It is a consequence of the Peano arithmetic (Peano axioms plus definition of addition and multiplication), it is just not provable in this system.
Gödel's first incompleteness theorem [1] states this fact, that no theory above a certain expressiveness (read as can express natural numbers with addition and multiplication) can be consistent and complete. Assuming Peano arithmetic is consistent, it can not be complete and complete means you can prove all true facts expressible in the system within the system itself.
The (standard) proof of Goodstein's theorem uses ordinal numbers [2] which are outside of Peano arithmetic and the Kirby–Paris theorem proves that there is no proof inside Peano arithmetic [3].
Gödel's first incompleteness theorem [1] states this fact, that no theory above a certain expressiveness (read as can express natural numbers with addition and multiplication) can be consistent and complete. Assuming Peano arithmetic is consistent, it can not be complete and complete means you can prove all true facts expressible in the system within the system itself.
The (standard) proof of Goodstein's theorem uses ordinal numbers [2] which are outside of Peano arithmetic and the Kirby–Paris theorem proves that there is no proof inside Peano arithmetic [3].
[1] http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo...
[2] http://en.wikipedia.org/wiki/Ordinal_number
[3] http://en.wikipedia.org/wiki/Goodsteins_theorem#Proof_of_Goo...