Logic

A Tour Through Mathematical Logic (Carus Mathematical by Robert S. Wolf

By Robert S. Wolf

The rules of arithmetic contain mathematical good judgment, set idea, recursion idea, version concept, and Gödel's incompleteness theorems. Professor Wolf presents the following a consultant that any reader with a few post-calculus adventure in arithmetic can learn, get pleasure from, and study from. it might additionally function a textbook for classes within the foundations of arithmetic, on the undergraduate or graduate point. The ebook is intentionally much less dependent and extra hassle-free than normal texts on foundations, so can be appealing to these outdoor the school room surroundings desirous to find out about the topic.

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But the obvious truth of this equivalence may be considered a nonrigorous proof, acceptable in most circumstances. Definitions. A theory T is called consistent if no contradiction can be derived from it. A formula P is said to be independent of T if neither P nor ∼ P can be proved from T . T is called complete if it is consistent and no sentence of its language is independent of it. In other words, T is complete if T P or T ∼ P, but not both, for every sentence P of the language of T . ) The subject that deals with first-order languages and theories is called first-order predicate logic, or simply first-order logic.

Example 13. Translate the statement “Every person likes somebody who doesn’t like him or her” into symbols, by defining a propositional variable for each atomic substatement and specifying the domain of every mathematical variable used. 22 Predicate Logic Solution. The statement is based on the property of one person liking another. So let L(x, y) stand for “x likes y,” where x and y are “people variables”—that is, their domain is the set of all people. The statement can then be written symbolically as ∀x ∃y[L(x, y)∧ ∼ L(y, x)].

Also, if t1 , t2 , . . , tn are terms and f is an n-ary function symbol, then the string of symbols f (t1 , t2 , . . , tn ) is also a term. ) Having defined terms, we can then define the well-formed formulas, or simply formulas, of a first-order language. First of all, if t1 , t2 , . . , tn are terms and R is an n-ary relation symbol, then the string of symbols R(t1 , t2 , . . , tn ) is a formula. Equations of the form t1 = t2 are also formulas. ) Inductively, if P and Q are any two formulas, then so are (P ∧ Q), (P ∨ Q), ∼ P, (P → Q), (P ↔ Q), ∀vn P, and ∃vn P (where n can be any natural number).

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