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The satisfaction of a formula ϕ under an evaluation h in A will be a ternary relation of our metatheory. If this relation between A, ϕ , and h holds, then we shall write A |= ϕ [h] (pronounced: “ϕ holds in A under h”, “ϕ is true in A under h”, or “ϕ is satisfied by A under h”); if this relation does not hold, then we shall write A |= ϕ [h]. This relation will likewise be defined by recursion on the construction of formulae, starting with the simplest formulae, the atomic formulae, and indeed simultaneously for all evaluations.

Here we call Σ an axiom system for T if T = { α ∈ Sent(L) | Σ α }. Observe that the set { α ∈ Sent(L) | Σ α }, which we wish to denote also by Ded(Σ ), is deductively closed. Indeed, if α1 , . . , αn ∈ Ded(Σ ) and {α1 , . . , αn } α , then we have n proofs of α1 , . . , αn from Σ , and a proof of α from {α1 , . . , αn }; from these n + 1 proofs we can easily assemble a proof of α directly from Σ . g. finite. Appendix A will explain more precisely how the concept of “effective enumerability” can be given a definition.

N from Σ , and a proof of α from {α1 , . . , αn }; from these n + 1 proofs we can easily assemble a proof of α directly from Σ . g. finite. Appendix A will explain more precisely how the concept of “effective enumerability” can be given a definition. In the following examples we shall write down the corresponding axiom systems concretely. Usually the most interesting case of a possible axiomatization of the L-theory of a class M is that in which M has exactly one L-structure A. 1) simplifies to Th(A) = { α ∈ Sent(L) | A |= α }.