Infinity and (un-)countable sets
Ronald Bruck
In article <335011...@ping.be>, Guido Wuyts <Guido...@ping.be> wrote:
:I have this problem with the distinction between countable-noncountable
:infinities.
:What does a countable series mean anyway? And yet, some theorems such as
:Gödel's eagerly call for countable sets, eg of theorems.
:In my opinion countability suffers at least two big restrictions of
:applicability:
:1. it just is a the result of a particular way of ordering a set, albeit
:a happy one, amidst a sea of less such happy ways;
:2. it postpones the enumeration of elements with ever more complicated
:description, ever more ahead (and in the end scrambles the border of
:distinction with related domains of noncountable elements).
:
:The set of natural numbers is the basic model, as a matter of fact the
:criterion for comparison of countable sets. However, it takes a minimal
:mix-up to transform them in kind of an unoverseeable, uncountable
:desert. Take as a rule of enumeration: first the evens, then the odds.
:The "then" implies a never ending enumeration that has to precede a next
:one.
:The rational numbers are proven countable thanks to a cunning way of
:diagonal counting. But taken in the natural order, any "finitely
:describable" number has no such neighbour, so there is no enumeration in
:finite terms along this order.
:In the case of natural and rational numbers we don't worry about that,
:since we know some ways "that work". But what with establishing a
:supposedly coutable set of theorems as Gödel would like?
etc.
Look: if I throw eighteen marbles in a jar and shake it up, and then reach
in and draw them out one at a time, I get... eighteen marbles.
And if I throw aleph-nought marbles into the jar, then draw them out one at
a time, I'll get aleph-nought marbles. You object, Draw out the red ones
first, then the green ones, then the purple ones, then the... etc etc; OK;
I do; I draw out aleph-nought reds, aleph-nought greens, aleph-nought
purples, etc., for all of your aleph-nought colors you give me. I still
get aleph-nought marbles.
The point is, you can't force me to draw out MORE than aleph-nought marbles.
The DEFINITION of "countably infinite" is that there EXISTS a 1-1
correspondence with the natural numbers. Showing a 1-1 correspondence with
a set having a DIFFERENT order is irrelevant (except that you can deduce
that the other set is also in a 1-1 correspondence with the naturals).
Cardinality has nothing to do with order.
--Ron Bruck
Now 100% ISDN from this address
Guido Wuyts
I have this problem with the distinction between countable-noncountable
infinities.
What does a countable series mean anyway? And yet, some theorems such as
Gödel's eagerly call for countable sets, eg of theorems.
In my opinion countability suffers at least two big restrictions of
applicability:
1. it just is a the result of a particular way of ordering a set, albeit
a happy one, amidst a sea of less such happy ways;
2. it postpones the enumeration of elements with ever more complicated
description, ever more ahead (and in the end scrambles the border of
distinction with related domains of noncountable elements).
The set of natural numbers is the basic model, as a matter of fact the
criterion for comparison of countable sets. However, it takes a minimal
mix-up to transform them in kind of an unoverseeable, uncountable
desert. Take as a rule of enumeration: first the evens, then the odds.
The "then" implies a never ending enumeration that has to precede a next
one.
The rational numbers are proven countable thanks to a cunning way of
diagonal counting. But taken in the natural order, any "finitely
describable" number has no such neighbour, so there is no enumeration in
finite terms along this order.
In the case of natural and rational numbers we don't worry about that,
since we know some ways "that work". But what with establishing a
supposedly coutable set of theorems as Gödel would like?
Let us try a "clever" way of ordering theorems, by grouping for instance
theorems that would, given a number A, perform an operation with A+n.
Each number n (natural or, worse, real) would determine a theorem Tn. In
doing so, we obtain a series T0, T1, T2, ... , Tn, ... from which there
is no escape before being allowed to continue the enumeration with
theorems of a "next" type.
OK then, let's try a "dumb" method. We accept all ASCII codes, ordered
in a definable way and then proceed by eliminating the ones which do not
represent theorems. First, how definable may that ordering be, eg how
long may the lines become, and how many...? Second, we now have to
establish whether a code is a theorem, which seems to counter our
intention to produce theorems mechanically. Well, at least we can
establish whether it respects syntax. My point is that this way we will
doubtlessly stumble upon endless subsets of meaningless code before
encountering a "next" code-with-a-theorem.
My argument leads to the assertion that there is no such thing as a
well-ordered set of theorems, supposedly representing the human mind
about some theory, and out of which Gödel would then playfully pick a
non-provable theorem to prove incompleteness. The end result of his
statement remains the same of course, but it is looked upon in another
way.
The noncountability of real numbers raises its own problems. The "power
of infinity" (I don't know the proper English term) of R defined as a
set of all decimals, or of R defined as a straight line in algbraic
geometry, is fundamentally arbitrary, in that you make it as great or as
small as you wish to, and moreover both representations of R are
mutually independent. The power of the "real", physical line in space is
what it is, probably unprobable, and independent of our mathematical
representations of R (however neatly they seem to settle down on it).
My last problem is with limits, one of the tricks to approach the reals.
