to be ordinarily uncontroversial).
However, we must at least address my "formalistic" rebuttal at least
first, so to recognize that in the sense of set theory as a formal
theory, whether we call call cardinality 'size' or 'zize'.
That said, however, still I don't take the import of set theory to be
to settle such questions except as they pertain to mathematical
inquiry, in which sense the notion of cardinality does seem to be
viable. This is so especially since at least I don't know of a
challenging formalization of the notion of size in the infinite case
that is any more intuitive than set theoretic cardinality, as at least
set theoretic cardinality is motivated by the intuitive notion of one-
to-one correspondence but also taken in the infinite case. More
specifically, Orlow himself has not presented any theory that
satisfies EITHER criteria: (1) formally coherent, (2) more intuitive
than set theoretic cardinality, as Orlow's notions are even LESS
intuitive than set theoretic cardinality. That he personally finds his
own ruminations more intuitive is not sufficient philosophical
motivation for patiently waiting for him to someday put his floating
ideas into the form of a formal theory.
MoeBlee
MoeBlee
> You cannot disagree with the fact that the rationals are a dense set
> whereas the naturals are sparse,
WRONG, WRONG, WRONG. I've explained this to you several times, and you
persist to ignore.
A set is dense with respect to some ORDERING. There IS a dense
ordering of the set of natural numbers, and there is an ordering of
the set of rationals that is NOT dense.
What you can say is that the set of natural numbers is not dense with
regard to the usual ordering on the set of natural numbers and that
the set of rational numbers is dense with regard to the usual ordering
on the set of rational numbers.
> with a countably infinite number of
> rationals lying quantitatively between any two given naturals, can
> you? Do you not see that in some sense there appear to be more
> rationals than naturals?
Yes, but some investigation shows that that is a superficial view,
since we see also that we can order the naturals densely and order the
rationals discreetly.
MoeBlee
MoeBlee
> On Jun 4, 1:42 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> > which we find in the logical literature? There is of course no
> > suggestion the set of theorems of PA + Con(PA) is uncountable.
>
> This also sounds very fascinating, and I'd like to know more.
It's not fascinating. Aatu is just reminding of the dry fact that in
conversational mathematics, we do use the word 'more' in a sense not
confined to cardinality. A more prosaic example: If I have a countably
infinite set of symbols, but "add on" another symbol, then I say "now
I have more symbols" though I don't mean that in the literal sense
that the "new" set has greater cardinality.
MoeBlee
David R Tribble
>> It makes perfect sense to say there are more rationals than integers, [...]
>
Pubkeybreaker <pubkey...@aol.com> writes:
>> Another moron weighs in.
>
Aatu Koskensilta wrote:
> The supply of morons in this world is inexhaustible.
> So naturally you object also to the statement
> PA + Con(PA) proves more true statements about naturals than PA.
I don't follow this.
If the number of true (constructible) statements proved by PA and
the number of true statements proved by PA + Con(PA) are both
countable, does it not follow that those two sets have the same
cardinality?
(Or can we not place all provable statements into sets?)
David R Tribble
>> The uncountability of the reals is simply based on the fact, that
>> there are more rational numbers than there are the natural numbers.
>
Tony Orlow wrote:
> The rationals have the same cardinality as the naturals, though there
> are surely more rationals per unit interval of the real line than
> there are naturals.
Yes, but it's a curious fact that there are the same number of
rationals as naturals on the _entire_ real number line.
kunzmilan wrote:
>> The rational numbers are defined as the natural number i divided by
>> the natural number j. Thus, rational numbers can be greater than 1,
>> too.
>
Tony Orlow wrote:
> One might think there were something like aleph_0^2 rationals, but
> that's not standard theory.
Actually, there are Aleph_0^2 rationals. And Aleph_0^2 = Aleph_0.
-drt
MoeBlee
> Tony Orlow wrote:
> > One might think there were something like aleph_0^2 rationals, but
> > that's not standard theory.
>
> Actually, there are Aleph_0^2 rationals. And Aleph_0^2 = Aleph_0.
Orlow can't be bothered to learn such basics.
MoeBlee
Tony Orlow
> On Jun 3, 7:34 am, Tony Orlow <t...@lightlink.com> wrote:
>
> > mathematicians misrepresent the theory,
> > claiming that every conclusion they draw "follows logicallly from the
> > axioms." This claim is simply not true, as cardinality is not
> > mentioned in the axioms, much less anything about omega or the alephs,
> > except for a declaration that something homomorphic to the naturals
> > exists, which is ultimately not a set, but a sequence.
>
> No, all of those terms are given by DEFINITIONS, which may take the
> form of definitional AXIOMS, which provide a conservative extension of
> the theory in its primnitive form. This has been explained to you at
> least a hundred times already, but you choose just to ignore.
Can you please provide a complete list of your definitional axioms for
my inspection? They are not part of ZFC, are they?
>
> Moreover, we don't just show that there exists a set homomorphic to
> the naturals (and their customary ordering) but rather that we define
> 'w' (omega) to be the least set that has 0 as a member and is closed
> under successsorship. Moreover, in set theory, sequences are sets.
Sequences are sets with order. Sets in general have no order. If set
theory would like to hide the recursive nature of infinite sets in
order to draw precarious conclusions about sequences as if they "just
exist", then it behooves the mathematical community to encourage more
investigation into the distinction and draw conclusions from that
information. Do you disagree?
>
> > By demanding to
> > know with which axioms a non-transfinitologist disagrees,
> > mathematicians obscure the question.
>
> WHAT question?
MoeBlee eat some meat or take vitamins. The question is, "Why do you
object to set theory, O Crackpot?" Isn't that what you've been asking
all along?
>
> > After some years of this
> > discussion have arisen some obvious culprits: the von Neumann ordinals
> > as some complete set,
>
> You're confused. Ordinary set theory does not assert that there is a
> set of all the von Neumann ordinals. However, in ordinary set theory
> it is proven that there exists a set of all the FINITE von Neumann
> ordinals. But fine if you reject that there exists a set of all finite
> von Neumann ordinals. Then, if you accept that there is an empty set
> and the pairing axiom, then you only (if I'm not missing a detail
> here) need reject the axiom of infinity to decline that there is a set
> that has all the von Neumann ordinals.
Here you have a minor point. I meant the finite von Neumann ordinals
as a complete set, and the non-finite limit ordinals as anything but
phantoms. You may call N a set, but it is really a sequence that is
never complete at any point. Did you miss the word, "complete"? That
was a kind of important word. As a mathematician (or whatever) you
shouldn't be leaving out symbols, much less whole words. I do not
discount the existence of N, but the Axiom of infinite is not about a
set, really, but a sequence without end. You do get the distinction,
no? Perhaps you're confused.
>
> > and the simplistic assumption of equivalence of
> > "size" based on bijection alone.
>
> I've presented to you at least a hundred times the answer (in
> variation) that instead of the word 'size' we could use the word
> 'zize'. Then your protest disappears. You've never dealt with that
> point.
