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:
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.
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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