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Topic: Answer to Dik T. Winter
Replies: 441   Last Post: Feb 5, 2013 6:25 AM

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Jack Markan

Posts: 6,964
Registered: 2/26/05
Re: Answer to Dik T. Winter
Posted: Jul 2, 2009 7:45 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Jul 2, 4:12 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> On Jul 2, 9:28 am, MoeBlee <jazzm...@hotmail.com> wrote:
>

> > On Jul 2, 12:06 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> > wrote:

>
> > > countable choice is a theorem of ZF.  
>
> > No, it's not.

> The definition of countable is there existing a bijective function
> between some set and the natural integers or here their ordinals.  


A definition of countable is ['w' stands for the set of natural
numbers]:

x is countable iff (there exists a bijection from x onto a natural
number of there exists a bijection from x onto w)

> A
> choice function is representable by any such function bijecting an
> ordinal as a set to a set.  


I have no idea what you're trying to say.

['P' stands for the power set operation:]

A choice function for x is a function f whose domain is Px\{0} and
such that for all y in Px\{0}, we have f(y) in y.

I guess sometimes people also refer to a choice function for x as
being a function whose domain is x\{0} and such that for all y in x
\{0} we have f(y) in y.

Context should make clear which sense is intended.

> Then, where countable choice means there
> are choice functions for countable sets,


The axiom of countable choice is:

For all x such that 0 is not in x, there exists a function f whose
domain is x and such that for all y in x we have f(y) in y.

Since finite sets are not at issue anyway, we can narrow:

For all x such that x is denumerable, there exists a function f whose
domain is x\{0} and such that for all y in x\{0} we have f(y) in y.

> otherwise there are
> "countable" sets with no provable bijection to the naturals,


You don't know what you're talking about. We don't require any choice
principles just to prove that there exists a bijection between a
countable set and either a natural number or the set of natural
numbers. The result follows simply by the DEFINITION of 'countable'.

> where if
> that's undecideable then so is whether they're countable, not
> disproving the definition.


Get help. Start with a book on working in the first order predicate
calculus.

