Abstract
Publisher Summary This chapter focuses on nonstandard models and related developments. To the average mathematician, the adjective “nonstandard” attaches itself to the noun “analysis.” However, there are also nonstandard models of first-order arithmetic, second-order arithmetic, and even set theory. While the study of nonstandard models of set theory postdates that of nonstandard models of the reals, that of second-order arithmetic stems from about the same time as nonstandard analysis, and the existence, if not the serious study thereof, of nonstandard models of arithmetic predates nonstandard analysis by several decades. There are three aspects to the study of nonstandard models: (1) the study of the models themselves, (2) the use of the models to study formal systems, and (3) the application of such outside logic to ordinary mathematics. The standard models of set theory have been widely used and studied.
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