Re: Peano's second axiom.

On Jun 29, 6:36 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 29, 5:16 pm, Zaljo...@xxxxxxxxx wrote:

How is exactly Peano's second axiom writtin in FOL?

The axiom reads: The successor of a number is a number.

That's not an axiom of first order PA as first order PA is usually
defined in modern times, since 'is a number' is not a predicate of
first order PA as first order PA is usually defined in modern times.

Here's an axiomatization of first order PA:

First order PA is an extension of first order identity theory, in a
language whose non-logical symbols are:

0 [0-place fcn symbol]
S [1-place fcn symbol]
+ [2-place fcn symbol]
* [2-place fcn symbol]

Axioms:

An ~Sn = 0
Ank(Sn = Sk -> n = k)
An n+0 = n
Ank n+Sk = S(n+k)
An n*0 = 0
Ank n*Sk = (n*k)+n

Axiom schema:
If P is a formula, n is a variable not free in P, x is a variable, and
n is free for x in P, then all closures of

(P[0|x] & An(P[n|x] -> P[Sn|x])) -> AnP[n|x]

are axioms.

MoeBlee

Nice.

But why not the following:

Primitives: 0,Number,S
Axiom: 0 is a number
Axiom: Ax( x is a number -> Sx is a number )
Axiom: Axyz( (z=Sx & z=Sy) -> x=y )
Axiom: Ax ( ~0=Sx )
Axiom: Ax (( Ay( yex -> y is a number ) & 0ex & Ay( yex->Syex ) ) ->
Az( z is a number -> zex ) ).

Zuhair


.

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