4.6.10 Prove Lemma 168 generically for any deductive system D.. Hint for ND: suppose ΓD⊥ in L+by a derivation Ξ.. It can be shown, by inspection of the rules of D, thatΞ is a valid deriv
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4.6.9 Prove Lemma 166 for:
(a) H; (b) ST; (c) ND; (d) RES.
4.6.10 Prove Lemma 168 generically for any deductive system D.
4.6.11 Prove Lemma 170
4.6.12 Complete the proof of Lemma 172 for:
(a) H; (b) ST; (c) ND; (d) RES.
4.6.13 Prove Lemma 174 for:
(a) H; (b) ST; (c) ND; (d) RES.
(Hint for ND: suppose ΓD⊥ in L+by a derivation Ξ LetΞ be the result of replacing each free occurrence of a new constantc iin Ξ by a new variablex inot
occurring in Ξ It can be shown, by inspection of the rules of D, thatΞ is a valid derivation of ΓD⊥ in L.)
4.6.14 Prove Lemma 175 for:
(a) H; (b) ST; (c) ND; (d) RES.
4.6.15 Prove that the deductive compactness property stated in Lemma 175 is equivalent
to the following: a first-order theory Γ is D-consistent iff every finite subset of Γ
is D-consistent.
4.6.16 Complete the generic proof details of Theorem 176 for any deductive system D.
4.6.17 Assuming soundness and completeness of H, prove soundness and completeness
of each of ST, ND, and RES by using a first-order analog of Proposition 76.
4.6.18 Assuming soundness and completeness of ST, prove soundness and
complete-ness of each of H, ND, and RES by using a first-order analog of Proposition 76.
4.6.19 Assuming soundness and completeness of ND, prove soundness and
complete-ness of each of ST, H, and RES by using a first-order analog of Proposition 76.
4.6.20 Assuming soundness and completeness of RES, prove soundness and
complete-ness of each of H, ST, and ND by using a first-order analog of Proposition 76.
4.6.21 Prove the equivalence of the semantic compactness theorems 179 and 180