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ABSTRACTION PRINCIPLES AND THE SIZE OF REALITY

Part of: Set theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  19 August 2025

BOKAI YAO
BOKAI YAO*
Affiliation:
DEPARTMENT OF PHILOSOPHY AND RELIGIOUS STUDIES PEKING UNIVERSITY BEIJING 100871 CHINA Url: https://bokaiyao.com
*
E-mail: bkyao@pku.edu.cn
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Abstract

The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic principles. We investigate how the size of reality—i.e., the number of urelements—interacts with these principles. We show that Basic Law V implies that for some well-ordered cardinal $\kappa $, there is no set of urelements of size $\kappa $. Building on recent work by Hamkins [10], we show that, under certain additional axioms, Basic Law V holds if and only if the urelements form a set. We construct models of urelement set theory in which the Reflection Principle holds while Hume’s Principle fails for sets. Additionally, assuming the consistency of an inaccessible cardinal, we produce a model of Kelley–Morse class theory with urelements that has a global well-ordering but lacks a definable map satisfying Hume’s Principle for classes.

Keywords

Abstraction principlesHume’s principleset theory with urelementsZFUZFA

MSC classification

Primary: 03A05: Philosophical and critical 03E25: Axiom of choice and related propositions 03E30: Axiomatics of classical set theory and its fragments 03E35: Consistency and independence results

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 18 , Issue 3 , September 2025 , pp. 812 - 825
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Copyright
© The Author(s) 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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