If we state eg, that the limit of a series 1/2, 1/3, 1/4, ..., 1/n, ...
equals zero, then I say to myself, no, it is at best a nest of real
numbers that couldn't be disentangled, as crowded as you like, and
amongst which zero is the only member describable in finite terms.
Greetings, Guido.
Ilias Kastanas
>infinities.
>What does a countable series mean anyway? And yet, some theorems such as
>Gödel's eagerly call for countable sets, eg of theorems.
>In my opinion countability suffers at least two big restrictions of
>applicability:
>1. it just is a the result of a particular way of ordering a set, albeit
>a happy one, amidst a sea of less such happy ways;
For some sets it can be done, for others no. That's the point.
>2. it postpones the enumeration of elements with ever more complicated
>description, ever more ahead (and in the end scrambles the border of
If there is one way to enumerate, there are many, yes.
>description, ever more ahead (and in the end scrambles the border of
>distinction with related domains of noncountable elements).
Ah!-- call the countable anything else, ... but not "uncountable"!
>The set of natural numbers is the basic model, as a matter of fact the
>criterion for comparison of countable sets. However, it takes a minimal
>mix-up to transform them in kind of an unoverseeable, uncountable
Hey... you said the "U" word again...
>desert. Take as a rule of enumeration: first the evens, then the odds.
>The "then" implies a never ending enumeration that has to precede a next
>one.
One mentally goes over the first part, then the next... -- like you
just did; you didn't stumble or get confused either.
Besides, it is the same set as before; there is a "simple" enumeration
for it. Somebody rearranged it this or that way. So what??
>The rational numbers are proven countable thanks to a cunning way of
>diagonal counting. But taken in the natural order, any "finitely
Q was countable before Cantor as well. It doesn't change, cunning
or no cunning.
>describable" number has no such neighbour, so there is no enumeration in
>finite terms along this order.
True, and irrelevant. We have an enumeration for Q; that settles
the matter.
>In the case of natural and rational numbers we don't worry about that,
>since we know some ways "that work". But what with establishing a
>supposedly coutable set of theorems as Gödel would like?
>
>Let us try a "clever" way of ordering theorems, by grouping for instance
>theorems that would, given a number A, perform an operation with A+n.
>Each number n (natural or, worse, real) would determine a theorem Tn. In
>doing so, we obtain a series T0, T1, T2, ... , Tn, ... from which there
>is no escape before being allowed to continue the enumeration with
>theorems of a "next" type.
"Dovetailing". Alternate between work on the T's and work on
the next part of the search.
>OK then, let's try a "dumb" method. We accept all ASCII codes, ordered
>in a definable way and then proceed by eliminating the ones which do not
>represent theorems. First, how definable may that ordering be, eg how
>long may the lines become, and how many...? Second, we now have to
Fully definable. The ASCII collating sequence is written in stone;
it orders strings of ASCII codes lexicographically; we use just one line
(string) with no bound on its length.
>establish whether a code is a theorem, which seems to counter our
>intention to produce theorems mechanically. Well, at least we can
>establish whether it respects syntax. My point is that this way we will
Good way of putting it. We look for strings YX where Y is a
formal deduction of statement X; verifying this does amount to just
"syntax" checking in a fully mechanical way.
>doubtlessly stumble upon endless subsets of meaningless code before
>encountering a "next" code-with-a-theorem.
It might be centuries before the first one shows up; the search
never ends; but every proof+theorem will appear at some finite time.
>My argument leads to the assertion that there is no such thing as a
>well-ordered set of theorems, supposedly representing the human mind
>about some theory, and out of which Gödel would then playfully pick a
>non-provable theorem to prove incompleteness. The end result of his
>statement remains the same of course, but it is looked upon in another
>way.
The axioms of PA (Peano Arithmetic) are specified without the sli-
ghtest ambiguity as strings of Predicate Calculus symbols, and so are PA's
theorems. Yes, they are well ordered, as strings or, if you Goedel-number
them, as integers by size. They don't change, any more than the set of
primes does.
They don't represent "the human mind";... just PA. And you can
cut down the work of finding a Q such that PA proves neither Q nor ~Q.
Take Q = "PA does not prove 0=1" ( <=> "PA is consistent").
>The noncountability of real numbers raises its own problems. The "power
>of infinity" (I don't know the proper English term) of R defined as a
>set of all decimals, or of R defined as a straight line in algbraic
>geometry, is fundamentally arbitrary, in that you make it as great or as
>small as you wish to, and moreover both representations of R are
The reals, whether defined by decimals (R1), or Dedekind cuts (R2), or
Cauchy sequences (R3), or integer sequences (R4)... etc have the exact
same cardinality; there is a 1-1 onto mapping between any two R's.
>mutually independent. The power of the "real", physical line in space is
>what it is, probably unprobable, and independent of our mathematical
>representations of R (however neatly they seem to settle down on it).
The "physical" line probably inspired R, but it has no effect on
R now, no influence or anything.
>My last problem is with limits, one of the tricks to approach the reals.
>If we state eg, that the limit of a series 1/2, 1/3, 1/4, ..., 1/n, ...
>equals zero, then I say to myself, no, it is at best a nest of real
>numbers that couldn't be disentangled, as crowded as you like, and
>amongst which zero is the only member describable in finite terms.
