arXiv subjects

Bokai Yao

Publications and source records attributed to Bokai Yao.

7 recordsLinked to original sources

arXiv2026

The Iterative Conception Reconsidered

We investigate the iterative conception of set in its most general form, allowing urelements without assuming that they form a set. We formulate this conception in two ways, as stage theory and as level theory, and develop a general theory of levels with urelements. Unlike their pure-set counterparts, the resulting stage and level theories are not set-theoretically equivalent; moreover, second-order level theory with urelements is not weakly quasi-categorical. We then consider further principles governing stages and levels, motivated by directedness, unboundedness, and reflection. Some of these principles restore set-theoretic equivalence between the corresponding theories, while their level-theoretic versions yield forms of quasi-categoricity. These principles form strict implication hierarchies, thereby revealing distinct stronger conceptions of set beyond the basic iterative conception.

math.LO↗

arXiv2026

Reflection Principles in ZFU

We separate the Collection Principle, the Reflection Principle, and the Partial Reflection Principle in ZF with urelements (ZFU), despite their equivalence under the Axiom of Choice. In particular, Collection and the Partial Reflection Principle are independent of one another, and Collection together with Partial Reflection does not imply the Reflection Principle. We show that Reflection and Collection are equivalent assuming either the Tail axiom or Small Violations of Choice.

math.LO↗

arXiv2025

Plenitudinous Urelements and the Definability of Cardinality

The Axiom of Plenitude asserts that every ordinal is equinumerous with a set of urelements, while its stronger form, Plenitude$^+$, extends it to all sets. We investigate these two axioms within ZF set theory with urelements. Assuming that cardinality is definable, Plenitude$^+$ together with the Collection Principle implies the Reflection Principle. If either cardinality is representable or Small Violations of Choice (SVC) holds, Plenitude$^+$ implies the Reflection Principle. In contrast, Plenitude is considerably weaker: SVC + Plenitude does not prove the Collection Principle, and SVC + Plenitude + Reflection Principle does not prove Plenitude$^+$.

math.LO↗

arXiv2025

Abstraction Principles and the Size of Reality

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 $κ$, there is no set of urelements of size $κ$. 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.

math.LO↗

arXiv2024

Axiomatization and Forcing in Set Theory with Urelements

In the first part of this paper, we consider several natural axioms in urelement set theory, including the Collection Principle, the Reflection Principle, the Dependent Choice scheme and its generalizations, as well as other axioms specifically concerning urelements. We prove that these axioms form a hierarchy over $\ZFCUR$ (ZFC with urelements formulated with Replacement) in terms of direct implication. The second part of the paper studies forcing over countable transitive models of $\ZFUR$. We propose a new definition of $¶$-names to address an issue with the existing approach. We then prove the fundamental theorem of forcing with urelements regarding axiom preservation. Moreover, we show that forcing can destroy and recover certain axioms within the previously established hierarchy. Finally, we demonstrate how ground model definability may fail when the ground model contains a proper class of urelements.

math.LO↗

arXiv2023

Set Theory with Urelements

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also explored. In Chapter 3, I investigate forcing with urelements and develop a new approach that addresses a drawback of the existing machinery. I demonstrate that forcing can preserve, destroy, and recover the axioms isolated in Chapter 2 and discuss how Boolean ultrapowers can be applied in urelement set theory. Chapter 4 delves into class theory with urelements. I first discuss the issue of axiomatizing urelement class theory and then explore the second-order reflection principle with urelements. In particular, assuming large cardinals, I construct a model of second-order reflection where the principle of limitation of size fails.

math.LO↗

arXiv2022

Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal

After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove, second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal $κ$ is supercompact if and only if every $Π^1_1$ sentence true in a structure $M$ (of any size) containing $κ$ in a language of size less than $κ$ is also true in a substructure $m\prec M$ of size less than $κ$ with $m\capκ\inκ$.

math.LO↗