It is known that Goodstein theorem is not provable in PA, but my question is if it is know if the theorem is true in the standard model of arithmetic.
I mean, Gödel's first incompleteness theorem shows us that there are sentences in PA that are not provable (in PA) but true in the standard model of arithmetic. My question is then if Goodstein theorem is, in fact, a concrete example of the later statement.
I know there are several questions about this topic but I don't think they really answer my question.
Best Answer
Yes. It has pretty much the same status as Godel's sentence or Con(PA). It is not provable in PA, but it is provable in other systems generally accepted as sound (ZF or even second-order arithmetic is major overkill) and is thus generally accepted as true.