Eh... isn't 1/2, or any 1/n finitely describable?! And 0 is
not one of them anyway; it just happens to satisfy the definition of
limit. I don't know what you are objecting to.
Ilias
Guido Wuyts
Thank you for your remarks on my cry for help. I think that I became a little more convinced
of a countable theorem construction set than I did with my up to now litterature. Some
points remain obscure though, or I cannot find the proper way to formulate pro or contra
them. Some of them I reply, as you did, within the text.
Ilias Kastanas wrote:
>
> If there is one way to enumerate, there are many, yes.
>
> Ah!-- call the countable anything else, ... but not "uncountable"!
But any "successful" enumeration has to push ahead in its tasklist the elements
whose description becomes increasingly complex, eg the rationals that become too
"real"-like, the elements bordering the non-rational domain...
>
>
> One mentally goes over the first part, then the next... -- like you
> just did; you didn't stumble or get confused either.
But you cannot mechanize this letting your program "mentally" jump over those
tasks...
> "Dovetailing". Alternate between work on the T's and work on
> the next part of the search.
Double-clever way. Still remains, how to order the theorems according to their
operations...
>
> It might be centuries before the first one shows up; the search
> never ends; but every proof+theorem will appear at some finite time.
When I said "endless" I meant infinite of the kind "all the evens", and without a
mechanical way of jumping out of this routine...
>
> The reals, whether defined by decimals (R1), or Dedekind cuts (R2), or
> Cauchy sequences (R3), or integer sequences (R4)... etc have the exact
> same cardinality; there is a 1-1 onto mapping between any two R's.
I can "mentally crowd" my geometrical line with many more dots than needed for R1 or
her sisters, or on the contrary think it a bit scarce for them.
>
> Eh... isn't 1/2, or any 1/n finitely describable?! And 0 is
> not one of them anyway; it just happens to satisfy the definition of
> limit. I don't know what you are objecting to.
Sorry, but I do not mean members from the series 1/n, I do mean the creatures
waiting at the end of that. They crowd a nest out of which we "observe" only zero. Or could
you give me a series whose limit is "the closest neighbour of zero"? No, that last one hides
in the nest...
>
> Ilias
(Sorry I did not join the original pars on which you commented, but this software would not
accept a reply with more included text than new text - how could you anyway?)
Greetings,
Guido
Jon Haugsand
ika...@alumnae.caltech.edu (Ilias Kastanas) writes:
> >My last problem is with limits, one of the tricks to approach the reals.
> >If we state eg, that the limit of a series 1/2, 1/3, 1/4, ..., 1/n, ...
> >equals zero, then I say to myself, no, it is at best a nest of real
> >numbers that couldn't be disentangled, as crowded as you like, and
> >amongst which zero is the only member describable in finite terms.
>
>
>
> Eh... isn't 1/2, or any 1/n finitely describable?! And 0 is
> not one of them anyway; it just happens to satisfy the definition of
> limit. I don't know what you are objecting to.
What he probably means is that the *limit* of the sequence 1/2, 1/3,
... is a «nest of real numbers that couldn't be disentangled». The
number 0 is the only number in the «nest» describable in finite terms.
However, I challange the original poster to describe this nest a
little bit further.
--
Jon Haugsand
Dept. of Informatics, Univ. of Oslo, Norway, mailto:jon...@ifi.uio.no
http://www.ifi.uio.no/~jonhaug/, Pho/fax: +47-22852441/+47-22852401
ilias kastanas 08-14-90
In article <335148...@ping.be>, Guido Wuyts <Guido...@ping.be> wrote:
@Thank you for your remarks on my cry for help. I think that I became a little more convinced
@of a countable theorem construction set than I did with my up to now litterature. Some
@points remain obscure though, or I cannot find the proper way to formulate pro or contra
@them. Some of them I reply, as you did, within the text.
@
@Ilias Kastanas wrote:
@>
@> If there is one way to enumerate, there are many, yes.
@> Ah!-- call the countable anything else, ... but not "uncountable"!
@But any "successful" enumeration has to push ahead in its tasklist the elements
@whose description becomes increasingly complex, eg the rationals that become
@too "real"-like, the elements bordering the non-rational domain...
The zig-zag through m/n's ? The higher the m, n the later you get
there; why is that a problem?
If every x in A shows up as an f(k), f is an enumeration of A; if not,
it isn't. There aren't any shades of grey in between! (Maybe you have in
mind an example where it is not easy to tell -- but it still is a yes or no
issue).
@> One mentally goes over the first part, then the next... -- like you
@> just did; you didn't stumble or get confused either.
@ But you cannot mechanize this letting your program "mentally" jump
@over those tasks...
There are highly complicated wellorderings of w ( = {0, 1, ...} ).
Of course if you _know_ X is a w.o. of w then you know it is countable
without performing any tasks to "read" it! If not, then sure, it may take
some effort to recognize what it is. "Mechanize" is a separate matter,
though. If you only use computable functions on w, you can recognize/
describe some of the w.o.'s of w, but not all; countable w.o.'s beyond a
certain level of complexity are out of reach.
Here we have strayed off cardinality, and gotten to ordinals... and
effectiveness.
