# Abstraction Principles and the Size of Reality

## TL;DR

This paper explores how the size of the universe of urelements influences Fregean abstraction principles within set theory, revealing conditions under which these principles imply size restrictions or equivalences.

## Contribution

It establishes new connections between Fregean abstraction principles and the size of urelements, including conditions for Basic Law V and models where Hume's Principle fails.

## Key findings

- Basic Law V implies no set of urelements of certain sizes
- Under certain axioms, Basic Law V holds iff urelements form a set
- Models where Reflection holds but Hume's Principle fails

## 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 \cite{hamkins2022fregean}, 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.