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Topic: WM's axioms for the naturals numbers
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Dan Christensen

Posts: 2,433
Registered: 7/9/08
WM's axioms for the naturals numbers
Posted: Mar 11, 2014 4:22 PM
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WM recently put forward here his alternative to Peano's axioms for the natural numbers. WM defines the set of natural numbers N to be the intersection of all sets M such that 1 in M and for all x in M, we have x+1 in M. Using my DC Proof notation, that is:

ALL(a):[a e N <=> ALL(M):[1 e M & ALL(b):[b e M => b+1 e M] => a e M ]]

where e is epsilon (set membership)

We get off to a good start, proving:

1 e n
1+1 e n
1+1+1 e n
etc.

For all x e n, there exists y e n such that y=x+1.

2=1+1
3=2+1
4=3+1
etc.

See my formal proof at http://dcproof.com/WM1.htm (72 lines)

So far, so good.

Then the wheels fall off when you realize that this N could also apply to N = {1} and 1=1+1. All the above numbers could be the SAME number! 1=2=3= .... Without additional axioms, there is no way to be certain.

See my formal proof at http://dcproof.com/WM2.htm (41 lines)

It really gives you an appreciation for thought that must have gone into Peano's axioms.

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my new math blog at http://www.dcproof.wordpress.com


Date Subject Author
3/11/14
Read WM's axioms for the naturals numbers
Dan Christensen
3/11/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/11/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/11/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/12/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/12/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/12/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/12/14
Read Re: WM's axioms for the naturals numbers
Mutt Buncher
3/11/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/11/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/11/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/11/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/11/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/13/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/11/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/11/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/11/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/13/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/13/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
John Gabriel
3/12/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/13/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com
3/13/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/13/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/13/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/13/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/13/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/14/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/14/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/14/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/13/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/13/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/13/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/13/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/12/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/12/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/13/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/13/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/13/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/13/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/13/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/13/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/13/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/13/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/13/14
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Virgil
3/12/14
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3/12/14
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thenewcalculus@gmail.com
3/12/14
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Dan Christensen
3/12/14
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3/13/14
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3/13/14
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3/13/14
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3/13/14
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3/14/14
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3/15/14
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wolfgang.mueckenheim@hs-augsburg.de
3/15/14
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3/16/14
Read Re: WM's axioms for the naturals numbers
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3/16/14
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3/17/14
Read Re: WM's axioms for the naturals numbers
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3/17/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/17/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/17/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/17/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/17/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/17/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/17/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/17/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/18/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/18/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/18/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/18/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/18/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/20/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/20/14
Read Re: WM's axioms for the naturals numbers
Richard Tobin
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/21/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/21/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/22/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/22/14
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Dan Christensen
3/22/14
Read Re: WM's axioms for the naturals numbers
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3/20/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/20/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/20/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/20/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/20/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/20/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/20/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/20/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/17/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/17/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/18/14
Read Re: WM's axioms for the naturals numbers
Tanu R.
3/18/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/18/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/18/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/17/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/17/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/17/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/17/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/17/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/19/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
wolfgang.mueckenheim@hs-augsburg.de
3/19/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/19/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/22/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/22/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/22/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/22/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/22/14
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Dan Christensen
3/23/14
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mueckenh@rz.fh-augsburg.de
3/23/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/23/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/23/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/23/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/23/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/23/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/23/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/23/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/23/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/23/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/24/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/24/14
Read Re: WM's axioms for the naturals numbers
Dan Christensen
3/24/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/24/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/24/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/24/14
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3/25/14
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3/26/14
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3/27/14
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3/28/14
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3/29/14
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3/30/14
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3/30/14
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3/31/14
Read Re: WM's axioms for the naturals numbers
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3/31/14
Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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Read Re: WM's axioms for the naturals numbers
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3/24/14
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Read Re: WM's axioms for the naturals numbers
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3/24/14
Read Re: WM's axioms for the naturals numbers
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3/23/14
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3/23/14
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mueckenh@rz.fh-augsburg.de
3/24/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/23/14
Read Re: WM's axioms for the naturals numbers
FredJeffries@gmail.com
3/24/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/23/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/23/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/22/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/22/14
Read Re: WM's axioms for the naturals numbers
mueckenh@rz.fh-augsburg.de
3/22/14
Read Re: WM's axioms for the naturals numbers
Virgil
3/31/14
Read Re: WM's axioms for the naturals numbers
thenewcalculus@gmail.com

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