Abstract
Publisher Summary This chapter explains set theory and higher-order logic. Several mutual applications of set theory and higher-order logic are developed. Second-order logic is used to discover the standard models of Zermelo–Fraenkel set theory. Consideration of standard models leads to the introduction of new systems of set theory; one of these systems is applied in finding a definition of truth for higher-order sentences and Zerrnelo–Fraenkel set theory with individuals is given a philosophical justification as logically true within higher-order logic. The chapter considers three well-known first-order theories. The first, called “Peano's arithmetic,” has the non-logical constants. The second theory, called “the theory of real closed fields” has the non-logical constants. The last principle is called the “continuity schema” and plays a role analogous to that of the induction schema.
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