@> "Dovetailing". Alternate between work on the T's and work on
@> the next part of the search.
@
@ Double-clever way. Still remains, how to order the theorems according
@to their operations...
You don't really _need_ dovetailing for that; you have an algorithm
that produces any theorem at some finite time... so as they come up you
classify them. You cannot do better; no algorithm can tell you whether
an arbitrary formula is or isn't a theorem.
Again, this is not just cardinality.
@> It might be centuries before the first one shows up; the search
@> never ends; but every proof+theorem will appear at some finite time.
@ When I said "endless" I meant infinite of the kind "all the evens", and without a
@mechanical way of jumping out of this routine...
Dovetailing does show you don't have to be stuck. There may be
infinitely many tasks t1, t2, ... to perform; carry out 1 step of t1,
then 1 of t1 and 1 of t2, then 1 of t1, 1 of t2, 1 of t3...
@> The reals, whether defined by decimals (R1), or Dedekind cuts (R2), or
@> Cauchy sequences (R3), or integer sequences (R4)... etc have the exact
@> same cardinality; there is a 1-1 onto mapping between any two R's.
@ I can "mentally crowd" my geometrical line with many more dots than needed for R1 or
@her sisters, or on the contrary think it a bit scarce for them.
No doubt, but your line wouldn't then be the reals.
@> Eh... isn't 1/2, or any 1/n finitely describable?! And 0 is
@> not one of them anyway; it just happens to satisfy the definition of
@> limit. I don't know what you are objecting to.
@ Sorry, but I do not mean members from the series 1/n, I do mean the creatures
@waiting at the end of that. They crowd a nest out of which we "observe" only zero. Or could
@you give me a series whose limit is "the closest neighbour of zero"? No, that last one hides
@in the nest...
Not in R; 0 is the only limit here (all limits in R are unique). You
can build nonstandard models extending R, where every x in R has a "cloud"
around it, x + o, for o in I, the Infinitesimals. Maybe that is the 'nest'!
But you are not in R then. And there still isn't a "closest neighbor of 0"
...
One might find wilder constructions yet... and might like such things
and enjoy studying them. This doesn't change the properties of R, however,
or make them problematic.
Ilias
G. Wuyts
Jon Haugsand <jon...@ejkthyrnir.ifi.uio.no> writes:
> ika...@alumnae.caltech.edu (Ilias Kastanas) writes:
>
> > >My last problem is with limits, one of the tricks to approach the reals.
> > >If we state eg, that the limit of a series 1/2, 1/3, 1/4, ..., 1/n, ...
> > >equals zero, then I say to myself, no, it is at best a nest of real
> > >numbers that couldn't be disentangled, as crowded as you like, and
> > >amongst which zero is the only member describable in finite terms.
> >
> >
> >
>
> ... is a «nest of real numbers that couldn't be disentangled». The
> number 0 is the only number in the «nest» describable in finite terms.
>
> However, I challange the original poster to describe this nest a
> little bit further.
>
> --
> Jon Haugsand
The original poster thanks you for almost exactly producing the
correction as he put it in a previous reply, which he/I repeat hereafter:
" Sorry, but I do not mean members from the series 1/n, I do mean the creatures
waiting at the end of that. They crowd a nest out of which we "observe" only zero. Or could
you give me a series whose limit is "the closest neighbour of zero"? No, that last one hides
in the nest... "
So the limit is the nest, not the series. What I mean is that the "real real"
numbers, which are not describable in finite terms, do seem to me neither
describable in terms of a series, infinite though it be, of elements with finite
description each.
You could say, but look at, say, a series defining pi. So what, pi is describable in
finite geometrical terms, but not as a decimal number: it is no less a nest
than my "zero" example... My point is that the decisive distinction between
(two "nearest") real numbers starts only after the aleph0-th decimal position,
that is at the very point where the tentative series comes to an end!
Greetings, Guido
Guido Wuyts
Jon Haugsand <jon...@ejkthyrnir.ifi.uio.no> writes:
> ika...@alumnae.caltech.edu (Ilias Kastanas) writes:
>
> > >If we state eg, that the limit of a series 1/2, 1/3, 1/4, ..., 1/n, ...
> > >equals zero, then I say to myself, no, it is at best a nest of real
> > >numbers that couldn't be disentangled, as crowded as you like, and
> > >amongst which zero is the only member describable in finite terms.
> >
> >
> >
> > Eh... isn't 1/2, or any 1/n finitely describable?! And 0 is
> > not one of them anyway; it just happens to satisfy the definition of
> > limit. I don't know what you are objecting to.
>
ilias kastanas 08-14-90
In article <5j0su1$g...@news2.belgium.eu.net>,
G. Wuyts <Guido...@ping.be> wrote:
@Jon Haugsand <jon...@ejkthyrnir.ifi.uio.no> writes:
@> ika...@alumnae.caltech.edu (Ilias Kastanas) writes:
@>
@> > >My last problem is with limits, one of the tricks to approach the reals.
@> > >If we state eg, that the limit of a series 1/2, 1/3, 1/4, ..., 1/n, ...
@> > >equals zero, then I say to myself, no, it is at best a nest of real
@> > >numbers that couldn't be disentangled, as crowded as you like, and
@> > >amongst which zero is the only member describable in finite terms.
