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A Foundation for the Core Mathematician

This paper proposes a new axiomatic foundation and a definite model for core mathematics based on the real numbers, aiming to resolve the indeterminacy of traditional set theory by assigning a unique truth value to every core mathematical assertion.

Original authors: David Mumford, Sy-David Friedman

Published 2026-05-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: David Mumford, Sy-David Friedman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine mathematics as a massive, towering skyscraper. For decades, most mathematicians have agreed that the foundation of this building is a specific set of rules called ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). They believe this foundation is solid enough to support every floor, from the basement of simple numbers to the penthouse of complex theories.

However, the authors of this paper, David Mumford and Sy-David Friedman, argue that while ZFC works for the "penthouse," it's actually a bit shaky and full of weird, unexplainable quirks for the "ground floor"—the part of math that deals with real numbers, time, space, and the physical world. They propose a new, sturdier foundation specifically designed for the "core mathematician" who works with the real world.

Here is the breakdown of their argument using simple analogies:

1. The Problem: The "Magic Trick" Foundation

The current foundation (ZFC) relies heavily on a rule called the Axiom of Choice. Think of this axiom as a magical ability to pick one item from an infinite number of boxes, even if you can't see inside them or describe how you picked them.

The Issue: While this magic trick helps mathematicians prove things about abstract, infinite structures, it creates "ghosts." It allows for the existence of sets that are so weird and chaotic they have no connection to reality (like the Banach-Tarski paradox, where you can cut a ball into pieces and reassemble them into two balls of the same size).
The Result: Because of these "ghosts," mathematicians can't be 100% sure that every question they ask has a single, definite "True" or "False" answer. The foundation feels less like a solid floor and more like a foggy landscape where different rules apply in different places.

2. The Proposal: A Foundation Based on Reality

The authors suggest we stop trying to build a foundation for everything (including the ghosts) and instead build a foundation specifically for the Core of Mathematics: the Real Numbers ( $\mathbb{R}$ ), the Integers, and the structures built from them.

They propose three main pillars for this new foundation:

Pillar A: The Real World is Real. They treat the set of real numbers (like the numbers on a ruler) as a "given" fact, just like we accept that we can count apples. They don't try to construct the real numbers from scratch; they just say, "Here they are, they exist."
Pillar B: Ditch the Magic, Keep the Randomness. They reject the "Axiom of Choice" because it creates those weird ghosts. Instead, they adopt Freiling's Axiom.
- The Analogy: Imagine two people throwing darts at a board in space. If they throw them at the exact same time from different planets (so no signal can travel between them fast enough to cheat), and they pick their spots completely randomly, there is no way one thrower's spot could be "predictable" based on the other's.
- This intuition leads to a rule that says: "If you pick two random numbers, they shouldn't be locked into some weird, pre-determined pattern." This rule naturally kills the "Axiom of Choice" and the weird sets it creates.
Pillar C: Stop the Infinite Ladder. In standard math, you can keep building bigger and bigger infinities forever. The authors say, "Stop." They propose building a model where we only go up as high as necessary to handle the real numbers and their subsets, but we stop before we reach the "unworldly" infinities that have no physical meaning.

3. The Two Models: The "Standard" and the "Minimalist"

The paper describes two ways to visualize this new foundation:

Model 1: The "Real World" Model. This is a version of math where the real numbers are exactly what we think they are. In this world, every set of real numbers behaves nicely (they are all "measurable," meaning you can calculate their size/area without paradoxes). It's a clean, logical world where every math problem has a definite answer.
Model 2: The "Minimalist" Model. This is a clever trick using a concept from physics and logic. Imagine a tiny, countable universe that thinks it is huge. Inside this tiny universe, the "real numbers" look like an infinite, uncountable ocean. But from the outside, we know it's actually just a small, finite list.
- Why do this? This model is "complete." In standard math (ZFC), Gödel proved that there are always questions you can't answer. In this minimalist model, because the universe is so tightly controlled, every single question has a definite answer. It's like a puzzle where every piece fits perfectly, and there are no missing pieces.

