By Stefan Bilaniuk

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**Extra info for A problem course in mathematical logic : is a freeware mathematics text**

**Example text**

1. M = ({5}) is a structure for L= , so Th(M) is a maximally consistent set of sentences. Since it turns out that Th(M) = Th ({ ∀x ∀y x = y }), this also gives us an example of a set of sentences Σ = { ∀x ∀y x = y } such that Th(Σ) is maximally consistent. 7. If Σ is a maximally consistent set of sentences, τ is a sentence, and Σ τ , then τ ∈ Σ. 8. Suppose Σ is a maximally consistent set of sentences and τ is a sentence. Then ¬τ ∈ Σ if and only if τ ∈ / Σ. 9. Suppose Σ is a maximally consistent set of sentences and ϕ and ψ are any sentences.

3) Commutative groups. (4) Fields of characteristic 5. CHAPTER 7 Deductions Deductions in first-order logic are not unlike deductions in propositional logic. Of course, some changes are necessary to handle the various additional features of propositional logic, especially quantifiers. In particular, one of the new axioms requires a tricky preliminary definition. Roughly, the problem is that we need to know when we can replace occurrences of a variable in a formula by a term without letting any variable in the term get captured by a quantifier.

Such a structure for a given language should supply most of the ingredients needed to interpret formulas of the language. Throughout this chapter, let L be an arbitrary fixed countable firstorder language. All formulas will be assumed to be formulas of L unless stated otherwise. 1. A structure M for L consists of the following: (1) A non-empty set M, often written as |M|, called the universe of M. (2) For each constant symbol c of L, an element cÅ of M. e. a k-place function on M. e. a k-place relation on M.

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