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Founding Mathematics on Semantic ConventionsClassical Mathematics and Plenitudinous Combinatorialism

Founding Mathematics on Semantic Conventions: Classical Mathematics and Plenitudinous... [This and the following two chapters discuss and criticize some existing systems of mathematics. This one is about classical mathematics as inspired by Georg Cantor, and especially its use of undefinable sets. The author refutes some arguments for actual infinity, and suggests that the picture the Cantorians provide of a realm of higher infinities is so unclear that there are good reasons to doubt they are even describing a possibility. More specifically, this chapter’s topics include the relationship between large cardinal axioms and theorems of arithmetic; Cantor’s two principles of generation for the transfinite ordinals; his proof that there are “more” real numbers than natural numbers; and the continuum hypothesis.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Founding Mathematics on Semantic ConventionsClassical Mathematics and Plenitudinous Combinatorialism

Part of the Synthese Library Book Series (volume 446)
Hansen, Casper Storm
Founding Mathematics on Semantic Conventions — Nov 5, 2021
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Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-88533-5
Pages
9 –27
DOI
10.1007/978-3-030-88534-2_2
Publisher site
See Chapter on Publisher Site

Abstract

[This and the following two chapters discuss and criticize some existing systems of mathematics. This one is about classical mathematics as inspired by Georg Cantor, and especially its use of undefinable sets. The author refutes some arguments for actual infinity, and suggests that the picture the Cantorians provide of a realm of higher infinities is so unclear that there are good reasons to doubt they are even describing a possibility. More specifically, this chapter’s topics include the relationship between large cardinal axioms and theorems of arithmetic; Cantor’s two principles of generation for the transfinite ordinals; his proof that there are “more” real numbers than natural numbers; and the continuum hypothesis.]

Published: Nov 5, 2021

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