4. Why This Matters

The authors aren't trying to change how physicists or engineers do their work. They are offering a "cleaner" operating system for the mathematician who studies the nature of numbers and shapes.

The Old Way: Uses a powerful but messy engine (ZFC) that sometimes produces "ghosts" and leaves some questions unanswered.
The New Way: Uses a specialized engine built on randomness and reality. It removes the ghosts, ensures that every set of numbers has a clear size, and guarantees that every mathematical question has a True or False answer.

Summary

Think of the current foundation of math as a massive, chaotic library where some books are written in invisible ink and some shelves are floating in mid-air. Mumford and Friedman are saying: "Let's build a new library specifically for the books we actually read (the real numbers). We'll throw out the invisible ink (the Axiom of Choice) and the floating shelves (the weird infinities). In our new library, every book is visible, every shelf is solid, and every story has a clear ending."

They prove that this new library is possible to build, provided we accept a few reasonable assumptions about the nature of infinity, and it offers a much more satisfying home for the "core" of mathematics.

Technical Summary: A Foundation for the Core Mathematician

Problem Statement
The authors, David Mumford and Sy-David Friedman, argue that the standard Zermelo-Fraenkel set theory with Choice (ZFC) is an inadequate foundation for the "core" of mathematical research. While ZFC is widely accepted, the authors contend that it fails to provide a "firm and fixed" basis because it admits a proliferation of models (the "multiverse" view) and relies on the Axiom of Choice (AC) in ways that generate non-constructive, "unworldly" sets (e.g., via the Banach-Tarski paradox or non-measurable sets). Furthermore, the Platonist belief in a unique universe of sets is undermined by the independence of the Continuum Hypothesis and the existence of large cardinal axioms.

The paper posits that the "central core" of mathematical experience concerns the natural numbers ( $\mathbb{N}$ ), the real numbers ( $\mathbb{R}$ ), and structures derived from them. The authors seek a foundation that:

Treats $\mathbb{R}$ as a given, "real-world" object (a Platonistic view specific to $\mathbb{R}$ , not arbitrary sets).
Ensures that all conjectures in this core have a definite truth value (a unique model).
Avoids the pathologies introduced by uncountable Choice while retaining sufficient strength for standard analysis.

Methodology and Axiomatic Framework
The paper proposes a specific axiomatic system $T$ and two distinct models to realize it.

1. The Axioms ( $T$ ):
The proposed system $T$ is defined as $(ZF^-) + \mathbb{R} + A^{\text{null}}_N + DC$ .

$ZF^-$ : Zermelo-Fraenkel set theory without the Power Set axiom and without uncountable Choice. It includes Extensionality, Foundation, Pairing, Union, Infinity, Comprehension, Collection, and Replacement.
Given $\mathbb{R}$ : The set of real numbers is taken as a primitive, given entity, modeling the physical continuum.
Freiling's Axiom ( $A^{\text{null}}_N$ ): This is the central innovation. Based on a "dart-throwing" thought experiment involving independent random variables and relativity theory (to rule out information leakage), it asserts that for any function $f: [0,1]^{\mathbb{N}} \to \mathcal{N}$ $f : [0, 1]^{N} \to N$ (mapping sequences to null sets), there exists a sequence of independent random reals $\{x_i\}$ ${x_{i}}$ such that $x_i \notin f(\sigma(i))$ $x_{i} \in / f (σ (i))$ for all $i$ $i$ .
- Consequence: This axiom contradicts the uncountable Axiom of Choice. It implies that $\mathbb{R}$ cannot be well-ordered and that the Continuum Hypothesis is false.
Dependent Choice (DC): The system retains the Axiom of Dependent Choice (introduced by Bernays), which is sufficient for most standard analysis (e.g., Baire Category Theorem, existence of limits) but weaker than full AC.