@> >
@> >
@> > Eh... isn't 1/2, or any 1/n finitely describable?! And 0 is
@> > not one of them anyway; it just happens to satisfy the definition of
@> > limit. I don't know what you are objecting to.
@>
@> What he probably means is that the *limit* of the sequence 1/2, 1/3,
@> ... is a «nest of real numbers that couldn't be disentangled». The
@> number 0 is the only number in the «nest» describable in finite terms.
@>
@> However, I challange the original poster to describe this nest a
@> little bit further.
@>
@> --
@> Jon Haugsand
@
@The original poster thanks you for almost exactly producing the
@correction as he put it in a previous reply, which he/I repeat hereafter:
@" Sorry, but I do not mean members from the series 1/n, I do mean the creatures
@waiting at the end of that. They crowd a nest out of which we "observe" only zero. Or could
@you give me a series whose limit is "the closest neighbour of zero"? No, that
@last one hides in the nest... "
@
@So the limit is the nest, not the series. What I mean is that the "real real"
@numbers, which are not describable in finite terms, do seem to me neither
@describable in terms of a series, infinite though it be, of elements with
@finite description each.
Since 0.a1, 0.a1 a2, ... does describe x = 0.a1 a2 a3 ..., you must
be thinking of sequences of a's longer than x.
@You could say, but look at, say, a series defining pi. So what, pi is describable in
@finite geometrical terms, but not as a decimal number: it is no less a nest
Pi is 2q, 0 < q < 2, 1 - (q^2)/2! + (q^4)/4! - ... = 0 (cos(q) = 0)
and its decimal digits can of course be found.
@than my "zero" example... My point is that the decisive distinction between
@(two "nearest") real numbers starts only after the aleph0-th decimal position,
@that is at the very point where the tentative series comes to an end!
Well, there is no aleph-0 position; aleph-0 means "all a_n, n in N".
But you may consider aleph-0 "plus 1": W = 0.a1 a2 ...; b1 -- or more b's.
What intuition leads you to this? (if in fact this is what you are saying)
For one thing, are W and V = 0.a1 a2 ...; c1 "nearest" reals?
It is not clear how such things work. What is 0. ...; 6 + 0. ...; 7
say?
Sequences longer than aleph-0 (especially binary ones) have their
place; but why should _they_ be "the reals"?
Ilias
Guido Wuyts
ilias kastanas 08-14-90 wrote:
>
> In article <5j0su1$g...@news2.belgium.eu.net>,
> G. Wuyts <Guido...@ping.be> wrote:
> @Jon Haugsand <jon...@ejkthyrnir.ifi.uio.no> writes:
> @> ika...@alumnae.caltech.edu (Ilias Kastanas) writes:
> @(two "nearest") real numbers starts only after the aleph0-th decimal position,
> @that is at the very point where the tentative series comes to an end!
>
> Well, there is no aleph-0 position; aleph-0 means "all a_n, n in N".
> But you may consider aleph-0 "plus 1": W = 0.a1 a2 ...; b1 -- or more b's.
> What intuition leads you to this? (if in fact this is what you are saying)
> For one thing, are W and V = 0.a1 a2 ...; c1 "nearest" reals?
>
> It is not clear how such things work. What is 0. ...; 6 + 0. ...; 7
> say?
>
> Sequences longer than aleph-0 (especially binary ones) have their
> place; but why should _they_ be "the reals"?
>
> Ilias
The problem which I argued occurs "after" aleph0, as a matter of fact
dooms whilest the decimal series unrolls: it raises there the question
whether two series like, say:
.1 .01 .001 .0001 .... and .9 .09 .009 .0009 ...
converge "as a matter of fact" to the same number, zero;
statement which I allow myself to put in question, as doing so seems to
me the only way in the decimal technique to keep a perspective for
numbers that are irrational and yet keep their hold in an immediate,
ever so tiny as you wish, vicinity around that very zero.
David Ullrich
You keep talking about this cloud of real numbers clustering
around 0, in spite of the fact that everybody keeps telling you that
there simply are no such (real) numbers. What makes you so certain
that these numbers exist????
--
David Ullrich
?his ?s ?avid ?llrich's ?ig ?ile
(Someone undeleted it for me...)
Guido Wuyts
David Ullrich wrote:
(...............)
> > The problem which I argued occurs "after" aleph0, as a matter of fact
> > dooms whilest the decimal series unrolls: it raises there the question
> > whether two series like, say:
> > .1 .01 .001 .0001 .... and .9 .09 .009 .0009 ...
> > converge "as a matter of fact" to the same number, zero;
> >
> > statement which I allow myself to put in question, as doing so seems to
> > me the only way in the decimal technique to keep a perspective for
> > numbers that are irrational and yet keep their hold in an immediate,
> > ever so tiny as you wish, vicinity around that very zero.
>
> You keep talking about this cloud of real numbers clustering
> around 0, in spite of the fact that everybody keeps telling you that
> there simply are no such (real) numbers. What makes you so certain
> that these numbers exist????
> --
> David Ullrich
Well, nobody seems to be able to accept the fact that there are,
rational as well as irrational, numbers that you can pinpoint by some
description (geometric or decimal, or as a converging series...), but
that there are (to my taste many more) others that you can't.
