No Identity Bijection for Omega

On Sep 3, 10:28 pm, logic...@comcast.net wrote:
> Let B be the identity bijection of w+1 = w U {w} .
> B is an ordered set of ordered pairs.
>
> B = < <0,0>, <1,1>, <2,2>, <3,3>, ..., <w,w> >
>

And clearly the identity bijection for w is

A = < <0,0>, <1,1>, <2,2>, <3,3>, ... >

So your proof that A does not exist must be flawed.
Hint. A does not have a last element.

> For each element, B_i, let C_i be the set of
> all B_i that appear in the bijection before B_i.
> Let C_i also includes B_i. (Skeptics - this is the one to attack.)
>
> Let D_i be the set that C_i is a bijection for.
> (D_i is the set of all natural numbers that appear in C_i).
>
> B_0 = <0,0>
> B_1 = <1,1>
> B_2 = <2,2>
> ...
>
> B_w = <w,w>
>
> C_0 = < <0,0> >
> C_1 = < <0,0>, <1,1> >
> C_2 = < <0,0>, <1,1>, <2,2> >
> ...
> C_w = < <0,0>, <1,1>, <2,2>, ..., <w,w> >
>
> D_0 = {0}
> D_1 = {0,1}
> D_2 = {0,1,2}
> ...
> D_w = {0,1,2,...,w}
>
> Assume D_k = w for some k.
> D_k = {0,1,2,...,k} and k must be the largest natural number.
> Therefore, no D_k = w.
>
> Assume there exists a C_k that is a bijection for w.
> Then there exists D_k.
>
> Assume there exists an identity bijection, A, for w.
> A must be an proper subset of B.
> A must also be an initial segment of B.

Ok to this point.

> If A is an initial segment of B then A = C_k for some k.

No. This is simply false.

If A is an initial segment of B *that has a last element*
then A = C_k for some k.


If A is an initial segment of B *that does not have a last element*
then A does not equal C_k for any k.

- William Hughes

On Sep 3, 10:28 pm, logic...@comcast.net wrote:
> Let B be the identity bijection of w+1 = w U {w} .
> B is an ordered set of ordered pairs.
>
> B = < <0,0>, <1,1>, <2,2>, <3,3>, ..., <w,w> >
>
> For each element, B_i, let C_i be the set of
> all B_i that appear in the bijection before B_i.
> Let C_i also includes B_i. (Skeptics - this is the one to attack.)
>
> Let D_i be the set that C_i is a bijection for.
> (D_i is the set of all natural numbers that appear in C_i).
>
> B_0 = <0,0>
> B_1 = <1,1>
> B_2 = <2,2>
> ...
>
> B_w = <w,w>
>
> C_0 = < <0,0> >
> C_1 = < <0,0>, <1,1> >
> C_2 = < <0,0>, <1,1>, <2,2> >
> ...
> C_w = < <0,0>, <1,1>, <2,2>, ..., <w,w> >
>
> D_0 = {0}
> D_1 = {0,1}
> D_2 = {0,1,2}
> ...
> D_w = {0,1,2,...,w}
>
> Assume D_k = w for some k.
> D_k = {0,1,2,...,k} and k must be the largest natural number.
> Therefore, no D_k = w.
>
> Assume there exists a C_k that is a bijection for w.
> Then there exists D_k.
>
> Assume there exists an identity bijection, A, for w.
> A must be an proper subset of B.
> A must also be an initial segment of B.
> If A is an initial segment of B then A = C_k for some k.
>
> There is no identity bijection for Omega.
>
> Russell
> - 2 many 2 count

da

On Sep 3, 7:28 pm, logic...@comcast.net wrote:
> Let B be the identity bijection of w+1 = w U {w} .
> B is an ordered set of ordered pairs.
>
> B = < <0,0>, <1,1>, <2,2>, <3,3>, ..., <w,w> >

No, if B is the identity function on wu{w}, then

B = {<0 0> <1 1> ... <w w>}.

I.e., curly braces on the outside.

> For each element, B_i, let C_i be the set of
> all B_i that appear in the bijection before B_i.

Then C_i = i = B_i.

> Let C_i also includes B_i. (Skeptics - this is the one to attack.)

Then C_i is not as you first described, but rather:

C_i = i+1.

> Let D_i be the set that C_i is a bijection for.
> (D_i is the set of all natural numbers that appear in C_i).

What does "that appear in" mean? If you mean 'that are elements of',
then D_i = C_i.

> B_0 = <0,0>
> B_1 = <1,1>
> B_2 = <2,2>
> ...
>
> B_w = <w,w>

That is NOT the identity function on wu{w}.

So, now we see that B is not as you first described, but rather

B = {<0 <0 0>> <1 <1 1>> ... <w <w w>>}.

Wow, why are you still not competent to just say what you mean from
the start? You've been doing your little "refute set theory" act for
how many years now? And you still don't know how to make utterly
simple formulations.

> C_0 = < <0,0> >
> C_1 = < <0,0>, <1,1> >
> C_2 = < <0,0>, <1,1>, <2,2> >
> ...
> C_w = < <0,0>, <1,1>, <2,2>, ..., <w,w> >

Again, you're using '< >' on the outside rather than curly braces.
That would be okay, except when you get to C_w, we have to know what
you mean by an infinite < >. Ordinarily, it would be a function as
opposed to a plain n-tuple. Or maybe you mean curly braces:

C_0 = {<0 0>}
etc.

So, we have C_i = range(B restricted to i+1).

> D_0 = {0}
> D_1 = {0,1}
> D_2 = {0,1,2}
> ...
> D_w = {0,1,2,...,w}

Then D_i = iu{i}.

