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4 - Extending the Iterative Conception of Set: A Height-Potentialist Perspective

from Part I - Structuralism, Extendability, and Nominalism

Published online by Cambridge University Press:  26 January 2021

Geoffrey Hellman
Geoffrey Hellman
Affiliation:
University of Minnesota
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Summary

After reviewing shortcomings of Boolos’ presentation of the “Iterative conception of set,” we formulate simple axioms on “stages” of set formation, using modal logic and the logic of plurals, that imply both the Axiom of Infinity and the Axiom of Replacement. (Two routes to these results are presented, the second less open to charges of circularity than the first.) We then present two other advantages of the Height-Potentialist framework, pertaining to motivating the smallest large cardinals (strongly inaccessible and Mahlo), and to furnishing attractive resolutions of the set-theoretic paradoxes. Our routes to these results are not available from within the Height-Actualist framework.

Type
Chapter
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Mathematics and Its Logics
Philosophical Essays
 , pp. 61 - 73
Publisher: Cambridge University Press
Print publication year: 2021

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References

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