TITLE: Paul Cohen on genesis of method of forcing and mathematical similarities
QUESTION [9 upvotes]: We have on record Paul Cohen's comments on being inspired by issues of formalizing algorithms in number theory (this needs to be verified as per comment) as well as related remarks on computability. Beyond such "inspiration", are there mathematical similarities between developments in those fields and Cohen's work on forcing? A related question was posed at MSE with limited success.

REPLY [3 votes]: Regarding your question about the connection between forcing and computability theory, the notion of a Cohen generic oracle is used in computational complexity; see for example An oracle builder's toolkit by Fenner et al.  For a different kind of connection, see Chapter 13 ("Forcing and Category") of S. Barry Cooper's book Computability Theory (unfortunately I don't think that the complete text is available online).
As for the connection to arithmetic, see
Forcing over set theory versus forcing over arithmetic