A Problem Course in Mathematical Logic by Stefan Bilaniuk

Logic

By Stefan Bilaniuk

An issue path in Mathematical common sense is meant to function the textual content for an advent to mathematical good judgment for undergraduates with a few mathematical sophistication. It provides definitions, statements of effects, and difficulties, besides a few factors, examples, and tricks. the assumption is for the scholars, separately or in teams, to benefit the fabric through fixing the issues and proving the consequences for themselves. The booklet may still do because the textual content for a path taught utilizing the transformed Moore-method.

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STRUCTURES AND MODELS interpret the language’s 2-place relation symbol. Q is not the only possible structure for LO : (R, <), ({0}, ∅), and (N, N2) are three others among infinitely many more. (Note that in these cases the relation symbol < is interpreted by relations on the universe which are not linear orders. ) On the other hand, (R) is not a structure for LO because it lacks a binary relation to interpret the symbol < by, while (N, 0, 1, +, ·, |, <) is not a structure for LO because it has two binary relations where LO has a symbol only for one, plus constants and functions for which LO lacks symbols.

Then x occurs free in a formula ϕ of L is defined as follows: (1) If ϕ is atomic, then x occurs free in ϕ if and only if x occurs in ϕ. (2) If ϕ is (¬α), then x occurs free in ϕ if and only if x occurs free in α. (3) If ϕ is (β → δ), then x occurs free in ϕ if and only if x occurs free in β or in δ. (4) If ϕ is ∀vk ψ, then x occurs free in ϕ if and only if x is different from vk and x occurs free in ψ. An occurrence of x in ϕ which is not free is said to be bound . A formula σ of L in which no variable occurs free is said to be a sentence.

Tk is a formula. (2) If t1 and t2 are terms, then = t1 t2 is a formula. (3) If α is a formula, then (¬α) is a formula. (4) If α and β are formulas, then (α → β) is a formula. (5) If ϕ is a formula and vn is a variable, then ∀vn ϕ is a formula. (6) Nothing else is a formula. Formulas of form 1 or 2 will often be referred to as the atomic formulas of L. 3 are borrowed directy from propositional logic. As before, we will exploit the way 28 5. LANGUAGES formulas are built up in making definitions and in proving results by induction on the length of a formula.

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