Re: Peano's second axiom.

On Jun 30, 2:54 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 30, 2:45 pm, Zaljo...@xxxxxxxxx wrote:





On Jun 30, 12:54 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 30, 11:05 am, Zaljo...@xxxxxxxxx wrote:

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

You used the symbol 'e' but neither declared it as primitive nor
defined it.

One thing I don't really understand is why 0 is among the list of
primitives in PA.

Why not having only two primitives: Number and Successor 'S'.

having the axiom:

Axiom of first number : E!x(number(x) & Ay(number(y) -> ~x=S(y))).

and then defining 0 as

x=0 <-> (number(x) & Ay(number(y) -> ~x=S(y))).

There is no need to put 0 among the list of primitives.

In this way we'll have four axioms and two primitives.

In first order PA, there is no need to have 'is a number' as a
primitive. And your theory doesn't cover addition and multiplication.
And your symbol 'e' is unaccounted for.

Yes, I know that. I was thinking of PA as a part of a set theory in
FOL with identity theory, that contain other axioms beside these.

Then why didn't you say so? I mean, what's the point of calling your
thing a first order PA if it's got all kinds of set theory axioms too?
And what's the point of asking about whether your thing accomplishes
the axiomatization I mentioned if your thing has all kinds of set
theory axioms too?

Sheesh. Over and over, you waste people's time by your negligence to
simply say what it is you intend.

MoeBlee

I learned something new. To me it is not a waste of time. Most of
people including you have supplied me with very good information, and
to me it was never a waste of time. Thanks to all who coaperated here
including angery Moe of course., by the way what does Sheesh
mean? LOL



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