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Sets and Point-Sets: Five Grades of Set-Theoretic Involvement in Geometry

Published online by Cambridge University Press:  28 February 2022

John P. Burgess
John P. Burgess*
Affiliation:
Princeton University
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Extract

Cantor was the founder of not one but two theories: an earlier theory of point-sets and a later theory of sets. How are they related? How much more about points and sets of points can be proved with the assumption of sets of sets of points, sets of sets of sets of points, and so on, than without? The present paper is a semi-popular (nontechnical except for presupposing some familiarity with first-order logic) survey of partial answers obtained by logicians during the last decades.

Tarski (1959) has shown how classical geometry (of any number of dimensions, say for definiteness three) can be formalized in a first-order language, here to be called L0, as a first-order theory, here to be called G0. L0 has variables x,y,z, and so on for points, and predicates for a couple of geometric relations among points. G0 has a dozen axioms and one scheme.

Type
Part XIV. Set Theory
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1988 , Issue 2: Volume Two: Symposia and Invited Papers 1988 , 1988 , pp. 456 - 463
Copyright
Copyright © 1989 by the Philosophy of Science Association

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