MoeBlee


Date Subject Author
5/27/09
Read Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
5/27/09
Read Re: Answer to Dik T. Winter
Dik T. Winter
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6/25/09
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mueckenh@rz.fh-augsburg.de
6/25/09
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Virgil
6/30/09
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Dik T. Winter
6/30/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/30/09
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MeAmI.org
6/30/09
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Virgil
7/1/09
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Dik T. Winter
7/2/09
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G. Frege
7/2/09
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Dik T. Winter
7/2/09
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herb z
7/2/09
Read Re: Answer to Dik T. Winter
Dik T. Winter
7/2/09
Read Re: Answer to Dik T. Winter
herb z
7/2/09
Read Re: Answer to Dik T. Winter
G. Frege
7/3/09
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Aatu Koskensilta
7/3/09
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Daryl McCullough
7/2/09
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G. Frege
7/2/09
Read Re: Answer to Dik T. Winter
G. Frege
7/2/09
Read Re: Answer to Dik T. Winter
G. Frege
7/2/09
Read Re: Answer to Dik T. Winter
Jack Markan
7/2/09
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G. Frege
7/2/09
Read Re: Answer to Dik T. Winter
Jack Markan
7/2/09
Read Re: Answer to Dik T. Winter
G. Frege
7/2/09
Read Re: Answer to Dik T. Winter
Jack Markan
7/3/09
Read Re: Answer to Dik T. Winter
Dik T. Winter
7/3/09
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G. Frege
7/3/09
Read Re: Answer to Dik T. Winter
Dik T. Winter
7/3/09
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G. Frege
7/4/09
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Aatu Koskensilta
7/6/09
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Dik T. Winter
7/6/09
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G. Frege
7/6/09
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Dik T. Winter
7/6/09
Read Re: Answer to Dik T. Winter
Jesse F. Hughes
7/3/09
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mueckenh@rz.fh-augsburg.de
7/3/09
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Virgil
7/3/09
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Ralf Bader
7/2/09
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G. Frege
7/2/09
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Virgil
7/2/09
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mueckenh@rz.fh-augsburg.de
7/2/09
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ross.finlayson@gmail.com
7/2/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
7/2/09
Read Re: Answer to Dik T. Winter
Jack Markan
7/2/09
Read Re: Answer to Dik T. Winter
Jack Markan
7/2/09
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Virgil
7/2/09
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ross.finlayson@gmail.com
7/2/09
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Jack Markan
7/2/09
Read Re: Answer to Dik T. Winter
ross.finlayson@gmail.com
7/6/09
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Jack Markan
7/6/09
Read Re: Answer to Dik T. Winter
ross.finlayson@gmail.com
7/6/09
Read Re: Answer to Dik T. Winter
Jack Markan
7/6/09
Read Re: Answer to Dik T. Winter
ross.finlayson@gmail.com
7/6/09
Read Re: Answer to Dik T. Winter
Jack Markan
6/8/09
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mueckenh@rz.fh-augsburg.de
6/8/09
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Virgil
6/9/09
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Jack Markan
6/9/09
Read Re: Answer to Dik T. Winter
Jack Markan
6/9/09
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Virgil
6/11/09
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mueckenh@rz.fh-augsburg.de
6/11/09
Read Re: Answer to Dik T. Winter
Virgil
6/11/09
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William Hughes
6/11/09
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Virgil
6/11/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/11/09
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Virgil
6/11/09
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mueckenh@rz.fh-augsburg.de
6/11/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/11/09
Read Re: Answer to Dik T. Winter
Virgil
6/11/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/11/09
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Virgil
6/12/09
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herb z
6/12/09
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Virgil
6/12/09
Read Re: Answer to Dik T. Winter
Owen Jacobson
6/12/09
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Virgil
6/12/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/12/09
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Virgil
6/11/09
Read Re: Answer to Dik T. Winter
Virgil
6/11/09
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Dik T. Winter
6/11/09
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William Hughes
6/11/09
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mueckenh@rz.fh-augsburg.de
6/11/09
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YBM
6/11/09
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Virgil
6/11/09
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Dik T. Winter
6/12/09
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mueckenh@rz.fh-augsburg.de
6/12/09
Read Re: Answer to Dik T. Winter
Virgil
6/12/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/12/09
Read Re: Answer to Dik T. Winter
Virgil
6/19/09
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Dik T. Winter
6/19/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/19/09
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Virgil
6/22/09
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Dik T. Winter
6/25/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/25/09
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Virgil
6/30/09
Read Re: Answer to Dik T. Winter
Dik T. Winter
6/30/09
Read Re: Answer to Dik T. Winter
Virgil
6/30/09
Read Re: Answer to Dik T. Winter
Guest
6/30/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
7/1/09
Read Re: Answer to Dik T. Winter
Dik T. Winter
6/11/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/11/09
Read Re: Answer to Dik T. Winter
Virgil
6/3/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/3/09
Read Re: Answer to Dik T. Winter
Virgil
6/3/09
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Dik T. Winter
6/7/09
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mueckenh@rz.fh-augsburg.de
6/7/09
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Virgil
6/11/09
Read Re: Answer to Dik T. Winter
Dik T. Winter
6/3/09
Read Re: Answer to Dik T. Winter
george
5/29/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
5/29/09
Read Re: Answer to Dik T. Winter
Virgil
5/30/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
5/30/09
Read Re: Answer to Dik T. Winter
Virgil
5/31/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
5/31/09
Read Re: Answer to Dik T. Winter
Virgil
5/31/09
Read Re: Answer to Dik T. Winter
Guest
5/27/09
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Virgil
5/28/09
Read Who introduced false logic?
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5/28/09
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5/30/09
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5/30/09
Read Re: Who introduced false logic?
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5/30/09
Read Re: Who introduced false logic?
Virgil
5/30/09
Read Re: Answer to Dik T. Winter
Virgil
5/30/09
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Virgil
5/30/09
Read Re: Answer to Dik T. Winter
Ralf Bader
5/27/09
Read Natural numbers are uniquely defined
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5/27/09
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David C. Ullrich
5/27/09
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Virgil
5/27/09
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5/27/09
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5/28/09
Read Re: Natural numbers are uniquely defined
mueckenh@rz.fh-augsburg.de
5/28/09
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Virgil
5/29/09
Read Re: Natural numbers are uniquely defined
Peter Webb
5/29/09
Read Re: Natural numbers are uniquely defined
Virgil
5/29/09
Read Re: Natural numbers are uniquely defined
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5/29/09
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Virgil
5/29/09
Read Re: Natural numbers are uniquely defined
Peter Webb
5/30/09
Read Re: Natural numbers are uniquely defined
mueckenh@rz.fh-augsburg.de
5/30/09
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Virgil
5/30/09
Read Re: Natural numbers are uniquely defined
Peter Webb
5/31/09
Read Re: Natural numbers are uniquely defined
mueckenh@rz.fh-augsburg.de
5/31/09
Read Re: Natural numbers are uniquely defined
Virgil
5/31/09
Read Re: Natural numbers are uniquely defined
mueckenh@rz.fh-augsburg.de
5/31/09
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Peter Webb
5/31/09
Read Re: Natural numbers are uniquely defined
Virgil
6/1/09
Read Re: Natural numbers are uniquely defined
mueckenh@rz.fh-augsburg.de
6/1/09
Read Re: Natural numbers are uniquely defined
David Bernier
6/1/09
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Jack Markan
6/1/09
Read Re: Natural numbers are uniquely defined
Virgil
6/2/09
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george
6/1/09
Read Re: Natural numbers are uniquely defined
mueckenh@rz.fh-augsburg.de
6/1/09
Read Re: Natural numbers are uniquely defined
Virgil
6/2/09
Read Re: Natural numbers are uniquely defined
george
6/2/09
Read Re: Natural numbers are uniquely defined
george
6/2/09
Read Re: Natural numbers are uniquely defined
george
6/3/09
Read Re: Natural numbers are uniquely defined
george
6/12/09
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2/5/13
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6/14/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/14/09
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Virgil
6/14/09
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mueckenh@rz.fh-augsburg.de
6/14/09
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Virgil
6/15/09
Read Re: Answer to Dik T. Winter
Owen Jacobson
6/15/09
Read Re: Answer to Dik T. Winter
Virgil
6/15/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/15/09
Read Re: Answer to Dik T. Winter
Owen Jacobson
6/15/09
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Virgil
6/15/09
Read Re: Answer to Dik T. Winter
Virgil
6/16/09
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MeAmI.org
6/16/09
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mueckenh@rz.fh-augsburg.de
6/16/09
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Virgil
6/18/09
Read Re: Answer to Dik T. Winter
mueckenh@rz.fh-augsburg.de
6/18/09
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Virgil
6/20/09
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