# No Identity Bijection for Omega - Theory and Concepts

This is a discussion on No Identity Bijection for Omega - Theory and Concepts ; lwalke3@lausd.net wrote: &gt; &gt; On Sep 4, 4:08 pm, MoeBlee &lt;jazzm...@hotmail.com&gt; wrote: &gt; &gt; On Sep 4, 4:02 pm, lwal...@lausd.net wrote: &gt; &gt; &gt; &gt; &gt; When it comes to the various so-called &quot;cranks,&quot; &gt; &gt; &gt; I usually try ...

1. ## Re: No Identity Bijection for Omega

lwalke3@lausd.net wrote:
>
> On Sep 4, 4:08 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > On Sep 4, 4:02 pm, lwal...@lausd.net wrote:
> >
> > > When it comes to the various so-called "cranks,"
> > > I usually try to find some axiomatization that
> > > may suit their intuitions better.

> >
> > What axiomatizations have you found?
> >
> > MoeBlee

>
> For RE, none. Some so-called "cranks" make statements
> that resemble other axiomatizations, but in the case of
> RE, who refers to sets which lack identity bijections,
> no axiomatization will help him there.
>
> I suspect that what RE wants is to argue that set
> theorists should use some other criterion other than the
> existence of a bijection between two sets to define a
> nonstandard notion of set size. But of course, whenever
> anyone suggests this, the standard mathematicians
> this by trying to show that there exists sets which lack
> identity bijections. Thus RE "proves" that there's a set
> which doesn't have the same set size as itself, which is
> certainly a nonintuitive idea. So therefore one should
> not use bijections to determine set size. Of course,
> you already point out that RE's "proof" fails in Z.
>
> The argument given by "cranks" other than RE is that
> some sets, such as omega, are Dedekind infinite -- that
> is, they admit bijections between themselves and their
> own proper subsets. This is contrary to their intuition
> that a set should have a strictly larger set size than all
> of its proper subsets. We are reminded of Euclid's
> fifth axiom (not his fifth _postulate_, but _axiom_):
>
> -- The whole is greater than the part.
>
> Let us abbreviate Euclid's fifth axiom by EA5. So the
> "cranks" (including possibly RE) want a set theory in
> which EA5 holds, yet ~EA5 is a theorem of ZFC --
> since any Dedekind infinite set isn't _greater_ than
> some of its parts (proper subsets), but may have the
> same cardinality.
>
> Notice that EA5 is a theorem of ZF-Infinity+~Infinity,
> known as the Pigeonhole Principle. So to some
> finitists, ZF-I+~I is more intuitive than ZF. Other
> "cranks" want there to exist sets that are larger than
> any standard finite set, yet EA5 still holds -- that is,
> for there to exist sets larger than any finite set, yet
> for _every_ set, including these sets larger than any
> finite set, to have a strictly larger size than all of its
> proper subsets.
>
> All of this discussion about "internal sets" and
> hyperreals and the Transfer Principle is an attempt
> to construct a set theory in which there exist sets
> larger than any finite set, yet every set is larger
> than its proper subsets. In other words, to finally
> formalize Euclid's fifth axiom.

A lucid contribution to the field of crankology.

--
hz

2. ## Re: No Identity Bijection for Omega

On Sep 5, 4:10 pm, herbzet <herb...> wrote:

> A lucid contribution to the field of crankology.

While I think it very much obscures and misses the point of the
question that motivated it, which is a question that I believe to go
right to the heart of the subject itself, especially to the heart of
the poster's own contention about the subject.

MoeBlee

3. ## Re: No Identity Bijection for Omega

MoeBlee wrote:
>
> herbzet wrote:
>
> > A lucid contribution to the field of crankology.

>
> While I think it very much obscures and misses the point of the
> question that motivated it, which is a question that I believe to go
> right to the heart of the subject itself, especially to the heart of
> the poster's own contention about the subject.

What do you think motivates set-theory type cranks?

--
hz

4. ## Re: No Identity Bijection for Omega

On Sep 5, 9:51 pm, herbzet <herb...> wrote:
> MoeBlee wrote:
>
> > herbzet wrote:

>
> > > A lucid contribution to the field of crankology.

>
> > While I think it very much obscures and misses the point of the
> > question that motivated it, which is a question that I believe to go
> > right to the heart of the subject itself, especially to the heart of
> > the poster's own contention about the subject.

>
> What do you think motivates set-theory type cranks?