I used "Bigulosity", remember? Surely you do, you rascal.
>
> > If bijection determines equal
> > cardinality that's fine, but to equate cardinality with set size
> > simply does not work for infinite sets to the satisfaction of most
> > people's intuitions.
>
> Most people haven't studied the subject. And, again, everywhere in
> discussions about set theory you may substitute 'size' with 'zize'. No
> substantive matter to the FORMAL theory.
"The" formal theory. I like that. You should really become a Hare
Krishna devotee.
>
> > To claim that they are being illogical by
> > objecting to the theory without identifying an axiom at fault is
> > disingenuous, since the axioms are clearly not the problem.
>
> The problem is that you demand that the word 'size' be used in a way
> that does not conflict with your ordinary notions about size. But set
> theory does not represent that such words are used in a way that
> conforms in every sense to such everyday notions.
Well, perhaps it satisfies your "everyday" experiences with infinity.
Or, perhaps, you don't have any. If you don't call it "size" I won't.
You say "cardinality", and I sqay "bigulosity".
>
> > The von Neumann limit ordinals are
> > simply bunk.
>
> Fine, then probably you want to throw out the axiom of infinity.
It says nothing about the non-finite limit ordinals. See above.
>
> > Everything Jesus said was true and wise, but the Trinity
> > and the Immaculate Conception are Christianity's von Neumann limit
> > ordinals, thrown in later for Constantine's political goal of state
> > religion incorporating pagan beliefs and christian.
>
> So who is the Jesus of set theory before statist limit ordinals
> corrupted the whole thing?
Good question. Perhaps Cantor and Dedekind in modern times, though
Galileo certainly explored bijection long before, and decided there
was no real answer, and Aristotle first made the distinction between
potential (countable) and actual (uncountable) infinities. This is
some of the ancestry of transfinitology. Good enough? Why do you ask?
>
> MoeBlee
Keep on truckin'
ToeKnee
Aatu Koskensilta
> It's not fascinating. Aatu is just reminding of the dry fact that in
> conversational mathematics, we do use the word 'more' in a sense not
> confined to cardinality.
Well, I was also pointing out there are in the logical literature
numerous not at all nonsensical attempts at defining a general notion of
size on which e.g. the rationals are of a larger size than the naturals.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
Tony Orlow
> On Jun 4, 11:16 am, Tony Orlow <t...@lightlink.com> wrote:
>
> > other
> > measures of sets can also exist, consistent with the axioms, but
> > inconsistent with cardinality.
>
> It is not ordinarily disputed that definitions may be given
> stipulatively. So what?
"stipulative - no dictionary results" - dictionary.com
>
> > > How does TO define 'sequence' without reference to something like the
> > > SET of naturals as indices?
>
> > It is a set wherein every element is either before or after (not
> > immediately) every other element. That's one possible definition,
>
> Yes, the objection is that it's more general than the ordinary
> definition, thus loses the special sense obtained from the ordinary
> definition. But again, if you don't like the ordinary definition of
> 'sequence', then just imagine everywhere 'sequence' appears in set
> theory discussion that the word is 'zequence'.
>
> MoeBlee
Do you not remember the thread not long ago where you methamaticians
started arguing amonst yourselves? What is the "ordinary" definition,
again? Never mind. It's a sidebar, as always with you.
Take a deep breath.
Love,
Tony
Tony Orlow
> Tony Orlow <t...@lightlink.com> writes:
> > It's really sad when those who claim to be so smart have sacrificed
> > their creativity and imagination to reach such an "esteemed position".
> > Perhaps you should partake in some lamb's bread and then reconsider
> > the issue.
>
> I think you should reconsider your choice in rhetoric gambits.
I'm sorry Aatu. Do you mean I should choose more carefully which posts
to respond to? Or, perhaps, you mean I should watch my tone. In either
case, and you are not among them, there are certain folk who seek to
correct and dominate rather than discuss and think, and sometimes I
feel they must have proper feedback, for their good and that of the
world. Perhaps that makes me one of those that seeks to correct and
dominate?
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Please advise.
Love,
Tony
Tony Orlow
Hi David
I expected you to excoriate me, but you kind of complimented me.
Thanks. I'm glad I made you think. I've already been corrected several
times in this thread on my original post, and freely accept it. It was
more of a question than a challenge, and my "hypothesis" stood
corrected, until I told it to sit down and shut up. I'm printing out
your web page and will peruse it with anticipation. :)
Take care,
Tony
David R Tribble
> Sequences are sets with order. Sets in general have no order.
I get what you're trying to say, but to be pedantic, sequences are
not sets at all. Consider the sequence S = 1, 1, 1, 1, ... .
> If set
> theory would like to hide the recursive nature of infinite sets in
> order to draw precarious conclusions about sequences as if they "just
> exist",
You often seem to have problems with this "just exists" concept.
If we say that the set E of all even naturals exists, we mean that
E = {0, 2, 4, ...} is there, in abstract mental idea space, in its
entirety, all at once. A set (or any other mathematical entity)
simply exists, all at once, fully formed.
Such entities are not processes that have to "execute" in order
to be "finished". Irrational numbers, for instance, are not incomplete
sequences of digits continuously growing by some mysterious
mathematical digit-appending daemon.
-drt
Michael Stemper
>On Jun 4, 1:12=A0pm, Pubkeybreaker <pubkeybrea...@aol.com> wrote:
>> On Jun 4, 12:20=A0pm, Tony Orlow <t...@lightlink.com> wrote:
>> > On Jun 3, 11:40=A0pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> > > There are not more rational numbers than naturals -- that is, |N|=|Q|.
>>
>> > > Even Tony knows that.
>> > Just because I concede that both sets are countably infinite and
>> > therefore of the same cardinality, nevertheless the sparse proper
>> > subset of the rationals called the naturals should not be equated in
>> > size with its dense proper superset.
>>
>> You have been given an explicit injection from Q to N.
>>
>> If you think it is wrong, explain why.
>>
>> Otherwise, explain this idiocy in which you believe that there are
>> more rationals than
>> integers.
>I have stated clearly that I understand there is a bijection between
>the countable infinities of N and Q. Are you daft? Why correct
>something which isn't wrong? Sure, they have the same cardinality. So,
>what?
>
>You cannot disagree with the fact that the rationals are a dense set
>whereas the naturals are sparse, with a countably infinite number of
>rationals lying quantitatively between any two given naturals, can
>you? Do you not see that in some sense there appear to be more
>rationals than naturals? Are you incapable of considering any other
>system of set measurement besides what was drilled into you in class?
Y'know, if you want, you're perfectly free to propose a different
way to compare two sets. Nobody's ever said that such comparisons
need to be based on the notion of cardinality.
