218 Logic as a Tool Undecidability Theorem? Alas, no The catch is that there may be an infinite number of such candidate sets of ground instances to check References for further reading For more detailed proofs of soundness and completeness of deductive systems for first-order logic, see: van Dalen (1983) and Chiswell and Hodges (2007) for a completeness proofs for ND; Shoenfield (1967), Hamilton (1988), Mendelson (1997), and Enderton (2001) for completeness of H; Nerode and Shore (1993), Fitting (1996), and Ben-Ari (2012) for completeness of ST and RES; Smullyan (1995) and Smith (2003) for completeness of ST; Ebbinghaus et al (1996) and Hedman (2004) for completeness of RES; and Boolos et al (2007) for completeness of the Sequent calculus presented there For more on Herbrand’s Theorem see Shoenfield (1967), Nerode and Shore (1993), and Fitting (1996) For expositions and discussions of Gödel’s Incompleteness Theorems, see Jeffrey (1994), Ebbinghaus et al (1996), Enderton (2001), Hedman (2004), and Boolos et al (2007) Exercises In the following exercises, FO stands for first-order logic, H, ND, ST, and RES refer to the respective deductive systems for FO introduced here, and D refers to any of these deductive systems All references to definitions and results in Section 2.7 now refer to the respective definitions and results for FO 4.6.1 Prove Proposition 60 for FO 4.6.2 Prove Proposition 62 for: (a) H; (b) ST; (c) ND; (d) RES 4.6.3 Prove Proposition 63 for: (a) H; (b) ST; (c) ND; (d) RES 4.6.4 Prove Proposition 68 generically for each deductive system D 4.6.5 Prove Proposition 70 generically for each deductive system D 4.6.6 Prove Theorem 157 for: (a) H; (b) ST; (c) ND; (d) RES 4.6.7 Prove Theorem 158 for: (a) H; (b) ST; (c) ND; (d) RES 4.6.8 Prove Lemma 164