A Friendly Introduction to Mathematical Logic by Christopher C. Leary PDF

By Christopher C. Leary

ISBN-10: 1942341326

ISBN-13: 9781942341321

On the intersection of arithmetic, computing device technology, and philosophy, mathematical good judgment examines the facility and barriers of formal mathematical pondering. during this growth of Leary's ordinary 1st version, readers with out earlier examine within the box are brought to the fundamentals of version thought, facts concept, and computability idea. The textual content is designed for use both in an top department undergraduate lecture room, or for self examine. Updating the first Edition's therapy of languages, constructions, and deductions, resulting in rigorous proofs of Gödel's First and moment Incompleteness Theorems, the elevated 2d version incorporates a new advent to incompleteness via computability in addition to options to chose workouts.

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We will say that v is free in φ if 1. φ is atomic and v occurs in (is a symbol in) φ, or 2. φ :≡ (¬α) and v is free in α, or 3. φ :≡ (α ∨ β) and v is free in at least one of α or β, or 4. φ :≡ (∀u)(α) and v is not u and v is free in α. Thus, if we look at the formula ∀v2 ¬(∀v3 )(v1 = S(v2 ) ∨ v3 = v2 ), the variable v1 is free whereas the variables v2 and v3 are not free. A slightly more complicated example is (∀v1 ∀v2 (v1 + v2 = 0)) ∨ v1 = S(0). In this formula, v1 is free whereas v2 is not free.

Chaff: The next couple of proofs are proofs by induction on the complexity of terms or formulas. 2 on page 16 if you find these difficult. 6. Suppose that s1 and s2 are variable assignment functions into a structure A such that s1 (v) = s2 (v) for every variable v in the term t. Then s1 (t) = s2 (t). Proof. We use induction on the complexity of the term t. If t is either a variable or a constant symbol, the result is immediate. If t :≡ f t1 t2 . . tn , then as s1 (ti ) = s2 (ti ) for 1 ≤ i ≤ n by the inductive hypothesis, the definition of s1 (t) and the definition of s2 (t) are identical, and thus s1 (t) = s2 (t).

See Exercise 7. 1 Exercises 1. 3 that the truth or falsity of the sentences 1 + 1 = 2 and (∀x)(x + 1 = x) might not be automatic. Find a structure for the language discussed there that makes the sentence 1 + 1 = 2 true. Find another structure where 1 + 1 = 2 is false. Prove your assertions. Then show that you can find a structure where (∀x)(x + 1 = x) is true, and another structure where it is false. 2. Let the language L be {S, <}, where S is a unary function symbol and < is a binary relation symbol.

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A Friendly Introduction to Mathematical Logic by Christopher C. Leary

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