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EXPRESSIVE LIMITATIONS OF NAÏVE SET THEORY IN LP AND MINIMALLY INCONSISTENT LP.

Published online by Cambridge University Press:  21 February 2014

NICK THOMAS
NICK THOMAS*
Affiliation:
University of Connecticut
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CONNECTICUT, USA E-mail: [email protected]
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Abstract

We give some negative results on the expressiveness of naïve set theory (NS) in LP and in the four variants of minimally inconsistent LP defined in Crabbé (2011): ${\rm{L}}{{\rm{P}}_m},{\rm{L}}{{\rm{P}}_ = },{\rm{L}}{{\rm{P}}_ \subseteq }$, and ${\rm{L}}{{\rm{P}}_ \supseteq }$. We show that NS in LP cannot prove the existence of sets that behave like singleton sets, Cartesian pairs, or infinitely ascending linear orders. We show that NS is close to trivial in ${\rm{L}}{{\rm{P}}_m}$ and ${\rm{L}}{{\rm{P}}_ \subseteq }$, in the sense that its only minimally inconsistent model is a one-element model. We show that NS in ${\rm{L}}{{\rm{P}}_ = }$ and ${\rm{L}}{{\rm{P}}_ \supseteq }$ has the same limitations we give for NS in LP.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 2 , June 2014 , pp. 341 - 350
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

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