For starters, you might want to investigate the concept of "Lebesgue
measure". As I understand it (and I haven't actually reached this in
my formal studies yet), the Lebesgue measure of some uncountable
proper subsets of R is less than that of R. This might be close to
what you're seeking. On the other hand (again, as I understand it),
the Lebesgue measure of any countable subset of R is zero, so this
still gives N and Q being of the "same size".
However, it might be a starting point that you could use as a springboard
for developing a method of comparing the "sizes" of sets in some new
way. I assure you that, if you come up with a well-defined way to do
this that doesn't give ambiguous (as opposed to unintuitive) results,
the real mathematicians here would be interested.
Something to use as a test case for any comparison method that you come
up with is to compare following sets:
1. The positive rationals, R+
2. The expression of all positive rationals as decimal fractions
3. The expression of all positive rationals as octal fractions
(The last two sets are sets of strings.)
Be ready to show how your method of comparison treats each of these sets,
and answer which is "biggest", "smallest", and show how your method arrives
at those answers.
--
Michael F. Stemper
#include <Standard_Disclaimer>
This sentence no verb.
Virgil
<51fafd21-e48d-461b...@d8g2000yqf.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> On Jun 3, 3:55 pm, Virgil <Vir...@home.esc> wrote:
> > In article
> > <1bb4e64e-9dd5-45a2-8c42-9ffa430f8...@c7g2000vbc.googlegroups.com>,
> > Tony Orlow <t...@lightlink.com> wrote:
> >
> > > In the same vein, most objectors to transfinitology have a hard time
> > > identifying where exactly they disagree with the construction of the
> > > theory, but have at least a few examples of conclusions drawn from it
> > > which are completely incorrect when approached through other avenues.
> > > This is partly because mathematicians misrepresent the theory,
> > > claiming that every conclusion they draw "follows logicallly from the
> > > axioms." This claim is simply not true, as cardinality is not
> > > mentioned in the axioms, much less anything about omega or the alephs,
> > > except for a declaration that something homomorphic to the naturals
> > > exists, which is ultimately not a set, but a sequence.
> >
> > The issue is whether cardinality CAN BE defined within an axiom system,
> > not whether it is explicitly mentioned in some axiom(s) of that system.
> >
> > And it can be.
>
> And yet, it does not logically follow from the axioms.
If you mean that one can avoid deducing the properties of cardinality,
note that one can, by totally ignoring the system, refuse to deduce
anything from it at all.
> It is simply
> consistent in the sense of not contradicting them.
Meaning that it can be deduced therefrom.
> Therefore, other
> measures of sets can also exist, consistent with the axioms, but
> inconsistent with cardinality.
Incompatible measures perhaps, but not inconsistent with cardinality
unless inconsistent with itself.
>
> >
> > How does TO define 'sequence' without reference to something like the
> > SET of naturals as indices?
>
> It is a set wherein every element is either before or after (not
> immediately) every other element. That's one possible definition,
> though you undoubtedly have some objection.
Both the rationals and the reals, with their usual orders, satisfy YOUR
definition of sequences, and while the rationals, with a suitable but
different ordering may be a sequence, there is no ordering on the reals
which is known to make them into a sequence, at least for any generally
accepted definition of "sequence".
>
> Peace, Virgil.
>
> Tony
Virgil
<2ff234ce-4fc9-49fb...@c33g2000yqm.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> On Jun 3, 11:40�pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > kunzmilan <kunzmi...@atlas.cz> writes:
> > > The uncountability of the reals is simply based on the fact, that
> > > there are more rational numbers than there are the natural numbers.
> >
> > This is, of course, utterly butt-wrong.
> >
> > There are not more rational numbers than naturals -- that is, |N|=|Q|.
> >
> > Even Tony knows that.
> >
> > --
> > Jesse F. Hughes
> >
> > One is not superior merely because one sees the world as odious.
> > � � � � � � � � -- Chateaubriand (1768-1848)
>
> Hi Jesse -
>
> Just because I concede that both sets are countably infinite and
> therefore of the same cardinality, nevertheless the sparse proper
> subset of the rationals called the naturals should not be equated in
> size with its dense proper superset.
>
> Tony
Physical objects can have diverse measures of size, such as mass, volumn
and surface area, so why are you so violently opposed to having a
variety of "size" measures for non-physical objects?
K_h
"Tony Orlow" <to...@lightlink.com> wrote in message
news:dc7fa188-5dc1-4171...@c11g2000vbe.googlegroups.com...
The term sequence has a mathematical and non-mathematical
usage. In mathematics, an infinite sequence is defined as
follows:
A set S is a sequence of the elements of a set T if S is a
function with domain N and codomain T.
For a finite sequence the definition is the same except the
domain is some n in N and not all of N. Remember, in the ZF
approach to set theory, functions are defined as sets.
_
Virgil
<f1b57d32-97be-4811...@a20g2000vbc.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> On Jun 4, 12:37 pm, Tonico <Tonic...@yahoo.com> wrote:
> > On Jun 4, 7:16 pm, Tony Orlow <t...@lightlink.com> wrote:
> >
> >
> >
> >
> >
> > > On Jun 3, 3:55 pm, Virgil <Vir...@home.esc> wrote:
> >
> > > > In article
> > > > <1bb4e64e-9dd5-45a2-8c42-9ffa430f8...@c7g2000vbc.googlegroups.com>,
> > > > Tony Orlow <t...@lightlink.com> wrote:
> >
> > > > > In the same vein, most objectors to transfinitology have a hard time
> > > > > identifying where exactly they disagree with the construction of the
> > > > > theory, but have at least a few examples of conclusions drawn from it
> > > > > which are completely incorrect when approached through other avenues.
> > > > > This is partly because mathematicians misrepresent the theory,
> > > > > claiming that every conclusion they draw "follows logicallly from the
> > > > > axioms." This claim is simply not true, as cardinality is not
> > > > > mentioned in the axioms, much less anything about omega or the alephs,
> > > > > except for a declaration that something homomorphic to the naturals
> > > > > exists, which is ultimately not a set, but a sequence.
> >
> > > > The issue is whether cardinality CAN BE defined within an axiom system,
> > > > not whether it is explicitly mentioned in some axiom(s) of that system.
> >
> > > > And it can be.
> >
> > > And yet, it does not logically follow from the axioms. It is simply
> > > consistent in the sense of not contradicting them. Therefore, other
> > > measures of sets can also exist, consistent with the axioms, but
> > > inconsistent with cardinality.
> >
> > > > How does TO define 'sequence' without reference to something like the
> > > > SET of naturals as indices?
> >
> > > It is a set wherein every element is either before or after (not
> > > immediately) every other element. That's one possible definition,
> > > though you undoubtedly have some objection.
> >
> > > Peace, Virgil.
> >
> > > Tony-
> >
> > Yes, Virgil will surely object something...I don't know, he may even
> > ask you to be logical or to base your arguments...that bastard!
