[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.