A Foundation for the Core Mathematician

TL;DR

This paper proposes a new foundational framework for core mathematics based on a specific set-theoretic model that assigns definite truth-values to core mathematical statements.

Contribution

It introduces a system of axioms and a concrete model that encapsulate essentially all core mathematics, providing definitive truth-values for assertions.

Findings

A specific set-theoretic model captures core mathematics.

Core mathematical assertions have definite truth-values in this model.

The framework addresses variability in set-theoretic models.

Abstract

The foundations of mathematics have long been considered settled by the Zermelo-Fraenkel-Choice axioms. But set theory abounds in models with different truths and even classical questions such as the measurability of projective sets can vary between models. The core of mathematics resides in the study of structures built from the set R of real numbers. This paper proposes a foundation for core mathematics, with both a system of axioms and a definite model of those axioms, in which essentially all core mathematics is incorporated. This definite model delivers a definite truth-value, either true or false, to any core mathematical assertion.