> > So a sequence "is a set wherein every element is either before [sic]
> > or after (not immediately)[sic] every [sic] other element" ...leaving
> > aside that wonderful and startling "every" in that ""definition"" ,
> > would you mind to tell us what do you mean by an element of a general
> > set "being before or after" some other one?! I mean, this is the kind
> > of things Virgil, mathematicians and people used to think more or less
> > logically can object to...
> > For example, what'd mean to be "before" or "after" an element in, say
> > the set of all the red apples from California?
>
> Duh-uh....is there a sequence of apples? Sets are not necessarily
> sequences. I might as well bring up monkeys and typewriters in some
> dicussion of Shakespeare, and it would be no more of a nonsequitur
> than your comment.
Note that the reals with their standard order satisfy Tony's definition
of "sequence" though there is not even any explicit well-ordering of
them.
>
> >
> > Tonio- Hide quoted text -
> >
> > - Show quoted text -
>
> Look, "Tonico", you can cop whatever attitude you want with me, but in
> a conversation in this very newsgroup not long ago it became obvious
> that seasoned mathematicians don't even agree on what a sequence is,
> some considering it isomorphic to the naturals, and others to the
> entire class of von Neumann ordinals. Besides, this is unrelated to
> the topic, and merely finger exercise for your pallid digits and
> tangled neurons.
By one fairly standard definition, a sequence (or, more properly, an
infinite sequence) is a surjection from the set of naturals to any
non-null set. Any other definition has to be fairly much equivalent.
Virgil
<613255cf-d67c-4c71...@c22g2000vbb.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> Sequences are sets with order.
But sets with order often are not sequences. Consider the standard reals.
In terms of order, a set must be well-ordered with one and only one
non-successor element to be a sequence. But there are sequences which
are not well-ordered sets.
Pollux
Pollux
MoeBlee
> On Jun 4, 2:32 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Jun 3, 7:34 am, Tony Orlow <t...@lightlink.com> wrote:
>
> > > mathematicians misrepresent the theory,
> > > claiming that every conclusion they draw "follows logicallly from the
> > > axioms." This claim is simply not true, as cardinality is not
> > > mentioned in the axioms, much less anything about omega or the alephs,
> > > except for a declaration that something homomorphic to the naturals
> > > exists, which is ultimately not a set, but a sequence.
>
> > No, all of those terms are given by DEFINITIONS, which may take the
> > form of definitional AXIOMS, which provide a conservative extension of
> > the theory in its primnitive form. This has been explained to you at
> > least a hundred times already, but you choose just to ignore.
>
> Can you please provide a complete list of your definitional axioms for
> my inspection? They are not part of ZFC, are they?
The answer to your question is embedded in about a hundred
explanations I've already given you over the years, as well as in the
very paragraph you just quoted.
The definitional axioms are not themselves part of ZFC in its
primitive form. Each definitional axiom is a sentence added to ZFC and
introduces a new defined symbol. However, by the method of proper
definitions, definitional axioms do not add any theorems to the
primitive theory in the primitive language but rather add only
theorems that have defined symbols in them as such theorems can be
reduced to equivalent theorems with only primitive notation. All of
this is covered in many a textbook in mathematical logic, and is a
formalization of basic common sense notions in mathematics.
It would be impractical for me to post a list of all the definitions
I've entered just in my own compendium. And such a list would vary
from author to author anyway, as each author will adopt definitions as
he or she needs for his or her own purposes of exposition.
In any case, ordinarily the list starts with such symbols as '0',
'P' (for power set), 'U' (for union), etc.
> > Moreover, we don't just show that there exists a set homomorphic to
> > the naturals (and their customary ordering) but rather that we define
> > 'w' (omega) to be the least set that has 0 as a member and is closed
> > under successsorship. Moreover, in set theory, sequences are sets.
>
> Sequences are sets with order.
That is not the set theoretic definition of a sequence. A set with an
order is of the form:
<S R> where S is a set and R is an ordering on S.
A sequence (in a general sense) is a function whose domain is an
ordinal
> Sets in general have no order.
Every set can be partially ordered. With the axiom of choice, every
set can be well ordered. And with the axiom of choice (but weaker than
the axiom of choice), every set has a linear ordering.
> If set
> theory would like to hide the recursive nature of infinite sets
I have no idea what you mean by the "recursive nature". If there is
some specific mathematical statement or statement about mathematics
you wish to make, would you please find out how to couch in some
sensible way?
> > > By demanding to
> > > know with which axioms a non-transfinitologist disagrees,
> > > mathematicians obscure the question.
>
> > WHAT question?
>
> MoeBlee eat some meat or take vitamins. The question is, "Why do you
> object to set theory, O Crackpot?" Isn't that what you've been asking
> all along?
I had a top sirloin last night and later a big bottle of Pomegranate
vitamin water. Don't worry about my nutrition. If you are to nag me, I
wish you would nag me about something actually in need of attention,
such as I need to renew my bus pass this month rather than waste money
each day on individual fares.
> You may call N a set, but it is really a sequence that is
> never complete at any point.
I don't begrudge that you have that notion of N and of sequence and of
'complete', whatever that may mean. Let me know when you have such
notions evinced in a formal theory.
> the Axiom of infinite is not about a
> set, really, but a sequence without end.
You may have your own notions about things, but you're off-base when
you presume to command the import of notions quite independent of you.
The axiom of infinity, as a sentence in a formal language, "reads off"
in an ordinary way as asserting that there exists an successor
inductive set.
> > > and the simplistic assumption of equivalence of
> > > "size" based on bijection alone.
>
> > I've presented to you at least a hundred times the answer (in
> > variation) that instead of the word 'size' we could use the word
> > 'zize'. Then your protest disappears. You've never dealt with that
> > point.
>
> I used "Bigulosity", remember? Surely you do, you rascal.
You could use "jiggywiggywosity" or whatever you want. It doesn't
address the point I've made. That YOU have some neologism concerning
your own mathematical thought-cloud doesn't address that, as far as
mere formal ZFC is concerned, it is not substantive whether we say
'size' or 'zize'.
> > > If bijection determines equal
> > > cardinality that's fine, but to equate cardinality with set size
> > > simply does not work for infinite sets to the satisfaction of most
> > > people's intuitions.
>
> > Most people haven't studied the subject. And, again, everywhere in
> > discussions about set theory you may substitute 'size' with 'zize'. No
> > substantive matter to the FORMAL theory.
>
> "The" formal theory. I like that. You should really become a Hare
> Krishna devotee.
Oh please, Swami Sri Nondevotee, of course, I mean whatever particular
theory is under discussion at the time. That doesn't imply that I
don't recognize that there are INFINITELY many theories.
> > > To claim that they are being illogical by
> > > objecting to the theory without identifying an axiom at fault is
> > > disingenuous, since the axioms are clearly not the problem.
>
> > The problem is that you demand that the word 'size' be used in a way
> > that does not conflict with your ordinary notions about size. But set
> > theory does not represent that such words are used in a way that
> > conforms in every sense to such everyday notions.
