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Topic: Infinity: The Story So Far
Replies: 298   Last Post: Apr 7, 2014 7:30 AM

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mueckenh@rz.fh-augsburg.de

Posts: 15,238
Registered: 1/29/05
Re: Infinity: The Story So Far
Posted: Feb 26, 2014 12:43 PM
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On Wednesday, 26 February 2014 18:14:51 UTC+1, Dan Christensen wrote:
> On Wednesday, February 26, 2014 11:58:07 AM UTC-5, John Gabriel wrote:
>
>
>

> > > > > > In your write-up, next(a) is meaningless garbage.
>
> >
>
> > > > > It is a successor function on the set of natural numbers. Got a problem with that?
>
> >
>
> > > > Yes, because then you already assume the prior existence of natural numbers.
>
> >
>
> > > Again, unless you want to rule out the existence of infinite sets altogether, the existence of the natural numbers is not problematic. See my previous posting above.
>
> >
>
> >
>
> > This is what you claim:
>
> >
>
> >
>
> >
>
> > The Definition of the set of natural numbers is given by nothing more or less than Peano's Axioms.
>
> >
>
> >
>
> >
>
> > It is false. The Peano axioms assume the prior existence of natural numbers.
>
>
>
> The Peano axioms describe structures that are embedded in EVERY infinite set.


For matheologians who believe in uncountable sets, you should say more precisely: every countably infinite set. That is correct. The set of natural numbers however is not every countable set but a very special countable set. The Peano axioms fail to produce this set because they are too general. They produce the set 1, S1, SS1, ...
Some of the many sets produced are
1, 1s, 1ss, ...
1, 11, 111, ...
1, 1^1, 1^1^1, ...
and if someone happens to have defined, or simply to know, the natural numbers, then they produce also
1, 1/2, 1/3, ...
1, 1^2, 1^3, ...
and with good luck
1, 2, 3, ...

Regards, WM


Date Subject Author
2/25/14
Read Infinity: The Story So Far
Dan Christensen
2/25/14
Read Re: Infinity: The Story So Far
Brian Q. Hutchings
2/25/14
Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com
2/25/14
Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
Leo Sgouros
2/26/14
Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
William Elliot
3/1/14
Read Re: Infinity: The Story So Far
Dan Christensen
3/2/14
Read Re: Infinity: The Story So Far
Brian Q. Hutchings
2/26/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
Read Re: Infinity: The Story So Far
Virgil
2/26/14
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Dan Christensen
2/26/14
Read Re: Infinity: The Story So Far
Martin Shobe
2/26/14
Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com
2/27/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/27/14
Read Re: Infinity: The Story So Far
Martin Shobe
2/27/14
Read Re: Infinity: The Story So Far
mueckenh@rz.fh-augsburg.de
2/27/14
Read Re: Infinity: The Story So Far
Virgil
2/26/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
Read show a canonical base-one accounting by induction on base-eleven
Brian Q. Hutchings
2/26/14
Read Re: Infinity: The Story So Far
mueckenh@rz.fh-augsburg.de
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
Read Re: Infinity: The Story So Far
mueckenh@rz.fh-augsburg.de
2/26/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
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Brian Q. Hutchings
2/27/14
Read Re: Infinity: The Story So Far
mueckenh@rz.fh-augsburg.de
2/27/14
Read Re: Infinity: The Story So Far
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2/27/14
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mueckenh@rz.fh-augsburg.de
2/27/14
Read Re: Infinity: The Story So Far
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2/27/14
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mueckenh@rz.fh-augsburg.de
2/27/14
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Dan Christensen
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Dan Christensen
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Peter Percival
3/5/14
Read Re: Peano's rot in action for you!
thenewcalculus@gmail.com
3/5/14
Read Re: Peano's rot in action for you!
Martin Shobe
3/5/14
Read Re: Peano's rot in action for you!
Wizard-Of-Oz
3/5/14
Read Re: Peano's rot in action for you!
Wizard-Of-Oz
3/5/14
Read Re: Peano's rot in action for you!
Wizard-Of-Oz
3/5/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/6/14
Read Re: Infinity: The Story So Far
Dan Christensen
3/6/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/6/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Wizard-Of-Oz
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/11/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/12/14
Read Re: Infinity: The Story So Far
thenewcalculus@gmail.com
3/12/14
Read Re: Infinity: The Story So Far
Martin Shobe
3/12/14
Read Re: Infinity: The Story So Far
ross.finlayson@gmail.com

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