On 14 Feb 2012, at 14:53, Stephen P. King wrote:

On 2/14/2012 7:56 AM, Bruno Marchal wrote:

On 13 Feb 2012, at 16:54, Stephen P. King wrote:

Dear Bruno,

What limits are there on what can constitute the "constant" that defines a particular model of a non-standard Arithmetic?

Non standard integers are infinite objects.



Hi Bruno,

OK, I am studying this idea. But your answer is confusing. AFAIK, standard integers are infinite objects also, given that they can be defined as equivalence classes where the equivalence relation is "has the same value as X", where X is the integer in question.

Standard integers are finite object. The fact that you can represent them with infinite sets does not change the fact that their are finite object. You can represent the integers with elephants, but this will not make the integers into mamals.

So how are non-standard integers different?

They have to be "infinite", independently of the way you represent them. Non standard integers does not look at all to the integers that we all know well. A non standard natural number is bigger than any standard natural number. They have infinitely many predecessors. They exists because we cannot throw them away in a first order logical way. They obey to the PA axioms, but that's all. The non standard models don't play the role of consistent extensions, or histories.



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