>
> Well, perhaps it satisfies your "everyday" experiences with infinity.
No, of course not. I HAVE NO everyday experiences with infinity (at
least not in the sense of sensory experience). Don't twist me. I said
'notions' not 'experiences'. And I didn't even say that anyone has
everyday notions about infinity. I simply said that set theory does
not (in itself, I'll add) represent that certain words are used in an
everyday sense; indeed, since the everyday sense of size pertains to
finite.
> Or, perhaps, you don't have any. If you don't call it "size" I won't.
> You say "cardinality", and I sqay "bigulosity".
So what? You persist to miss my point even after I've stated it point
blank over a hundred times.
> > > The von Neumann limit ordinals are
> > > simply bunk.
>
> > Fine, then probably you want to throw out the axiom of infinity.
>
> It says nothing about the non-finite limit ordinals. See above.
So what? But along with other axioms it entails the existence of
infinite ordinals and of limit ordinals.
You're wasting our time.
> > > Everything Jesus said was true and wise, but the Trinity
> > > and the Immaculate Conception are Christianity's von Neumann limit
> > > ordinals, thrown in later for Constantine's political goal of state
> > > religion incorporating pagan beliefs and christian.
>
> > So who is the Jesus of set theory before statist limit ordinals
> > corrupted the whole thing?
>
> Good question. Perhaps Cantor and Dedekind in modern times, though
> Galileo certainly explored bijection long before, and decided there
> was no real answer, and Aristotle first made the distinction between
> potential (countable) and actual (uncountable) infinities. This is
> some of the ancestry of transfinitology.
As far as I know, neither Galileo nor Aristotle proposed what could
fairly be called a set theory.
Anyway, so everything Galileo and/or Aristotle said was true and wise?
> Good enough? Why do you ask?
Curious to see how you'd finish your half-baked analogy.
> Keep on truckin'
Don't do your knockin' where your sheeps are aflockin'.
MoeBlee
> ToeKnee
BoerEeng.
MoeBlee
MoeBlee
> MoeBlee <jazzm...@hotmail.com> writes:
> > It's not fascinating. Aatu is just reminding of the dry fact that in
> > conversational mathematics, we do use the word 'more' in a sense not
> > confined to cardinality.
>
> Well, I was also pointing out there are in the logical literature
> numerous not at all nonsensical attempts at defining a general notion of
> size on which e.g. the rationals are of a larger size than the naturals.
Okay, I stand corrected. In any case, the more prosaic sense applies
also, as you illustrated.
MoeBlee
MoeBlee
> On Jun 4, 2:37 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Jun 4, 11:16 am, Tony Orlow <t...@lightlink.com> wrote:
>
> > > other
> > > measures of sets can also exist, consistent with the axioms, but
> > > inconsistent with cardinality.
>
> > It is not ordinarily disputed that definitions may be given
> > stipulatively. So what?
>
> "stipulative - no dictionary results" - dictionary.com
Try 'stipulative definition' on the Internet.
> > > > How does TO define 'sequence' without reference to something like the
> > > > SET of naturals as indices?
>
> > > It is a set wherein every element is either before or after (not
> > > immediately) every other element. That's one possible definition,
>
> > Yes, the objection is that it's more general than the ordinary
> > definition, thus loses the special sense obtained from the ordinary
> > definition. But again, if you don't like the ordinary definition of
> > 'sequence', then just imagine everywhere 'sequence' appears in set
> > theory discussion that the word is 'zequence'.
> Do you not remember the thread not long ago where you methamaticians
> started arguing amonst yourselves? What is the "ordinary" definition,
> again?
I just gave it to you. So what that some people discovered that there
are different uses?
> Never mind. It's a sidebar, as always with you.
Because getting straight the subject matter on which you shoot your
big mouth off is always a side matter with you.
> Take a deep breath.
Deep advice.
> Love,
Someone should write a song or a poem on that subject one of these
days.
MoeBlee
Rob Johnson
I don't think it is the number of provable statements that one
considers important, but what those statements are. Although we may
be able put the statements proved by PA and the statements proved by
PA + Con(PA) into a 1-1 correspondence, there are probably statements
provable by PA + Con(PA) that are not provable by PA that are useful.
Although Z and 2Z can be put into 1-1 correspondence, there are times
when I really need to use a 1, and no matter how hard I look, I can't
seem to find one in 2Z.
Rob Johnson <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
Ludovicus
are surely more rationals per unit interval of the real line than
there are naturals.
No.
The Farey Series of order n have 3*(n/pi)^2 terms.
When n ---- infinte, we have all the rationals in
the unit interval, but the formula shows that
that quantity is countable.
That is, it have the same power of the naturals.
Ludovicus
Jesse F. Hughes
> Aatu Koskensilta wrote:
>> The supply of morons in this world is inexhaustible.
>> So naturally you object also to the statement
>> PA + Con(PA) proves more true statements about naturals than PA.
>
> I don't follow this.
>
> If the number of true (constructible) statements proved by PA and
> the number of true statements proved by PA + Con(PA) are both
> countable, does it not follow that those two sets have the same
> cardinality?
Aatu's point is simply that, in some contexts, we say that this set
has more elements than that set even though they have the same
cardinality. Sometimes, we use the word "more" to refer to the
superset relation.
--
Jesse F. Hughes
"I think the problem for some of you is that you think you are very
smart. I AM very smart. I am smarter on a scale you cannot really
comprehend and there is the problem." -- James S. Harris
Marshall
> On Jun 4, 1:22 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
> > It makes perfect sense to say there are more rationals than integers,
>
Please, this is an informal setting. There is no need to introduce
yourself before you speak.
Marshall
Tony Orlow
> On Jun 4, 2:50 pm, Tony Orlow <t...@lightlink.com> wrote:
>
> > You cannot disagree with the fact that the rationals are a dense set
> > whereas the naturals are sparse,
>
> WRONG, WRONG, WRONG. I've explained this to you several times, and you
> persist to ignore.
>
> A set is dense with respect to some ORDERING. There IS a dense
> ordering of the set of natural numbers, and there is an ordering of
> the set of rationals that is NOT dense.
>
> What you can say is that the set of natural numbers is not dense with
> regard to the usual ordering on the set of natural numbers and that
> the set of rational numbers is dense with regard to the usual ordering
> on the set of rational numbers.
That's usually what I mean. I'll be more specific if not.
>
> > with a countably infinite number of
> > rationals lying quantitatively between any two given naturals, can
> > you? Do you not see that in some sense there appear to be more
> > rationals than naturals?
>
> Yes, but some investigation shows that that is a superficial view,
> since we see also that we can order the naturals densely and order the
> rationals discreetly.
Perhaps simple bijection as a proof of equinumerosity is superficial.
That's also a possibility. :)
Peace,
TOny
>
> MoeBlee
Tony Orlow
> On Jun 4, 1:32 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Jun 3, 7:34 am, Tony Orlow <t...@lightlink.com> wrote:
> > > and the simplistic assumption of equivalence of
> > > "size" based on bijection alone.
