Algebraic Methods of Mathematical LogicAlgebraic Methods of Mathematical Logic focuses on the algebraic methods of mathematical logic, including Boolean algebra, mathematical language, and arithmetization. The book first offers information on the dialectic of the relation between mathematical and metamathematical aspects; metamathematico-mathematical parallelism and its natural limits; practical applications of methods of mathematical logic; and principal mathematical tools of mathematical logic. The text then elaborates on the language of mathematics and its symbolization and recursive construction of the relation of consequence. Discussions focus on recursive construction of the relation of consequence, fundamental descriptively-semantic rules, mathematical logic and mathematical language as a material system of signs, and the substance and purpose of symbolization of mathematical language. The publication examines expressive possibilities of symbolization; intuitive and mathematical notions of an idealized axiomatic mathematical theory; and the algebraic theory of elementary predicate logic. Topics include the notion of Boolean algebra based on joins, meets, and complementation, logical frame of a language and mathematical theory, and arithmetization and algebraization. The manuscript is a valuable reference for mathematicians and researchers interested in the algebraic methods of mathematical logic. |
Contents
9 | |
Chapter 2 THE LANGUAGE OF MATHEMATICS AND ITS SYMBOLIZATION | 25 |
Chapter 3 RECURSIVE CONSTRUCTION OF THE RELATION OF CONSEQUENCE | 38 |
Chapter 4 EXPRESSIVE POSSIBILITIES OF THE PRESENT SYMBOLIZATION | 70 |
Chapter 5 INTUITIVE AND MATHEMATICAL NOTIONS OF AN IDEALIZED AXIOMATIC MATHEMATICAL THEORY | 83 |
Chapter 6 THE ALGEBRAIC THEORY OF ELEMENTARY PREDICATE LOGIC | 101 |
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a n b according assertion axiomatic axioms basic instances basic Lindenbaum algebra belong binary binary relation Boolean expressions Boolean function Boolean indeterminates classes condition construction corresponding defined definition denote derived determined equality equivalent example exists finite sequences formal consequence formal proof formalized theory free Boolean algebra free individual indeterminates free occurrence holds homomorphism i-homomorphism i-prime over-ideals indexed Boolean algebra indexing homomorphism indexing ideal individual constant individual indeterminates inductive assumption infinite intuitive isomorphism joins and meets laws Lemma mapping mathematical language mathematical logic mathematical theory meaning metamathematical n-tuples natural numbers notion obviously occur free operations order theory positive integers possible precisely predicate constants predicate logic prime ideal properties propositional calculus prove quantifier quotient algebra real numbers recursive relation of consequence Remark replacement Rule satisfied Sect semantic sense sentence sentential expressions set theory set-theoretic similarly subalgebra subexpression subsets substitutively indexed algebra symbolic language symbolized mathematical syntactic valid variables