Q:

How is The strengthened finite Ramsey theorem known to be true for natural numbers?

http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem states that independence of "The strengthened finite Ramsey theorem" from PA is proved by implying consistency of PA.
But how do we know that it's actually true for the Model that's Natural Numbers. How was it even proved for Natural Numbers? Under what logical system is it proved, if not for Peano's Axioms? And how do we know that such a logical system on which it's proved actually models Natural Numbers?

A:

One of the common frameworks for mathematics is the axiomatic set theory ZFC. This system proves the consistency of second-order PA. In particular, we can show that the finite von Neumann ordinals form a model of this system.
In particular every model of second-order PA is a model of first-order PA.
The strengthened Ramsey theorem can be proved in second-order PA, as stated in the Wikipedia page linked in the question. Therefore ZFC proves that this theorem is true for the natural numbers.
As discussed in this question when we say that a theorem is "true for the natural numbers" we mean that it holds for the standard model of PA (which is the model of second-order PA).

