Mathematical Realism and the Theory of Sets

Dissertation, University of Notre Dame (1984)
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Abstract

Set theoretic platonism is the view that there exist objective, mind-independent abstract sets, and that set theory is the science of these entities. For the realist, this view offers the most natural semantical account of set theoretic discourse. Nonetheless, set theoretic platonism is beset by a number of serious difficulties. Chief among these, it turns out, is that it must deny the fundamental set theoretic intuition that any available objects can be collected into a further object. After a brief historical overview, I discuss this difficulty at length in Part I of my dissertation, and argue that three responses to it proffered respectively by Lear, Parsons, and Maddy are unsatisfactory. A comprehensive solution to this problem, I suggest, would have to consist in the development of a metaphysically and mathematically adequate alternative ontological framework. Necessary conditions of such a framework's metaphysical adequacy would be that it be rich enough to supplant the platonistic ontology of sets, and that it avoid the problem above. To be mathematically adequate it would have to preserve, in some strong sense, classical set theory . In Part II I develop just such an alternative framework. Beginning with Hao Wang's suggestive elaborations of some of Cantor's remarks, I develop a detailed account according to which set theory is based upon an idealization of our ability to collect objects together mentally. I then go on to show that this account is both metaphysically and mathematically adequate. An appendix is included in which the work of Cantor is discussed

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Christopher Menzel
Texas A&M University

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