dc.contributor.advisor Ullman, Arthur W. J. dc.contributor.author Segal, Arthur Cherny en_US dc.date.accessioned 2019-10-11T15:11:01Z dc.date.available 2019-10-11T15:11:01Z dc.date.created 1966 en_US dc.date.issued 1966 en_US dc.identifier aleph-255102 en_US dc.identifier.uri https://repository.tcu.edu/handle/116099117/33789 dc.description.abstract In Chapter I, some relationships are obtained between V^w , defined as the set of sequences of zeros and ones, and a Cantor set K having positive measure. First, a useful well-ordering of V^w is found. (The set of sequences corresponding to a fixed limit ordinal plus the natural numbers is called a block.) Then, K is defined arithmetically, and a 1-1 mapping R is defined on the collection of subsets of V^w onto the collection of subsets of K. After a consideration of some algebraic properties of V^w, an analog of Lebesgue's example of a nonmeasurable set is constructed in K. It follows immediately that if Z is a measure zero subset of K, then there are uncountably many blocks in which R^-1[Z] has no elements. A useful class of homeomorphisms of K onto itself is found, and their properties are used to obtain another sufficient condition for the R^-1 image of a subset of K to be void in uncountably many blocks, namely that the set be of the first category in K. In Chapter II, the set K is viewed as a model for a propositional logic containing infinitely long formulas. First, the syntax of the logic is discussed, and then it is seen that V^w can serve as a truth table for the logic, each sequence in V^w being regarded as a row in the truth table. A description is given of the distribution in V^w of rows on which fundamental formulas (conjuncts of variables and their negations) assume the value "true". Then, a mapping T is defined on the class of well-formed formulas, onto K, as follows: T[L] = R(A], where A is the set of rows on which the well-formed formula L assumes the value "true''. Topological and measure properties of T images of the types of fundamental formulas are established. It is next shown that images of finite fundamental formulas form a countable base for the topology of K, and as a consequence, it is found that B is a Borel subset of K if and only if there exists L such that T[L] =B and L is obtainable from the variables by at most countable applications of negation, conjunction, and disjunction. However, an example shows that the normal form (disjunct of fundamental formulas) of the T^-1 image of a closed set can involve a non-reducible uncountable disjunct of fundamental formulas. dc.format.extent 37 leaves, bound en_US dc.format.medium Format: Print en_US dc.language.iso eng en_US dc.relation.ispartof Texas Christian University dissertation en_US dc.relation.ispartof AS38.S44 en_US dc.subject.lcsh Logic, Symbolic and mathematical en_US dc.subject.lcsh Mathematics--Philosophy en_US dc.title A model for an extended logic en_US dc.type Text en_US etd.degree.department Department of Mathematics etd.degree.level Doctoral local.college College of Science and Engineering local.department Mathematics local.academicunit Department of Mathematics dc.type.genre Dissertation local.subjectarea Mathematics dc.identifier.callnumber Main Stacks: AS38 .S44 (Regular Loan) dc.identifier.callnumber Special Collections: AS38 .S44 (Non-Circulating) etd.degree.name Doctor of Philosophy etd.degree.grantor Texas Christian University
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