>
> > I've presented to you at least a hundred times the answer (in
> > variation) that instead of the word 'size' we could use the word
> > 'zize'. Then your protest disappears. You've never dealt with that
> > point.
>
> the question. That is to say, I don't begrudge that there may be a
> legitimate philosophic dispute over whether set theoretic cardinality
> does or does not capture our intuitive notion of size in regards
> infinite sets. (That set theoretic cardinality does capture our notion
> of size in finite cases I take to be ordinarily uncontroversial).
I really appreciate that philosophical concession, and freely admit
that cardinality captures the essence of set size with respect to
finite sets. No controversy there.
> However, we must at least address my "formalistic" rebuttal at least
> first, so to recognize that in the sense of set theory as a formal
> theory, whether we call call cardinality 'size' or 'zize'.
Okay, phrase missing, but more or less understood. Proceed....
>
> That said, however, still I don't take the import of set theory to be
> to settle such questions except as they pertain to mathematical
> inquiry, in which sense the notion of cardinality does seem to be
> viable. This is so especially since at least I don't know of a
> challenging formalization of the notion of size in the infinite case
> that is any more intuitive than set theoretic cardinality, as at least
> set theoretic cardinality is motivated by the intuitive notion of one-
> to-one correspondence but also taken in the infinite case. More
> specifically, Orlow himself has not presented any theory that
> satisfies EITHER criteria: (1) formally coherent, (2) more intuitive
> than set theoretic cardinality, as Orlow's notions are even LESS
> intuitive than set theoretic cardinality. That he personally finds his
> own ruminations more intuitive is not sufficient philosophical
> motivation for patiently waiting for him to someday put his floating
> ideas into the form of a formal theory.
>
> MoeBlee
Well, sure, MoeBlee. I'm not sure how easy you think it is to concoct
an original mathematical theory regarding infinity on one's own. In
some sense I have been working on this for thirty years. The past
several have been a revival, both of interest in understanding the
standard theory, and in improving on it. There will remain questions
unanswered when I'm done, I'm sure, but the continuum hypothesis is
not one of them.
Poll: would you rather see a collection of all axioms and theorem
derivations, or a more explanatory approach? I'm thinking the formal
axiom set and development is more of an appendix.
Best Regards,
ToeKnee
Tony Orlow
Piffle to you both. I already stated that very fact very early in this
thread. Don't start crying "quantifier dyslexia". You know better.
Tony
Tony Orlow
> Tony Orlow wrote:
> > Sequences are sets with order. Sets in general have no order.
>
> I get what you're trying to say, but to be pedantic, sequences are
> not sets at all. Consider the sequence S = 1, 1, 1, 1, ... .
Yes, granted. Monotonically increasing sequences are sets with order
of magnitude correlated positively with order of occurrence. Better?
That's the way the simplest infinite "sets" are defined, no?
>
> > If set
> > theory would like to hide the recursive nature of infinite sets in
> > order to draw precarious conclusions about sequences as if they "just
> > exist",
>
> You often seem to have problems with this "just exists" concept.
>
> If we say that the set E of all even naturals exists, we mean that
> E = {0, 2, 4, ...} is there, in abstract mental idea space, in its
> entirety, all at once. A set (or any other mathematical entity)
> simply exists, all at once, fully formed.
>
> Such entities are not processes that have to "execute" in order
> to be "finished". Irrational numbers, for instance, are not incomplete
> sequences of digits continuously growing by some mysterious
> mathematical digit-appending daemon.
>
> -drt
It's not so much that such a sequence does not exist, but that the
existence of each but the first is dependent on the existence of at
least one other. The size of such a structure simply cannot be
measured, except relatively with respect to some other countably
infinite sequence, such as N.
Peace,
Tony
Tony Orlow
wrote:
Yes, I've looked into Lebesgue measure, and the notion that any
countable (finite or infinite) set of contiguous points has a spatial
measure of zero is consistent with my theory, insofar as it refers to
standard measure. However, when assuming a unit infinitesimal into
account, as the measure of a point, the Lebesgue measures of these
sets might be extended so that they are distinguishable on that
scale.
>
> However, it might be a starting point that you could use as a springboard
> for developing a method of comparing the "sizes" of sets in some new
> way. I assure you that, if you come up with a well-defined way to do
> this that doesn't give ambiguous (as opposed to unintuitive) results,
> the real mathematicians here would be interested.
One can only hope. We'll see how it goes. :)
>
> Something to use as a test case for any comparison method that you come
> up with is to compare following sets:
>
> 1. The positive rationals, R+
That's a little tricky to quantify with respect to N if one wants to
keep all quantities unique regardless of expression, but I have a few
ganglia working on it.
> 2. The expression of all positive rationals as decimal fractions
> 3. The expression of all positive rationals as octal fractions
>
> (The last two sets are sets of strings.)
Yes, that's where I apply N=S^L. Good reminder!!
Of course, there's still that problem with the rationals in
general... :(
>
> Be ready to show how your method of comparison treats each of these sets,
> and answer which is "biggest", "smallest", and show how your method arrives
> at those answers.
Indeed. Thanks for the advice. I'm sure some questions will remain
after I'm "done". You guys figgered out that continuum thingy yet?
Mine don't gots dat problem... ;)
>
> --
> Michael F. Stemper
> #include <Standard_Disclaimer>
> This sentence no verb.- Hide quoted text -
>
> - Show quoted text -
Thanks,
TOny
Tony Orlow
> In article
> Tony Orlow <t...@lightlink.com> wrote:
>
>
>
>
>
> > On Jun 3, 3:55 pm, Virgil <Vir...@home.esc> wrote:
> > > In article
> > > <1bb4e64e-9dd5-45a2-8c42-9ffa430f8...@c7g2000vbc.googlegroups.com>,
> > > Tony Orlow <t...@lightlink.com> wrote:
>
> > > > In the same vein, most objectors to transfinitology have a hard time
> > > > identifying where exactly they disagree with the construction of the
> > > > theory, but have at least a few examples of conclusions drawn from it
> > > > which are completely incorrect when approached through other avenues.
> > > > This is partly because mathematicians misrepresent the theory,
> > > > claiming that every conclusion they draw "follows logicallly from the
> > > > axioms." This claim is simply not true, as cardinality is not
> > > > mentioned in the axioms, much less anything about omega or the alephs,
> > > > except for a declaration that something homomorphic to the naturals
> > > > exists, which is ultimately not a set, but a sequence.
>
> > > The issue is whether cardinality CAN BE defined within an axiom system,
> > > not whether it is explicitly mentioned in some axiom(s) of that system.
>
> > > And it can be.
>
> > And yet, it does not logically follow from the axioms.
>
> If you mean that one can avoid deducing the properties of cardinality,
> note that one can, by totally ignoring the system, refuse to deduce
> anything from it at all.
