[Kranar]

Not at all. There is no largest natural number to begin with even in the standard model. One way to conceptualize non-standard natural numbers would be to consider natural numbers with an infinite number of digits. Any such number would be greater than any natural number, and no first order model of arithmetic can exclude every possible way to express such numbers.

The main issue is that first order logic can't define the concept of finite. There is no way for a first order system to express a statement like "There are only finitely many x such that P(x) holds." Introducing such a finite quantifier or finite predicate will also introduce inconsistencies.

If it were possible then one could introduce an axiom along the lines of "For all x, x has a finite number of predecessors." and then we could eliminate all non-standard natural numbers.

[theteapot]

I think I'm philosophically a finitist / constructivist, which seems to be very 19th century and out of vogue with modern Mathematicians AFAICT.

> There is no largest natural number to begin with even in the standard model.

I'm aware of that. I get Peano Arithmetic more of less.

> One way to conceptualize non-standard natural numbers would be to consider natural numbers with an infinite number of digits. Any such number would be greater than any natural number, and no first order model of arithmetic can exclude every possible way to express such numbers.

I'm not sure you can argue such a number is "greater" than any natural number? They seem incomparable.