Detlefsen (1986, Appendix; see also  1990) has defended Hilbert’s Program against this argument as well. Detlefsen argues that “Hilbertian” instrumentalism escapes the argument from G1 by denying that ideal mathematics must be conservative over the real part; all that is required is real-soundness. Hilbertian instrumentalism requires only that the ideal theory not prove anything which is in conflict with the real theory; it is not required that it prove only real statements which the real theory also proves. (See Zach 2006 for more on the issue of conservativity and consistency, the relevant section in the entry on Gödel for further discussion, Franks 2009 for a related defense and re-evaluation of Hilbert’s project, and McCarthy 2016 for an alternative approach to the provability of consistency and G2 due to Gödel himself.)