That's a question that, in its more broad sense, has fascinated me for
a couple of years now. But I have only tentative thoughts that are

motivation or whether you've just asking what are the particular
objections cranks have to set theory. As to the latter, I think
there's enough said by the cranks themselves that might as well be
taken at face value as a true representation of what is on their mind
in that regard. Things such as failure of the part-whole principle
(which was mentioned by lwal), the "non-computational" nature of
infinitistic statements, etc. So, recognizing those kinds of
objections, as they are plainly enough stated, does answer the
question in that routine sense. But the question that more interests
me is why cranks choose to exercise their conceptual differences with
set theory in a crank way rather than in a mathematically coherent
way. In other words, I don't think it's crank just to challenge set
theory; but what is crank to me is the WAY cranks challenge set
theory. And the question of why cranks choose to challenge set theory
in crank ways seems to me to a be a psychological question, one
regarding human motivations that are more basic even than mere
disagreements about matters of mathematics or even of philosophy of
mathematics.

MoeBlee

5. ## Re: No Identity Bijection for Omega

On Sep 4, 7:15 am, Mitch <maha...> wrote:
> OK...I'll chime in, too.
>
> On Sep 3, 10:28 pm, logic...@comcast.net wrote:
>
>
>
> > There is no identity bijection for Omega.

>
> Why do you bother with this bijection stuff? Just cut out all the
> ordered pair confusion and prove that w doesn't exist. I think that's
> what your proof is really trying to say. Something like:
>
> Assume k = w for some k.
> k = k and k must be the largest natural number.
> Therefore, no k = w.
>
> Assume there exists a k that is w.
> Then there exists k.
> ...
> There is no Omega.
>
> I think that kind of proof is something we could all work with.
>
> Mitch

Actually, I presented a proof like that some time ago.
One can show in ZF-I that every set that is provably an ordinal
must have a largest element.

I was told that showing every ordinal has a largest element
is not enough to prove that some set,
a set that can't be proven to be an ordinal in ZF-I,
might be an ordinal without a largest element.

I find it incredulous ZF without the Axiom of Infinity
is incapable of proving every natural number is finite.
If there is such a proof in ZF-I, I can easily use it to prove
omega doesn't exist.

or at least very counter-intuitive results.

S = w U {w}.
a_i = the set of elements of S that are less than i.
b_i = the set of elements of S that are less than or equal to i.

a_0 = {}
a_1 = {0}
a_2 = {0,1}
....
a_w = {0,1,2,...}

b_0 = {0}
b_1 = {0,1}
b_2 = {0,1,2}
....
b_w = {0,1,2,...,w}

A = {0,1,2,...,w}
B = {1,2,3,...,w+1}

Let C be the set of all elements of A that are not in B
and let D be the set of all elements of B that are not in A:

C = {0, w}
D = (w+1}

(Omega can not be a member of B because if b_z = w,
z would be the largest natural number)

If sets A and B have the same number of elements,
how can there be two elements of A not in B,
but only one element of B not in A?
I can easily create an "identity" bijection between A and B.

Let H be the set of ordered pairs <i,j> such that
i = a_m, j= b_n and a_m = b_n.

H = { <a_1, b_0>, <a_2, b_1>, <a_3, b_2>, ... }
H = { <1,1>, <2,2>, <3,3>, ... }

a_0, a_w, and b_w are not members of any element of H.

Let H' = H U { <a_0, b_w> }
H' = { <0,w+1>, <1,1>, <2,2>, ... }

H' is not a bijection between A and B because a_w
is not a member of any element of H'.

How can H' not be a bijection?
Why isn't <w,z> a member of H'?

The usual excuse involves order types,
but I don't see how order types applies to this proof.
I don't care how my (almost) identity bijection is ordered.

Clearly, every element of B is paired with an element of A.
How can A have exactly one element left over?
How can there not be a member of B such that b_z = a_w?

Russell
- 2 many 2 count

6. ## Re: No Identity Bijection for Omega

logiclab@comcast.net wrote:
....
> I find it incredulous ZF without the Axiom of Infinity
> is incapable of proving every natural number is finite.
> If there is such a proof in ZF-I, I can easily use it to prove
> omega doesn't exist.

Remember that ZF-I merely deletes one axiom from ZF. It does not add
anything. If you can prove anything in ZF-I that you cannot prove
equally well in ZF you are doing something wrong.

> or at least very counter-intuitive results.

A contradiction and a counter-intuitive result are completely different
things.