Yeah, man, like, wow. Why deduce anything when you just know? Here, I
grew this flower for you, in my hair. Doesn't it smell nice? I'm sure
it will convince you..... ;)
>
> > It is simply
> > consistent in the sense of not contradicting them.
>
> Meaning that it can be deduced therefrom.
No, meaning that there does not arise any contradiction. That is to
say, no set of assumptions within the model can be used to derive the
opposite of any subset of the axioms. The axioms do not imply the
model. The model simply fails to directly contradict the axioms. I'm
sure you understand this, deep down inside your soul, man. Dig it. ;)
>
> > Therefore, other
> > measures of sets can also exist, consistent with the axioms, but
> > inconsistent with cardinality.
>
> Incompatible measures perhaps, but not inconsistent with cardinality
> unless inconsistent with itself.
>
I'm sure I won't be the only one to disagree with this statement so
I'll reserve comment for now.
>
>
> > > How does TO define 'sequence' without reference to something like the
> > > SET of naturals as indices?
>
> > It is a set wherein every element is either before or after (not
> > immediately) every other element. That's one possible definition,
> > though you undoubtedly have some objection.
>
> Both the rationals and the reals, with their usual orders, satisfy YOUR
> definition of sequences, and while the rationals, with a suitable but
> different ordering may be a sequence, there is no ordering on the reals
> which is known to make them into a sequence, at least for any generally
> accepted definition of "sequence".
>
Surely you remember the T-Riffics?
Love,
Tony
Tony Orlow
> In article
>
> - Show quoted text -
Do you really think I'm violent? (sniffle)
Tony
Tony Orlow
> In article
There are always the H-Riffics. Remember "Well Ordering the Reals"?
>
>
> > > Tonio- Hide quoted text -
>
> > > - Show quoted text -
>
> > Look, "Tonico", you can cop whatever attitude you want with me, but in
> > a conversation in this very newsgroup not long ago it became obvious
> > that seasoned mathematicians don't even agree on what a sequence is,
> > some considering it isomorphic to the naturals, and others to the
> > entire class of von Neumann ordinals. Besides, this is unrelated to
> > the topic, and merely finger exercise for your pallid digits and
> > tangled neurons.
>
> By one fairly standard definition, a sequence (or, more properly, an
> infinite sequence) is a surjection from the set of naturals to any
> non-null set. Any other definition has to be fairly much equivalent.- Hide quoted text -
>
> - Show quoted text -
Yeah. That's "fairly standard", at least in the sense of being fair to
middling. I mean, "fairly standard"? Isn't mathematics all about
precision? Or, perhpaps there is some kind of double standard?
It's all good,
TOny
David R Tribble
> Perhaps simple bijection as a proof of equinumerosity is superficial.
> That's also a possibility. :)
If you have two sets, and every single member of both are paired
together, in what logical sense can you say that they are not really
(only "superficially") equinumerous?
David R Tribble
>> Note that the reals with their standard order satisfy Tony's definition
>> of "sequence" though there is not even any explicit well-ordering of them.
>
Tony Orlow wrote:
> There are always the H-Riffics. Remember "Well Ordering the Reals"?
Yeah. Remember how several of us demonstrated that the H-riffics
is only a countably infinite set, and omits vast subsets of the reals
(e.g., all the multiples of powers of integers k, where k is not 2)?
David R Tribble
>> Both the rationals and the reals, with their usual orders, satisfy YOUR
>> definition of sequences, and while the rationals, with a suitable but
>> different ordering may be a sequence, there is no ordering on the reals
>> which is known to make them into a sequence, at least for any generally
>> accepted definition of "sequence".
>
Tony Orlow wrote:
> Surely you remember the T-Riffics?
Yeah. Surely you remember how you could never come up with
a self-consistent notation for them? Or a self-consistent definition
for incrementing from one T-riffic to the next? Or several other
missing critical pieces of your theory?
Virgil
<ed1b8d08-87d2-427b...@d8g2000yqf.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> Perhaps simple bijection as a proof of equinumerosity is superficial.
> That's also a possibility. :)
It is certainly adequate as such a proof of equal cardinality.
Virgil
<9083dd54-a2f1-46af...@k39g2000yqb.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
To declare, as TO does above, that the cardinality of the rationals
being equal to aleph_0^2 is NOT part of the standard theory, is just
plain wrong!
Virgil
<a459d982-9855-4f03...@j4g2000yqh.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> On Jun 4, 4:55 pm, David R Tribble <da...@tribble.com> wrote:
> > Tony Orlow wrote:
> > > Sequences are sets with order. Sets in general have no order.
> >
> > I get what you're trying to say, but to be pedantic, sequences are
> > not sets at all. Consider the sequence S = 1, 1, 1, 1, ... .
>
> Yes, granted. Monotonically increasing sequences are sets with order
> of magnitude correlated positively with order of occurrence. Better?
> That's the way the simplest infinite "sets" are defined, no?
The issue here being how SEQUENCES are defined, TO, as usual, goes off
on a tangent.
Virgil
<db5cbe4b-a8b8-4b6a...@i28g2000yqa.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> Yes, that's where I apply N=S^L.
Which is a wrong now as when first dropped on an unsuspecting world.
Virgil
<796c4157-f47d-4aba...@c10g2000yqi.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> > > It is a set wherein every element is either before or after (not
> > > immediately) every other element. That's one possible definition,
> > > though you undoubtedly have some objection.
> >
> > Both the rationals and the reals, with their usual orders, satisfy YOUR
> > definition of sequences, and while the rationals, with a suitable but
> > different ordering may be a sequence, there is no ordering on the reals
> > which is known to make them into a sequence, at least for any generally
> > accepted definition of "sequence".
> >
>
> Surely you remember the T-Riffics?
Does Tony Orlow really want to maintain that ANY part of his idiotic
"T-Riffics" was ->generally accepted<- ?
Virgil
<426447ef-9569-4bea...@g7g2000yqj.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
More like "violet".
Virgil
<5d346c04-9c57-4d64...@d37g2000yqm.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> >
> > > Look, "Tonico", you can cop whatever attitude you want with me, but in
> > > a conversation in this very newsgroup not long ago it became obvious
> > > that seasoned mathematicians don't even agree on what a sequence is,
> > > some considering it isomorphic to the naturals, and others to the
> > > entire class of von Neumann ordinals. Besides, this is unrelated to
> > > the topic, and merely finger exercise for your pallid digits and
> > > tangled neurons.
> >
> > By one fairly standard definition, a sequence (or, more properly, an
> > infinite sequence) is a surjection from the set of naturals to any
> > non-null set. Any other definition has to be fairly much equivalent.- Hide
> > quoted text -
> >
>
> Yeah. That's "fairly standard", at least in the sense of being fair to
> middling. I mean, "fairly standard"? Isn't mathematics all about
> precision? Or, perhpaps there is some kind of double standard?
