A Friendly Introduction to Mathematical Logic - download pdf or read online

Logic

By Christopher C. Leary

ISBN-10: 1942341326

ISBN-13: 9781942341321

On the intersection of arithmetic, laptop technological know-how, and philosophy, mathematical good judgment examines the ability and barriers of formal mathematical pondering. during this growth of Leary's common 1st version, readers with out prior learn within the box are brought to the fundamentals of version thought, evidence idea, and computability concept. The textual content is designed for use both in an top department undergraduate lecture room, or for self research. Updating the first Edition's therapy of languages, constructions, and deductions, resulting in rigorous proofs of Gödel's First and moment Incompleteness Theorems, the extended second version features a new advent to incompleteness via computability in addition to suggestions to chose workouts.

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Prove, by ordinary induction on the natural numbers, that 12 + 22 + · · · + n2 = n(n + 1)(2n + 1) . 6 2. Prove, by induction, that the sum of the interior angles in a convex n-gon is (n − 2)180◦ . ) 3. Prove by induction that if A is a set consisting of n elements, then A has 2n subsets. 4. Suppose that L is {0, f, g}, where 0 is a constant symbol, f is a binary function symbol, and g is a 4-ary function symbol. Use induction on complexity to show that every L-term has an odd number of symbols.

34 Chapter 1. Structures and Languages Suppose that A |= ∀x∃y¬(x = y). This sentence is, in fact, true in any structure A such that A has at least two elements. If we then proceed to replace the variable x by the variable u, we get the statement ∃y¬(u = y), which will still be true in A, no matter what value we give to the variable u. If, however, we take our original formula and replace x by y, then we find ourselves looking at ∃y¬(y = y), which will be false in any structure. So by a poor choice of substituting variable, we have changed the truth value of our formula.

What is the value of 5 + Ingrid? Is Bogie < 0? 3. Here is a language consisting of one constant symbol, one 3-ary function symbol, and one binary relation symbol: L is { , , }. Describe an Lmodel that has as its universe R, the set of real numbers. Describe another L-model that has a finite universe. 7. Truth in a Structure 27 4. Write a short paragraph explaining the difference between a language and a structure for a language. 5. Suppose that A and B are two L-structures. We will say that A and B are isomorphic and write A ∼ = B if there is a bijection i : A → B such that for each constant symbol c of L, i(cA ) = cB , for each n-ary function symbol f and for each a1 , .

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A Friendly Introduction to Mathematical Logic by Christopher C. Leary


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