> By number that cannot be written, I don't mean cannot be written due to lack of paper; I mean a value that there is no notation for.
That's true for most real numbers in ordinary mathematics too, so if you accept the existence of those numbers, no reason why you can't do the same for the natural numbers in these non-standard models.
> You cannot run a machine for "that many" steps because such a value is unreachable by steps. Non-standard models have a set of values that begin with a copy of the actual natural numbers, followed by some ordered set of copies of the integers in the sense that any values "beyond" the initial natural numbers have an infinite number of successors and an infinite number of predecessors, like integers do. By counting in steps it is not possible to move from a value in the initial natural number fragment to one of these non-standard values, because there are an infinite number of values in between them that you would be required to step through.
That's true for most real numbers in ordinary mathematics too, so if you accept the existence of those numbers, no reason why you can't do the same for the natural numbers in these non-standard models.
> You cannot run a machine for "that many" steps because such a value is unreachable by steps. Non-standard models have a set of values that begin with a copy of the actual natural numbers, followed by some ordered set of copies of the integers in the sense that any values "beyond" the initial natural numbers have an infinite number of successors and an infinite number of predecessors, like integers do. By counting in steps it is not possible to move from a value in the initial natural number fragment to one of these non-standard values, because there are an infinite number of values in between them that you would be required to step through.
Yes...