Even for the rationals, whatever the order of counting down you may
invent, some numbers will always hide beyond the "...etcetera" you find
yourself at last putting behind your list. And what about the
irrationals in their immediate vicinity...?
Guido
Dave Seaman
In article <E91Fu...@clw.cs.man.ac.uk>,
Charles Lindsey <c...@clw.cs.man.ac.uk> wrote:
>In <335690...@ping.be> Guido Wuyts <Guido...@ping.be> writes:
>
>>David Ullrich wrote:
>>> You keep talking about this cloud of real numbers clustering
>>> around 0, in spite of the fact that everybody keeps telling you that
>>> there simply are no such (real) numbers. What makes you so certain
>>> that these numbers exist????
>
>>rational as well as irrational, numbers that you can pinpoint by some
>>description (geometric or decimal, or as a converging series...), but
>>that there are (to my taste many more) others that you can't.
By the way, I didn't see any reply to this. There is nothing here that
is inconsistent with the standard real numbers, but I get the
impression that more was being claimed here than meets the eye -- such
as the existence of infinitesimals. Those exist in nonstandard
analysis, but not in the standard real numbers.
>Here is a related question that I have often wondered about.
>
>Consider the reals between 0 and 1 laid out on a straight line. Drag a pin
>along the line. Then seemingly you pass through ALL the real numbers: the
>rational numbers (all beth(0)[1] of them), the algebraic numbers (all beth(0)
>of them) and the transcendental numbers (all beth(1) of them).
>
>Now pick a number with your pin. Is it possible (indeed probable) that you
>will pick a transcendental one?
Yes. In fact, you will pick a transcendental with probability one
(assuming a uniform probability density). The algebraic numbers are
countable, and therefore have measure zero.
>Is it perhaps correct to argue that it depends on the diameter of the point of
>your pin? If that diameter is 1/beth(0), then maybe it is too broad to
>distinguish one transcendental number from another. If it is of diameter
>1/beth(1) then it might work.
I thought religious discussions were usually more concerned with the
other end of the pin. In mathematics, we don't use pins. We just
select a point at random, using a specified probability distribution.
The size of a point is exactly zero, not 1/beth(0) or 1/beth(1), since
those are not real numbers.
The probability of selecting a point p in [0,1] is the integral
/ p
|
P = | 1 dx = 0
|
/ p
exactly. There are no infinitesimals involved in the evaluation of the
integral.
>Essentially, I am proposing a series of ever more infinitesimal zeroes -
>1/beth(0), 1/beth(1), i/beth(2) ...
You have a bit of a problem here in that the transfinite cardinals
cannot be embedded in a field and still retain their usual rules such
as beth(0)+beth(0)=beth(0), but beth(0) != 0. That doesn't mean
infinitesimals can't be defined; they just don't have anything
particularly to do with the transfinite cardinals. Probability theory
doesn't need infinitesimals, and in previous incarnations of this
discussion, no one has described a way that infinitesimals could even
be made useful in probability theory.
--
Dave Seaman dse...@purdue.edu
++++ stop the execution of Mumia Abu-Jamal ++++
++++ if you agree copy these lines to your sig ++++
++++ see http://www.xs4all.nl/~tank/spg-l/sigaction.htm ++++
Charles Lindsey
>David Ullrich wrote:
>> You keep talking about this cloud of real numbers clustering
>> around 0, in spite of the fact that everybody keeps telling you that
>> there simply are no such (real) numbers. What makes you so certain
>> that these numbers exist????
> Well, nobody seems to be able to accept the fact that there are,
>rational as well as irrational, numbers that you can pinpoint by some
>description (geometric or decimal, or as a converging series...), but
>that there are (to my taste many more) others that you can't.
Here is a related question that I have often wondered about.
Consider the reals between 0 and 1 laid out on a straight line. Drag a pin
along the line. Then seemingly you pass through ALL the real numbers: the
rational numbers (all beth(0)[1] of them), the algebraic numbers (all beth(0)
of them) and the transcendental numbers (all beth(1) of them).
Now pick a number with your pin. Is it possible (indeed probable) that you
will pick a transcendental one?
Is it perhaps correct to argue that it depends on the diameter of the point of
your pin? If that diameter is 1/beth(0), then maybe it is too broad to
distinguish one transcendental number from another. If it is of diameter
1/beth(1) then it might work.
Essentially, I am proposing a series of ever more infinitesimal zeroes -
1/beth(0), 1/beth(1), i/beth(2) ...
Is this meaningful (i.e. is division of cardinal numbers sufficiently well
defined for my purpose)? Indeed, is this an already well-researched topic?
[1] beth(0) = aleph(0)
beth(n+1) = 2**beth(n)
--
Charles H. Lindsey ---------At Home, doing my own thing-------------------------
Email: c...@clw.cs.man.ac.uk Web: http://www.cs.man.ac.uk/~chl
Voice/Fax: +44 161 437 4506 Snail: 5 Clerewood Ave, CHEADLE, SK8 3JU, U.K.
