著者

Joel DAVID HAMKINS

出版者

科学基礎論学会

雑誌

Annals of the Japan Association for Philosophy of Science (ISSN:04530691)

巻号頁・発行日

vol.19, pp.37-55, 2011-05-15 (Released:2017-08-01)

参考文献数

11

被引用文献数

1 4

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Set theorists often take their subject as constituting a foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets. In this way, they regard the set-theoretic universe as the universe of all mathematics. And although many set-theorists affirm the Platonic view that there is just one universe of all sets, nevertheless the most powerful set-theoretic tools developed over the past half century are actually methods of constructing alternative universes. With forcing and other methods, we can now produce diverse models of ZFC set theory having precise, exacting features. The fundamental object of study in set theory has thus become the model of set theory, and the subject consequently begins to exhibit a category-theoretic second-order nature. We have a multiverse of set-theoretic worlds, connected by forcing and large cardinal embeddings like constellations in a dark sky. In this article, I will discuss a few emerging developments illustrating this second-order nature. The work engages pleasantly with various philosophical views on the nature of mathematical existence.