Read e-book online A Tour Through Mathematical Logic (Carus Mathematical PDF
By Robert S. Wolf
The rules of arithmetic comprise mathematical good judgment, set concept, recursion concept, version idea, and Gödel's incompleteness theorems. Professor Wolf offers right here a consultant that any reader with a few post-calculus event in arithmetic can learn, get pleasure from, and examine from. it will probably additionally function a textbook for classes within the foundations of arithmetic, on the undergraduate or graduate point. The publication is intentionally much less dependent and extra straight forward than ordinary texts on foundations, so can be appealing to these outdoor the school room surroundings desirous to know about the topic.
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Additional info for A Tour Through Mathematical Logic (Carus Mathematical Monographs)
The usual logical equivalences used to move negations through quantifiers (∼ ∀xP ↔ ∃x ∼ P, and ∼ ∃xP ↔ ∀x ∼ P) are derivable in first-order logic, and by applying them repeatedly to ∼ Q we get a prenex formula that is equivalent to P. If P has the form Q1 ∧ Q2 then, by the induction hypothesis, Q1 and Q2 have prenex equivalents Q1 and Q2 , respectively. 3), we may assume that none of the bound variables of Q1 appears in Q2 , and vice-versa. But if this is so, all of the quantifiers of both Q1 and Q2 can simply be moved to the front of the formula, creating a prenex equivalent of P.
We conclude this section with several examples of this translation process. We’ll begin with some nonmathematical statements, but even though the content of these statements may seem frivolous, the translation process is still a valuable mental exercise. It is important to bear in mind that when a statement is translated from one form to another, the original version and the translated version must have exactly the same free variables. 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.
Translate these statements into completely symbolic form. Use m, n, and k as integer variables and x, y, and z as real variables: (a) n is a prime number. ) (b) There is no largest real number. (c) For any two distinct integers, there is a real number between them. (d) For any two integers, there’s an integer between them if and only if they differ by more than 1. 4 First-order languages and theories We will now give a more rigorous and detailed treatment of the ideas introduced in the previous two sections.
A Tour Through Mathematical Logic (Carus Mathematical Monographs) by Robert S. Wolf