I knew a very counter-intuitive result in set theory when I was
fourteen. The result was the existence of a bijection between the
natural numbers and the even natural numbers, despite the fact that half
the natural numbers do not appear in the even natural numbers.

If I could prove a contradiction in ZF I would immediately apply to
switch my Ph.D. subject from computer science to mathematics and submit
the proof as my dissertation.

Patricia

7. ## Re: No Identity Bijection for Omega

On Sep 7, 12:14 am, logic...@comcast.net wrote:
> Actually, I presented a proof like that some time ago.

No, you didn't.

> One can show in ZF-I that every set that is provably an ordinal
> must have a largest element.

You don't know the DEFINITION of ordinal.
Ordinals BY DEFINITION include a kind of ordinal called a LIMIT
ordinal.
Limit ordinals BY DEFINITION do NOT have a largest element.
What is provable is IRRELEVANT: IT'S A *DEFINITION* !!

8. ## Re: No Identity Bijection for Omega

On 5 Sep., 08:33, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au>
wrote:
> "MoeBlee" <jazzm...@hotmail.com> wrote in message
>
>
> > On Sep 4, 4:02 pm, lwal...@lausd.net wrote:

>
> >> When it comes to the various so-called "cranks,"
> >> I usually try to find some axiomatization that
> >> may suit their intuitions better.

>
> > What axiomatizations have you found?

>
> > MoeBlee

>
> ZF minus the Axiom of Infinity seems to work for most of them. I have never
> seen a crank post on finite sets.

Recently, it was doubted whether {i,j} had a maximum (i,j natural
numbers).

9. ## Re: No Identity Bijection for Omega

On Sep 7, 12:14 am, logic...@comcast.net wrote:
> One can show in ZF-I that every set that is provably an ordinal
> must have a largest element.

That's a slight mis-location of "in". Showing that *IN* ZF-I
requires a lot of fancy encoding of proof-predicates.
What you mean is that you can show it ABOUT ZF-I,
using ZF+I+C.

> I was told that showing every ordinal has a largest element

Well, that is COMPLETELY irrelevant -- you CAN'T -- not in
Z, or ZFC, or Z-I, or anything like that, prove that every ordinal
has a largest element. RATHER, you can prove that every set
THAT YOU CAN *PROVE* IS AN ORDINAL has a largest element.
Something might BE an ordinal withOUT your being able to PROVE it was
one!

> is not enough to prove that some set,
> a set that can't be proven to be an ordinal in ZF-I,
> might be an ordinal without a largest element.

Exactly. Did you not understand what you were told?

> I find it incredulous ZF without the

No, you ARE increduLOUS. You find IT incredIBLE.

> Axiom of Infinity is incapable of proving every natural number is finite.

It's not any better WITH the axiom of infinity. JUST WHAT IS a
natural
number anyway? Just what does "finite" mean anyway? The main reason
why this is normally easy to prove is that in the Z context, "natural
number"
is simply DEFINED as "finite ordinal". If you have any other
definition for either
term then nobody, not ZFC OR ANYTHING ELSE EITHER, will EVER be able
to prove this.

> If there is such a proof in ZF-I, I can easily use it to prove
> omega doesn't exist.

As you've already had explained to you, if there is no proof from ZF,
then there certainly can't be a proof from any SMALLER axiom set
(like Z-I). Since there is no proof in first-order logic PERIOD,
though,
you have nothing to worry about.

10. ## different sizes of denumerable sets?

On Sep 7, 9:12 am, Patricia Shanahan <p...@acm.org> wrote:
> I knew a very counter-intuitive result in set theory when I was
> fourteen. The result was the existence of a bijection between the
> natural numbers and the even natural numbers, despite the fact that half
> the natural numbers do not appear in the even natural numbers.
>
> If I could prove a contradiction in ZF I would immediately apply to
> switch my Ph.D. subject from computer science to mathematics and submit
> the proof as my dissertation.

I honestly think the more legitimate path toward a math dissertation
would be coming up with some kind of new numbers that would allow a
size/metric to be established for infinite subsets of natnums,
something like
probability or density. Everyone wants to say that there are twice as
many
natnums as even natnums. They don't stop wanting it until convention
beats them
down by insisting that equipollence (which works fine for finite
collections) must
also be strictly applied as the size-equivalence definition for
infinite collections as
well. Somebody at some point has to say "I think we can do better
than that".
Has this point ever yet been reached?

Obviously any number of cranks may have tried it in a newsgroup, but
I am just asking, is anybody aware of any actual papers?