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Topic: No Identity Bijection for Omega
Replies: 116   Last Post: Sep 22, 2007 5:58 PM

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George Greene

Posts: 448
Registered: 12/12/04
Re: No Identity Bijection for Omega
Posted: Sep 7, 2007 9:54 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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?
Not reading ahead, I just made the mistake of telling it to you again.


> 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.



Date Subject Author
9/3/07
Read No Identity Bijection for Omega
Russell Easterly
9/3/07
Read Re: No Identity Bijection for Omega
Chris Menzel
9/4/07
Read Re: No Identity Bijection for Omega
William Elliot
9/4/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/4/07
Read Re: No Identity Bijection for Omega
William Elliot
9/4/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/4/07
Read Re: No Identity Bijection for Omega
David C. Ullrich
9/4/07
Read Re: No Identity Bijection for Omega
Mitch
9/7/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/7/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/7/07
Read different sizes of denumerable sets?
George Greene
9/7/07
Read Re: different sizes of denumerable sets?
Alan Smaill
9/12/07
Read Re: different sizes of denumerable sets?
George Greene
9/7/07
Read Re: different sizes of denumerable sets?
Jesse F. Hughes
9/8/07
Read Re: different sizes of denumerable sets?
tchow@lsa.umich.edu
9/12/07
Read Re: different sizes of denumerable sets?
George Greene
9/12/07
Read Re: different sizes of denumerable sets?
tchow@lsa.umich.edu
9/12/07
Read Re: different sizes of denumerable sets?
tchow@lsa.umich.edu
9/12/07
Read Re: different sizes of denumerable sets?
George Greene
9/7/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/7/07
Read Re: No Identity Bijection for Omega
George Greene
9/7/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/10/07
Read Re: No Identity Bijection for Omega
George Greene
9/11/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/11/07
Read Re: No Identity Bijection for Omega
George Greene
9/11/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/11/07
Read Re: No Identity Bijection for Omega
George Greene
9/10/07
Read Re: No Identity Bijection for Omega
George Greene
9/7/07
Read Re: No Identity Bijection for Omega
George Greene
9/7/07
Read Re: No Identity Bijection for Omega
George Greene
9/7/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/7/07
Read Re: No Identity Bijection for Omega
George Greene
9/7/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/7/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/8/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/8/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/8/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/8/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/8/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/8/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/8/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/9/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/9/07
Read Re: No Identity Bijection for Omega
Ross Finlayson
9/9/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/9/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/9/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/9/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/9/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/9/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/10/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/10/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/10/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/10/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/12/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/12/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/10/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/10/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/10/07
Read Re: No Identity Bijection for Omega
William Hughes
9/11/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/11/07
Read Re: No Identity Bijection for Omega
William Hughes
9/10/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/11/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/11/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/12/07
Read Re: No Identity Bijection for Omega
George Greene
9/12/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/12/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/12/07
Read Re: No Identity Bijection for Omega
William Hughes
9/13/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/13/07
Read Re: No Identity Bijection for Omega
William Hughes
9/13/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/13/07
Read Re: No Identity Bijection for Omega
William Hughes
9/13/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/13/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/17/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/17/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/17/07
Read Re: No Identity Bijection for Omega
William Hughes
9/18/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/19/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/19/07
Read Re: No Identity Bijection for Omega
William Hughes
9/19/07
Read Re: No Identity Bijection for Omega
George Greene
9/19/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/18/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/20/07
Read Re: No Identity Bijection for Omega
Russell Easterly
9/20/07
Read Re: No Identity Bijection for Omega
William Hughes
9/20/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/22/07
Read Re: No Identity Bijection for Omega
reasterly@gmail.com
9/13/07
Read Re: No Identity Bijection for Omega
George Greene
9/10/07
Read Re: No Identity Bijection for Omega
George Greene
9/10/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/8/07
Read Re: No Identity Bijection for Omega
Jesse F. Hughes
9/8/07
Read Re: No Identity Bijection for Omega
Patricia Shanahan
9/8/07
Read Re: No Identity Bijection for Omega
William Hughes
9/10/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/9/07
Read Re: No Identity Bijection for Omega
David R Tribble
9/4/07
Read Re: No Identity Bijection for Omega
George Greene
9/4/07
Read Re: No Identity Bijection for Omega
George Greene
9/4/07
Read Re: No Identity Bijection for Omega
William Hughes
9/4/07
Read Re: No Identity Bijection for Omega
Schroeder
9/4/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/4/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/4/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/4/07
Read Re: No Identity Bijection for Omega
G. Frege
9/5/07
Read Re: No Identity Bijection for Omega
Peter Webb
9/5/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/5/07
Read Re: No Identity Bijection for Omega
Daryl McCullough
9/5/07
Read Re: No Identity Bijection for Omega
G. Frege
9/6/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/7/07
Read Re: No Identity Bijection for Omega
hagman
9/7/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/7/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/7/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/5/07
Read Re: No Identity Bijection for Omega
lwalke3@lausd.net
9/5/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/5/07
Read Re: No Identity Bijection for Omega
herb z
9/5/07
Read Re: No Identity Bijection for Omega
Jack Markan
9/6/07
Read Re: No Identity Bijection for Omega
herb z
9/6/07
Read Re: No Identity Bijection for Omega
Jack Markan

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