Question

Say that S is a sentence of SL and Г is a set of sentences of SL, and Г is infinite Problem: Show that if Г that Δ S. S then there is some finite set Г such This is a consequence of the compactness theorem 6.4.12, but I want you to show how the statement follows directly from the completeness and soundness theorems. The argument takes just four or five lines. Hint: Derivations are only finitely many lines long.)

Answer 1

Solution:

Given that:

Gamma a set of sentences of SL, 'S' is sentence of SL completeness Theorem: - For a logic system S, under a sentences Gamma. For any triangle of the language of S Triangle is provable in S. If triang t Da tantology y under the semantics Gamma Therefore t_s Delta iff Gamma . Delta soundness Theorem: - For formal theory says: - for any formal 'Delta' of the language of the theory THS(S_L) \r\n If a formula 'Delta' is procuble is the theory THS(S_L) then 'Delta' is true in any model of the set of specific axiens S_L. language 'L' of s. L = (A, F) where, A = Alphabet F = Formula set Delta elementof F* , Delta = A_1, A_2, A_3 Ellipsis A_n v*(Delta) = v* (A_1, A_2, Ellipsis A_n) = v* (A_1) Union v* (A_2) Ellipsis Union v* (A_n) Delta = (A_1 Union A_2 Union Ellipsis A_n) A elementof E x_s A P R_S = {A elementof E: t_s A} \r\n Gamma t_s A if Gamma s Delta, then