PGP: 2C15F1A9 Fingerprint: 73 6D C2 51 93 A0 01 E7 65 E8 64 7E 14 A4 AB A5
Ilias Kastanas
In article <335690...@ping.be>, Guido Wuyts <Guido...@ping.be> wrote:
>David Ullrich wrote:
>(...............)
>> > The problem which I argued occurs "after" aleph0, as a matter of fact
>> > dooms whilest the decimal series unrolls: it raises there the question
>> > whether two series like, say:
>> > .1 .01 .001 .0001 .... and .9 .09 .009 .0009 ...
>> > converge "as a matter of fact" to the same number, zero;
They do; what would be an argument to the contrary?
>> > statement which I allow myself to put in question, as doing so seems to
>> > me the only way in the decimal technique to keep a perspective for
>> > numbers that are irrational and yet keep their hold in an immediate,
>> > ever so tiny as you wish, vicinity around that very zero.
>>
>> You keep talking about this cloud of real numbers clustering
>> around 0, in spite of the fact that everybody keeps telling you that
>> there simply are no such (real) numbers. What makes you so certain
>> that these numbers exist????
>> --
>> David Ullrich
> Well, nobody seems to be able to accept the fact that there are,
>rational as well as irrational, numbers that you can pinpoint by some
>description (geometric or decimal, or as a converging series...), but
>that there are (to my taste many more) others that you can't.
If you intend digit sequences "longer than aleph_0", simply _wanting_
them to exist in R is not enough. How do they add/subtract?
When this simplest of questions has no answer, you may draw some
conclusions.
>Even for the rationals, whatever the order of counting down you may
>invent, some numbers will always hide beyond the "...etcetera" you find
>yourself at last putting behind your list. And what about the
>irrationals in their immediate vicinity...?
You can map pairs (m, n), and hence the rationals, to 2^(m+1) *
* 3^(n+1), 1-1 into N and with room to spare. No etceteras... or rascals
hiding behind them!
Ilias
Mike McCarty
)rational as well as irrational, numbers that you can pinpoint by some
)description (geometric or decimal, or as a converging series...), but
)that there are (to my taste many more) others that you can't.
)
)Even for the rationals, whatever the order of counting down you may
)invent, some numbers will always hide beyond the "...etcetera" you find
)yourself at last putting behind your list. And what about the
)irrationals in their immediate vicinity...?
)
)Guido
The rationals may be explicitly listed in their entirety. No "etcetera"
tacked onto the end. In other words, one may explicitly give a function
f(.) such that for each non-negative integer n, f(n) is a rational
number, and furthermore that each rational number occurs as a value of
f(.) for exactly one non-negative integer. When that has been done, then
each rational is exactly pinpointed by the index number n associated
with it. Every one of them has been given exactly one name. Such
functions may be written as explicit formulas.
Mike
--
----
char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
I don't speak for DSC. <- They make me say that.
Dik T. Winter
In article <5jm8a8$c...@sun001.spd.dsccc.com> jmcc...@sun1307.spd.dsccc.com (Mike McCarty) writes:
> The rationals may be explicitly listed in their entirety. No "etcetera"
> tacked onto the end. In other words, one may explicitly give a function
> f(.) such that for each non-negative integer n, f(n) is a rational
> number, and furthermore that each rational number occurs as a value of
> f(.) for exactly one non-negative integer. When that has been done, then
> each rational is exactly pinpointed by the index number n associated
> with it. Every one of them has been given exactly one name. Such
> functions may be written as explicit formulas.
I challenge you to give such an explicit function f(.). I am afraid that
a Usenet article is too small to contain it. What can be done is write
a function f(.) that maps each integer onto a rational number, moreover,
all rational numbers will be mapped onto by at least one integer. So
each rational is pinpointed by at least one index number n, but there may
be more index numbers. This is, however, sufficient to prove that the
rationals are countable. (A similar argument applies to algebraic numbers.)
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Mike McCarty
In article <E97zI...@cwi.nl>, Dik T. Winter <d...@cwi.nl> wrote:
)In article <5jm8a8$c...@sun001.spd.dsccc.com> jmcc...@sun1307.spd.dsccc.com (Mike McCarty) writes:
) > The rationals may be explicitly listed in their entirety. No "etcetera"
) > tacked onto the end. In other words, one may explicitly give a function
) > f(.) such that for each non-negative integer n, f(n) is a rational
) > number, and furthermore that each rational number occurs as a value of
) > f(.) for exactly one non-negative integer. When that has been done, then
) > each rational is exactly pinpointed by the index number n associated
) > with it. Every one of them has been given exactly one name. Such
) > functions may be written as explicit formulas.
)
)I challenge you to give such an explicit function f(.). I am afraid that
)a Usenet article is too small to contain it. What can be done is write
)a function f(.) that maps each integer onto a rational number, moreover,
)all rational numbers will be mapped onto by at least one integer. So
)each rational is pinpointed by at least one index number n, but there may
)be more index numbers. This is, however, sufficient to prove that the
)rationals are countable. (A similar argument applies to algebraic numbers.)
)--
)dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
)home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Well, Dik, you are wrong. I have seen a function which does exactly what
I describe. It was not all that complex. However, I don't have a copy of
it handy. I could work one out, but I'm not that interested in expending
the necessary amount of energy.
