 For collections of sources on the classical foundational positions (finitism, intuitionism, predicativism) see van Heijenoort (1967), Ewald (1996), and Mancosu (1998). On finitism and intuitionism see the entries   Hilbert’s program and   intuitionism in mathematics.   On predicativity see Feferman (2005). On Paris-Harrington see Katz and Reimann 2018; on Goodstein’s theorem see the friendly presentation in Stillwell (2010). Stillwell (2010) also has a chapter on large cardinals; for recent directions see Woodin (2011) and Steel (2015). On the interplay between finite and infinite in recursion theory see Hirschfeldt (2015).