Re: Non-standard arithmetic

On Jun 20, 8:38 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
apoorv says...

If we know, a priori , that the axioms could have a model
with non-standard naturals,

We don't know it a priori, but we can prove it.

then the axiom of induction, as usually stated , runs counter
to all intuition -- every property of all standard naturals
automatically carries over to all non-standard naturlals.
We would perhaps have stated the Axiom of Induction as:
[P(0) & An (P(n)->P(S(n)))]->[n is standard -> P(n)]
( with additional axioms to define/distinguish standard and non-
standard naturals).

Yes, but the interesting thing about nonstandard analysis
is that if the formula P(x) is expressible without using
the predicate "standard", then induction holds for *all*
elements, not just the standard ones.

To me, the interesting feature of the axioms stated in the opening
post is that the theory incorprates addition and
multiplication and is omega inconsitent.

I don't see that it is very interesting. What you are basically
doing is enlarging the domain to talk about objects other than
naturals. Nobody had any doubt that that was possible. For example,
the theory of rational numbers or reals contains objects that
are not naturals. There is nothing surprising about that.

What *is* surprising (until you've seen a proof) is that
the axioms of Peano Arithmetic have nonstandard models
with *no* changes to the axioms.
It is indeed very surprising.Given that Standard(x)
->Standard(x+1) and Non-Standard(x)->Non-Standard(x+1), will we ever
have an instance of P(standard x)->P(non-standard(x+1))? How does
induction move from standard to non-standard? Otherwise P(x) could be
uniformly true for standard x and false for all non-standard x and yet
we would have Ax[(P(x)->P(x+1)] and conclude AxP(x).
How is the formula Sum i(i<=n)=n(n+1)/2 interpreted when n is non-
standard?
Indeed, induction as stated by you
'9. Phi(0) & (Forall x, Phi(x) -> Phi(S(x)))
-> Forall x, (standard(x) -> Phi(x))
where Phi is any formula whatsoever.'
appears to be in consonance with intuition and perception.
x has an unspecified domain which may or may not have non-standard
elements. From the premises we may infer the truth of P for any x that
can be reached step by step in an iterative manner; no non-standard x
could be so reached
and therefore AxP(x) would be an inappropriate inference.

Can there be a first order predicate that distinguishes
standard and non-standard? 'Equal to its own successor' is a good
example. If there is a x such that Sx=x, then that x is definitely non-
standard.(although there could be models with other non-standards for
which ~Sx=x and it is not immediately clear how one could exclude
them.)
-apoorv


.

Relevant Pages

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