Chapter

Set Theory

Geoffrey Hellman

in Mathematics without Numbers

Published in print November 1993 | ISBN: 9780198240341
Published online November 2003 | e-ISBN: 9780191597664 | DOI: http://dx.doi.org/10.1093/0198240341.003.0003

Series: Clarendon Paperbacks

 Set Theory

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The modal‐structural approach is extended to set theory, as an alternative to reifying sets as actual abstract objects, thinking instead of ‘possible results of selection’. Drawing inspiration from Zermelo's important paper of 1930 and suggestions of Putnam, it is shown how to incorporate modality into second‐order Zermelo–Fraenkel set theory, avoiding proper classes and a fixed universe of sets, instead sustaining extendability principles (‘any set‐domain can be part of a larger one’, and strengthenings of this). It is then shown how this naturally leads to important kinds of ‘large cardinals’ (inaccessibles, Mahlo cardinals), motivated ‘from below’ rather than ‘from above’.

Keywords: extendability principles; inaccessible cardinals; large cardinals; Mahlo cardinals; proper classes; set theory; Zermelo–Fraenkel set theory

Chapter.  17662 words. 

Subjects: Philosophy of Mathematics and Logic

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