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 June 23rd, 2014, 10:56 AM #1 Newbie   Joined: Jun 2014 From: here Posts: 2 Thanks: 0 H&B's 2nd axiom for successor function contradicts 1st? Hi all, I'm reading Charles Petzold's "The Annotated Turing" for fun, and I'm no mathematician, so this might be a silly question. I have a question about page 226, where Hilbert and Bernay's axioms for the successor function are defined. Specifically, the second axiom: (Ǝx)(y)-S(y,x) I don't agree with this axiom: what's an example of a number with no successor? Doesn't this directly contradict the first axiom: (x)(Ǝy)S(x,y) ? Thanks.
June 23rd, 2014, 01:12 PM   #2
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Quote:
 Originally Posted by User (Ǝx)(y)-S(y,x) I don't agree with this axiom: what's an example of a number with no successor?
There is an x such that for all y, x is not the successor of y. This is an axiom for the natural numbers: there is some element (usually 0 in modern formulations) which is not a successor.

Quote:
 Originally Posted by User Doesn't this directly contradict the first axiom: (x)(Ǝy)S(x,y)
For all x, there is some y such that y is the successor of x. Not the same as the above because of the order of quantifiers and the order of the arguments. 0 may have no predecessor but it and all other elements have successors.

 June 24th, 2014, 03:46 AM #3 Newbie   Joined: Jun 2014 From: here Posts: 2 Thanks: 0 Thanks. I didn't know these only applied to the natural numbers, so yes, they both make sense in that regard.
June 24th, 2014, 05:43 AM   #4
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 Originally Posted by User Thanks. I didn't know these only applied to the natural numbers, so yes, they both make sense in that regard.
You could say they define the natural numbers.

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