In any case, =whatever= the complexity of the explicit formula, it
exists. Which is all that I stated.
Dr D F Holt
In article <E97zI...@cwi.nl>,
d...@cwi.nl (Dik T. Winter) writes:
>In article <5jm8a8$c...@sun001.spd.dsccc.com> jmcc...@sun1307.spd.dsccc.com (Mike McCarty) writes:
> > The rationals may be explicitly listed in their entirety. No "etcetera"
>
I don't understand what the problem is here.
Suppose that you give me a surjective function g: N -> Q.
Then I define a bijective function f: N -> Q by
f(0) = g(0).
f(n+1) = g(m), where m is the least positive integer such that
g(m) is not equal to f(r) for any r with 0 <= r <= n.
Are you saying that this is not an explicit definition of f?
From the point of view of standard mathematics, based on set theory,
this is a perfectly valid recursive definition.
Maybe you were thinking of a non-recursive definition, but that is not
what the original poster claimed. Mind you, it would be an intersting
challenge to try and construct one.
Derek Holt.
Mike McCarty
In article <5k4fiv$n...@crocus.csv.warwick.ac.uk>,
Dr D F Holt <ma...@csv.warwick.ac.uk> wrote:
)In article <E97zI...@cwi.nl>,
) d...@cwi.nl (Dik T. Winter) writes:
)>In article <5jm8a8$c...@sun001.spd.dsccc.com> jmcc...@sun1307.spd.dsccc.com (Mike McCarty) writes:
)> > The rationals may be explicitly listed in their entirety. No "etcetera"
)> > tacked onto the end. In other words, one may explicitly give a function
)> > f(.) such that for each non-negative integer n, f(n) is a rational
)> > number, and furthermore that each rational number occurs as a value of
)> > f(.) for exactly one non-negative integer. When that has been done, then
)> > each rational is exactly pinpointed by the index number n associated
)> > with it. Every one of them has been given exactly one name. Such
)> > functions may be written as explicit formulas.
)>
)>I challenge you to give such an explicit function f(.). I am afraid that
)>a Usenet article is too small to contain it. What can be done is write
)>a function f(.) that maps each integer onto a rational number, moreover,
)>all rational numbers will be mapped onto by at least one integer. So
)>each rational is pinpointed by at least one index number n, but there may
)>be more index numbers. This is, however, sufficient to prove that the
)>rationals are countable. (A similar argument applies to algebraic numbers.)
)
)I don't understand what the problem is here.
)Suppose that you give me a surjective function g: N -> Q.
)Then I define a bijective function f: N -> Q by
)
)f(0) = g(0).
)f(n+1) = g(m), where m is the least positive integer such that
) g(m) is not equal to f(r) for any r with 0 <= r <= n.
)
)Are you saying that this is not an explicit definition of f?
)From the point of view of standard mathematics, based on set theory,
)this is a perfectly valid recursive definition.
)
)Maybe you were thinking of a non-recursive definition, but that is not
)what the original poster claimed. Mind you, it would be an intersting
)challenge to try and construct one.
Actually, I -have- seen an explicit formula f(n).
David Petry
d...@cwi.nl (Dik T. Winter) wrote:
>In article <5jm8a8$c...@sun001.spd.dsccc.com> jmcc...@sun1307.spd.dsccc.com (Mike McCarty) writes:
> > The rationals may be explicitly listed in their entirety. No "etcetera"
> > tacked onto the end. In other words, one may explicitly give a function
> > f(.) such that for each non-negative integer n, f(n) is a rational
> > number, and furthermore that each rational number occurs as a value of
> > f(.) for exactly one non-negative integer.
>I challenge you to give such an explicit function f(.). I am afraid that
>a Usenet article is too small to contain it.
Define f(A/B) as A^2 B^2 / product {p_i} where {p_i} is the set of prime
factors of B. Then f is a bijection between the integers and the rationals.
Ilias Kastanas
>> > The rationals may be explicitly listed in their entirety. No "etcetera"
>> > tacked onto the end. In other words, one may explicitly give a function
>> > f(.) such that for each non-negative integer n, f(n) is a rational
>> > number, and furthermore that each rational number occurs as a value of
>>
>
>
>f(n+1) = g(m), where m is the least positive integer such that
>
>
The question is how "simple" can f be; e.g. this or that weak fragment
of PA suffices.
For g: NxN -> N, it's as simple as it gets: g(m,n) = (m+n)^2 +3m+n /2,
just operations. For the inverses, let s() be the arithm. square root,
s(z)^2 <= z < (s(z)+1)^2. Then m(x) = 8x+1-s(8x+1)^2 /8 , and n(x) =
= s(8x+1)-(2m(x)+1) /2.
It takes little more to handle (NxN)xN, etc... and even N^(<N), all
finite sequences, by including in the code the sequence length.
Now Q+ seems to need some extra complexity. We might let r = m/n
with m, n rel. prime, and use prime factorization. I prefer reduction to
N^(<N), though; express r as a continued fraction, a0 +1/ a1+1/ ... a_n.
Some care is needed, as the a's are >=1 (not 0), and for uniqueness a_n >=2;
we shift them accordingly.
Ilias