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Topic: WM's axioms for the naturals numbers
Replies: 298   Last Post: Apr 2, 2014 1:43 PM

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 Dan Christensen Posts: 8,219 Registered: 7/9/08
WM's axioms for the naturals numbers
Posted: Mar 11, 2014 4:22 PM

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
Visit my new math blog at http://www.dcproof.wordpress.com

Date Subject Author
3/11/14 Dan Christensen
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3/11/14 Tanu R.
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3/11/14 Dan Christensen
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3/11/14 Virgil
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3/12/14 Virgil
3/11/14 Dan Christensen
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3/12/14 Dan Christensen
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3/12/14 wolfgang.mueckenheim@hs-augsburg.de
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3/12/14 Dan Christensen
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3/12/14 Dan Christensen
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3/12/14 Dan Christensen
3/12/14 Dan Christensen
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3/12/14 Dan Christensen
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