arXiv subjects

Elliot Glazer

Publications and source records attributed to Elliot Glazer.

6 recordsLinked to original sources

arXiv2026

Loose Inductive Set Theories

We prove that KP, with Full Set Induction, is neither solid nor tight. More generally, every consistent theory obtained from KPC by adding a quantifier bounded set of sentences has a model with a parameter-free definable class surjection from its natural numbers onto its universe. Using such enumerations to define generic extensions by relative interpretations, we show that theories with the stable definable ground property are not tight. Ground definability over MOST and continuum coding over ZFC^{Π_3} yield applications to weak inductive set theories with large cardinals and forcing axioms. Finally, solidity for quantifier bounded extensions of MOST with Full Set Induction implies a Ground Axiom formulated using a fixed finite level of genericity; under the same solidity hypothesis, the corresponding extension of full ZFC proves the ordinary Ground Axiom.

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

FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI

We introduce FrontierMath, a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians. The questions cover most major branches of modern mathematics -- from computationally intensive problems in number theory and real analysis to abstract questions in algebraic geometry and category theory. Solving a typical problem requires multiple hours of effort from a researcher in the relevant branch of mathematics, and for the upper end questions, multiple days. FrontierMath uses new, unpublished problems and automated verification to reliably evaluate models while minimizing risk of data contamination. Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community. As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.

cs.AI↗

arXiv2024

Global choice is not conservative over local choice for Zermelo set theory

We prove that Global Choice is not conservative over ZC and that ZF $-$ Union does not prove existence of $x \cup y$ for all $x$ and $y$. Each proof is by constructing a pathological model inside a symmetric extension of $L.$

math.LO↗

arXiv2023

A Topological Tennenbaum Theorem

We prove that in Borel models of arithmetic on an uncountable Polish space, neither addition nor multiplication is continuous. This is an analogue of Tennenbaum's Theorem for topological models of arithmetic. This answers a question of Enayat, Hamkins, and Wcisło.

math.LO↗

arXiv2022

A choiceless box game paradox

We identify a choiceless variation of the box game paradox, in which players predict unknown real numbers with near-perfect accuracy despite lacking any useful information. We also verify that choice is necessary in the solution of the original problem and consider some further variations of the game.

math.LO↗