Since To seems to think that my definition is less than perfectly
precise, I challenge TO to find a more precise definition expressed only
in words.
Mine: an infinite sequence is a surjection from the set of naturals
to a set (note the "non-null" is not actually needed)
Tony's: It is a set wherein every element is either before or after (not
immediately) every other element.
I'm sure Tony can improve on his present definition, but I doubt he can
improve on mine.
Tony Orlow
Hi David -
I can say so in the sense that one may be a proper subset of the
other, or is less dense in the natural quantitative order, or that
bijection is only part of a problem which is only properly solved by
taking into account the mapping function between the two sets. Take
your choice.
Smiles,
Tony
Tony Orlow
Sure when using 2 as a base, the numbers you mention are uncountably
distant from the beginning of the uncountable sequence. But then, I am
using "uncountable sequence" in a rather nonstandard way.
:) Tony
Tony Orlow
That doesn't ring a bell.
:) TOny
Tony Orlow
> In article
> Tony Orlow <t...@lightlink.com> wrote:
>
> > Perhaps simple bijection as a proof of equinumerosity is superficial.
> > That's also a possibility. :)
>
> It is certainly adequate as such a proof of equal cardinality.
Duly conceded.
:) Tony
Tony Orlow
> In article
>
> - Show quoted text -
I didn't say that was its cardinality, and if I had, it wouldn't
matter because aleph_0^2=aleph_0 in standard theory.
:) Tony
David R Tribble
>> Perhaps simple bijection as a proof of equinumerosity is superficial.
>> That's also a possibility. :)
>
David R Tribble wrote:
>> If you have two sets, and every single member of both are paired
>> together, in what logical sense can you say that they are not really
>> (only "superficially") equinumerous?
>
Tony Orlow wrote:
> I can say so in the sense that one may be a proper subset of the
> other, or is less dense in the natural quantitative order, ...
Yes. But how does that make them not equinumerous?
> .. or that
> bijection is only part of a problem which is only properly solved by
> taking into account the mapping function between the two sets. Take
> your choice.
The problem being, what, exactly? What we mean by "size of an
infinite set"?
David R Tribble
>> There are always the H-Riffics. Remember "Well Ordering the Reals"?
>
David R Tribble wrote:
>> Yeah. Remember how several of us demonstrated that the H-riffics
>> is only a countably infinite set, and omits vast subsets of the reals
>> (e.g., all the multiples of powers of integers k, where k is not 2)?
>
Tony Orlow wrote:
> Sure when using 2 as a base, the numbers you mention are uncountably
> distant from the beginning of the uncountable sequence. But then, I am
> using "uncountable sequence" in a rather nonstandard way.
It's more basic than that. Your H-riffic set completely omits most of
the reals, such as any multiple of any integral power of 3 (e.g., 3,
1/3,
27, etc., ad infinitum).
Besides, your set is only countable (which is obvious from its very
definition), so it can't possibly contain all the reals. It doesn't
even
contain all the rationals.
Virgil
<ae9e776b-2cad-4b09...@y12g2000vbr.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
So non-standard that not even Tony Orlow has any idea what he is talking
about.
Virgil
<8c60ceed-ef2c-4f49...@s41g2000vba.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
Does FOR me!
Virgil
<60e3188a-0cc8-4a43...@o15g2000vbb.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> On Jun 5, 12:53 pm, Virgil <Vir...@home.esc> wrote:
> > In article
> > <9083dd54-a2f1-46af-ab41-421b3c253...@k39g2000yqb.googlegroups.com>,
> > Tony Orlow <t...@lightlink.com> wrote:
> >
> >
> >
> >
> >
> > > On Jun 4, 4:24 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > > > On Jun 4, 3:20 pm, David R Tribble <da...@tribble.com> wrote:
> >
> > > > > Tony Orlow wrote:
> > > > > > One might think there were something like aleph_0^2 rationals, but
> > > > > > that's not standard theory.
> >
> > To declare, as TO does above, that the cardinality of the rationals
> > being equal to aleph_0^2 is NOT part of the standard theory, is just
> > plain wrong!
>
> I didn't say that was its cardinality, and if I had, it wouldn't
> matter because aleph_0^2=aleph_0 in standard theory.
Perhaps Tony should not have said "One might think there were something
like aleph_0^2 rationals, but that's not standard theory" when it is
PRECISELY standard theory that there are PRECISELY aleph_0^2 rationals.
Transfer Principle
> Tony Orlow wrote:
> >> There are always the H-Riffics. Remember "Well Ordering the Reals"?
> David R Tribble wrote:
> >> Yeah. Remember how several of us demonstrated that the H-riffics
> >> is only a countably infinite set, and omits vast subsets of the reals
> >> (e.g., all the multiples of powers of integers k, where k is not 2)?
I don't recall what exactly TO's H-riffics are, but this
post has piqued my interest.
At first, by Tribble's comments here, I thought that the
H-riffics were simply the elements of Z[1/2], the ring Z
with 1/2 appended to it. It consists of all rationals
with a finite binary expansion. Thus, for example, 1/3
is not an H-riffic.
> Tony Orlow wrote:
> > Sure when using 2 as a base, the numbers you mention are uncountably
> > distant from the beginning of the uncountable sequence. But then, I am
> > using "uncountable sequence" in a rather nonstandard way.
> It's more basic than that. Your H-riffic set completely omits most of
> the reals, such as any multiple of any integral power of 3 (e.g., 3,
> 1/3, 27, etc., ad infinitum).
> Besides, your set is only countable (which is obvious from its very
> definition), so it can't possibly contain all the reals. It doesn't
> even contain all the rationals.
But now Tribble writes that the H-riffics not only exclude
the value 1/3, but the value _3_ as well.
I can see the ring Z[1/2] being worth defending, but a set
excluding _3_ is indefensible. If 3 really isn't an
H-riffic, then I wonder what 2+1 is in the H-riffics.
But first, I'd like to see what exactly the H-riffics are
in the first place.
Transfer Principle
> In article
> Tony Orlow <t...@lightlink.com> wrote:
> > Surely you remember the T-Riffics?
> Does Tony Orlow really want to maintain that ANY part of his idiotic
> "T-Riffics" was ->generally accepted<- ?
Hold on a minute. Earlier, TO and Tribble were discussing
something called the H-riffics. Now Virgil is referring
to something called the T-riffics.
If by "T-riffics" Virgil is actually referring to the
"H-riffics" as mentioned by Tribble, then for once, I
actually agree with Virgil. For according to Tribble, the
H-riffics lack a value for 3. Even _I_ can't accept a
theory in which one can't even prove the existence of 3
(especially if it does prove the existence of 4, which,
being a power of two, does exist in this theory).
If the T-riffics are distinct from the H-riffics, then I
would like to learn more about the T-riffics before I
attempt to pass judgment. I don't mind learning more
about sets other than the classical real numbers (i.e.,
standard R) and standard set theories.