[IngoBlechschmid]

This is not quite true, Gödel's incompleteness theorems can luckily be formalized in extremely weak fragments of Peano arithmetic such as primitive recursive arithmetic (PRA) with its very limited induction principle. :-)

The only position on the philosophy of mathematics I know which does not accept PRA is ultrafinitism.

[ProfHewitt]

Problem with 1st-order PRA is that it is too weak to serve as

foundation of mathematics.

[erichahn]

Can you provide a paper or book on this?