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Published online by Cambridge University Press: 05 June 2012
This appendix contains some comments and explanations for Section 9.1. It also includes some missing proofs of the results stated there.
We start by explaining a few more terms. In Section 9.1 we said that a formula ϕ is a consequence of a theory T, T ⊢ ϕ, if there is a formal proof of ϕ from T. By a formal proof in this statement we mean a formalization of what we really do in proving theorems. Thus a proof of ϕ from T is a finite sequence ϕ0,…, ϕn of formulas such that ϕn = ϕ and a formula ϕk can appear in the sequence only because of one of following two reasons:
ϕk is an axiom, that is, it belongs to T or is a logic axiom; or
ϕk is obtained from some ϕi and ϕj (i, j < k) by a rule of detachment (also called a modus ponens rule), that is, if there exist i, j < k such that (ϕj has the form ϕi→ϕk.
Since any formal proof is a finite sequence of formulas, it can contain only finitely many sentences from T. Thus, if T ⊢ ϕ then there is a finite subtheory T0 ⊂ T such that T0 ⊢ ϕ.
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