A model for an extended logic

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.