2. The Proposed Model ( $M$ ):
The authors construct a model $M = L_{\alpha^*}(\mathbb{R})$ , where sets are constructed from $\mathbb{N}$ and $\mathbb{R}$ via transfinite induction up to a specific ordinal $\alpha^*$ .

Construction: Starting with $L_0(\mathbb{R}) = \mathbb{R}$ , the model iterates definable sets.
The Limit $\alpha^*$ : The induction stops at the first "fully $\mathbb{R}$ -admissible ordinal" ( $\alpha_{ZF^-}$ ), defined as the point where no new subsets of $\mathbb{R}$ are generated ( $L_{\alpha^*}(\mathbb{R}) \cap \mathcal{P}(\mathbb{R}) = L_{\alpha^*+1}(\mathbb{R}) \cap \mathcal{P}(\mathbb{R})$ ).
Properties: In this model, every statement has a "ground truth." The model satisfies $ZF^-$ , DC, and Freiling's axiom. The existence of such a model is shown to follow from the consistency of ZFC plus the existence of an inaccessible cardinal.

3. The Minimalist Alternative (Solovay's Model):
The paper also discusses a "minimalist" foundation based on Robert Solovay's model, which assumes the consistency of ZF + an inaccessible cardinal.

Construction: A countable transitive model is constructed where all subsets of $\mathbb{R}$ are Lebesgue measurable and have the Baire property.
Completeness: The authors define a theory $T^*$ (an extension of $T$ ) which includes a "minimality property" and a generic set $G$ . They claim that any two well-founded models of $T^*$ satisfy the same first-order sentences. This provides a form of "completeness" for well-founded models, acting as an antidote to Gödelian incompleteness for the core mathematician who rejects large, ill-founded sets.

Key Contributions and Results

Rejection of Uncountable Choice: The paper formally replaces the uncountable Axiom of Choice with Freiling's probabilistic axiom $A^{\text{null}}_N$ , arguing that the latter is more intuitive for the "real world" and avoids the construction of non-measurable sets.
Refinement of Second-Order Arithmetic: While acknowledging that Second-Order Arithmetic ( $Z_2$ ) suffices for much of analysis, the authors argue it is insufficient for defining the set of all Borel sets or topological spaces directly. Their proposed model extends $Z_2$ by explicitly including $\mathbb{R}$ and allowing constructions up to $\alpha^*$ .
Cardinality Limbo: The paper highlights a unique feature of their model: while there is a surjection from $\mathbb{R}$ onto $\omega_1$ (the first uncountable ordinal), there is no bijection due to Freiling's axiom. Thus, $\omega_1$ exists but lacks a clear cardinality relationship with $\mathbb{R}$ , existing in a "limbo" that differs from standard ZFC.
Consistency Proofs: The authors demonstrate that their axioms are consistent relative to the existence of an inaccessible cardinal. They utilize Solovay's forcing techniques to show that if ZFC + Inaccessible is consistent, then their system $T$ (and the stronger $T^*$ ) is consistent.

Significance and Claims
The paper claims to offer a "better foundation" specifically for the "core mathematician"—those working with reals, integers, and derived structures (e.g., analysis, PDEs, Banach spaces).

Definiteness: The primary significance claimed is the restoration of a "firm and fixed" basis where all conjectures have a unique answer, countering the "multiverse" of set theory.
Realism: By grounding the foundation in the physical intuition of random variables (Freiling's axiom) and the given nature of $\mathbb{R}$ , the authors argue their foundation is more aligned with mathematical practice than the abstract, non-constructive nature of full ZFC.
Antidote to Incompleteness: The "minimalist" foundation ( $T^*$ ) is presented as a solution for mathematicians who wish to avoid the independence phenomena of large cardinals while still maintaining a robust theory of the reals.

The authors remain modest, noting that their approach relies on the consistency of an inaccessible cardinal and that probabilistic methods might eventually disprove the measurability of projective sets in certain large cardinal contexts. However, they assert that their framework provides a coherent, non-controversial basis for the vast majority of mathematical research.