>At least in the Gold's formalization

It seems to be a persistent myth (possibly revived more recently due to Norvig?) that Chomsky's POS argument has some interesting connection to Gold's theorem. The two things have only a very loose logical connection (Gold's theorem is in no sense a formalization of any claim of Chomsky's), and Chomsky himself never based any of his arguments for innateness on Gold's theorem. Here is a secondary source making the same point (search for 'Gold'): https://stevenpinker.com/files/pinker/files/jcl_macwhinney_c...

The assumption that syntax is 'separate from semantics' also does not figure in any of Chomsky's POS arguments. Chomsky argued that syntax was separate from semantics only in the fairly uncontroversial sense that there are properly syntactic primitives (e.g. 'noun', 'chain', 'c-command') that do not reduce entirely to semantic or phonological notions. But even if that were untrue, it would not undermine POS arguments, which for the most part can be run without any specific assumptions about the syntax/semantics boundary. Indeed, semantic and conceptual knowledge provides an equally fertile source of POS problems.