You've taken us from B to C to D, just to get the successor function!

That's ridiculous!

> Assume D_k = w for some k.

That's an assumption that contradicts Z set theory.

So what is your point? By assuming something that contradicts Z set
theory, you're going to derive a contradiction with Z set theory? Wow,
that's deep.

> D_k = {0,1,2,...,k} and k must be the largest natural number.

Well, since you assumed a contradiction. Anything follows.

> Therefore, no D_k = w.

You didn't need to tell us that, but now that you have, are we still
under the assumption that for some k, we have D_k = w?

Looking down below, I don't see anything that came from your
assumption. It was pointless.

> Assume there exists a C_k that is a bijection for w.

Again, that assumption contradicts Z set theory. So what's the point?

> Then there exists D_k.

"There exists D_k" is not even a statement.

There exists a D_k SUCH THAT WHAT?

Anyway, since you've taken another assumption that contradicts Z set
theory, you can, with your assumption and Z set theory, derive any
statement whatsoever in the language of Z set theory.

> Assume there exists an identity bijection, A, for w.

Do you mean a bijection A from w onto w? In Z set theory we prove
there does exist such a bijection.

> A must be an proper subset of B.

No. NO member of bijection from w onto w is a member of B.

A bijection from w onto w looks like:

{<0 0> <1 1> ... }

No member of that is of the form <n <n n>>.

So maybe you have in mind a bijection A such that A is from w onto {<i
i> | i in w} and such that for each i in w, we have A_i = <i i>.

Then, yes, A is a proper subset of B.

However, if we are still under either of your previous assumptions
that contradict Z set theory, then anything follows with Z set theory
and that assumption.

> A must also be an initial segment of B.

Okay, with 'initial segment' suitably defined.

> If A is an initial segment of B then A = C_k for some k.

That's a complete non sequitur. You've shown no logic, no basis,
whatsoever.

However, if we are still under either of your previous assumptions
that contradict Z set theory, then anything follows with Z set theory
and that assumption.

And you're formulations of C and A AGAIN don't match themselves.

> There is no identity bijection for Omega.

You've not shown any such thing.

What you did is this:

(1) Take a long, unnecessary windup just to get to D, which is the
successor function on wu{w}.

(2) Make two separate assumptions that are contradictions with Z set
theory. Neither of those assumptions leading to anything at all.

(3) Make self-contradictory definitions.

(4) End by drawing a complete non sequitur.

MoeBlee

On Sep 3, 7:28 pm, logic...@comcast.net wrote:
> There is no identity bijection for Omega.

When it comes to the various so-called "cranks,"
I usually try to find some axiomatization that
may suit their intuitions better. But the
standard mathematicians have criticized me for
trying to find someone like the OP (Russell
Easterly or RE), who doesn't believe in the
existence of bijections between any given set
and itself.

Perhaps what RE wants is for something other
than bijections to be the basis of determining
set size or cardinality. Then he would be better
served by arguing that way, rather than disprove
the existence of an identity bijection.

The remainder of RE's post exploits the fact
that omega, unlike the finite nonzero ordinals,
is not the successor of any ordinal, and so the
successor function fails to be onto. Of course,
this doesn't actually prove what RE wants it to
prove at all.

Notice that in the hyperreals, it turns out
that every positive hypernatural has a
predecessor, so perhaps RE's problem is omega's
lack of a predecessor, not its lack of an
identity bijection.

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

"MoeBlee" <jazzmobe@hotmail.com> wrote in message
news:1188947334.374467.292630@50g2000hsm.googlegro ups.com...
> 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.

On Sep 4, 11:33 pm, "Peter Webb"
<webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
> "MoeBlee" <jazzm...@hotmail.com> wrote in message
>
> news:1188947334.374467.292630@50g2000hsm.googlegro ups.com...
>
> > 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.

Work to do what? How has a crank ever derived any form of analysis
from ZF-I?

Personally, I've not seen a crank use only ZF-I to express and argue
for his mathematical notions.

MoeBlee

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
would ask, what's wrong with bijections? So RE answers
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.

On Sep 5, 1:32 pm, lwal...@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,

As I said, I don't know of of any such axiomatization. And ZF-I+~I is
not such a one, since while it may be acceptable to certain cranks, it
is not ENOUGH to express and derive all that is encompassed in any of
their own views. ZF-I+~I is essentially first order PA. That doesn't
give the mathematics of real numbers in whatever form cranks work with
real numbers, nor of various algebraic structures, nor of probability
theory, etc., to whatever extent (and usually there is some extent)
cranks do work with such things.

Moreover, some cranks even explicity eschew that a theory should be
axiomatized (let alone formally, let alone recursively, axiomatized).

> 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.

I've not seen any crank show any such axiomatization that does not
include the axiom of infinity. As you know, both non-standard analysis
derived from mathematical logic and IST are based on set theory with
infinite sets and with some form of choice (or at least, as far as
choice is concerned, if I'm not mistaken, some non-constructive
principle such as existence of ultrafilters).

MoeBlee

MoeBlee says...

>Personally, I've not seen a crank use only ZF-I to express and argue
>for his mathematical notions.

I agree. The cranks don't reject the meaningfulness of infinite
sets, they usually believe in infinite sets, but believe weird
things *about* those sets. Often, they believe that anything that
is true of finite sets should also be true of infinite sets. Then
when it is pointed out that this leads to contradictions, they
start "patching" their ideas in an ad-hoc fashion, ending up
with incredibly convoluted rules that they can't even keep straight.

It would be much simpler if they just rejected the axiom of infinity.

--
Daryl McCullough
Ithaca, NY