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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: 15,051
Registered: 1/29/05
Re: Infinity: The Story So Far
Posted: Feb 28, 2014 5:57 AM
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On Friday, 28 February 2014 10:23:33 UTC+1, Virgil wrote:
> In article <89017e7c-bf29-4ca7-bd14-77f976c852ce@googlegroups.com>,
>
> mueckenh@rz.fh-augsburg.de wrote:
>
>
>

> > On Friday, 28 February 2014 08:00:03 UTC+1, Virgil wrote:
>
> > > In article <7alnvr.i2m.19.1@news.alt.net>, fom <fomJUNK@nyms.net>
>
> > >
>
> >
>
> > >
>
> > > > > It is only later, after all the basic axioms are satisfied, when
>
> > >
>
> > > > > defining addition that any "1" appears.
>
> > >
>
> > > > >
>
> > >
>
> > > >
>
> > >
>
> > > > That is actually problematic.
>
> > >
>
> > >
>
> > >
>
> > > NOT in the Peano axioms.
>
> >
>
> > And not in your head, Virgil. You accept individually undefinable "real"
>
> > numbers and axioms that are defining the natural numbers without defining 1.
>
>
>
> The Peano axioms never do define 1 or any natural numbers since the
> axioms are equally valid for sets which need not contain any natural
> numbers whatsoever.


Read the text. Peano in his second approach defined:
1) 1 is a natural number. (In his third approach he defined 0 is a natural number.)
2) Every number a in N has a certain follower a'.
3) No number has the follower 1.
4) a' = b' implies a = b.
5) Every set M of natural numbers containing 1 and for every number a also a' contains all natural numbers.

These axioms are correct. But they do not say what N is. I use also (5) in a slighly modified version. I use also (1). And I use (2) in a better version while (3) and (4) are self-evident.


> On the other hand, the set of natural numbers is often defined to be a
> set satisfying the Peano axioms.


No objection. But ...
It is useless to try to explain any thing to you.

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
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2/26/14
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ross.finlayson@gmail.com
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3/1/14
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Dan Christensen
3/2/14
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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
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thenewcalculus@gmail.com
2/26/14
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2/26/14
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Virgil
2/26/14
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Dan Christensen
2/26/14
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Martin Shobe
2/26/14
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ross.finlayson@gmail.com
2/27/14
Read Re: Infinity: The Story So Far
Dan Christensen
2/27/14
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Martin Shobe
2/27/14
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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
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Read Re: Infinity: The Story So Far
Dan Christensen
2/26/14
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2/26/14
Read Re: Infinity: The Story So Far
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2/26/14
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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
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2/26/14
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mueckenh@rz.fh-augsburg.de
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2/26/14
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2/27/14
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mueckenh@rz.fh-augsburg.de
2/27/14
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2/27/14
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mueckenh@rz.fh-augsburg.de
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mueckenh@rz.fh-augsburg.de
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mueckenh@rz.fh-augsburg.de
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2/27/14
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Dan Christensen
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3/2/14
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3/2/14
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3/2/14
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3/3/14
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3/4/14
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3/4/14
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3/5/14
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3/5/14
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thenewcalculus@gmail.com
3/5/14
Read Re: Peano's rot in action for you!
Dan Christensen
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!
Dan Christensen
3/5/14
Read Re: Peano's rot in action for you!
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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