A theorem which is true in every model is provable by Godel's completeness theorem. Since this theorem is true for the standard model of the natural numbers but not provable, it follows there are nonstandard models of the natural numbers for which it is false.
That is, there are models of Peano arithmetic which contain all of the natural numbers we know and love, and some other ones on top of that and there are some Goodstein sequences using those extra "non-standard" natural numbers which do not terminate at zero.
That is, there are models of Peano arithmetic which contain all of the natural numbers we know and love, and some other ones on top of that and there are some Goodstein sequences using those extra "non-standard" natural numbers which do not terminate at zero.
https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet...