A formula for any real number, maybe

TL;DR

The paper constructs three specific natural numbers such that for any real number, it is consistent with ZFC that the real equals a complex formula involving these numbers, and explores set-theoretic implications of large cardinals.

Contribution

It provides a method to encode any real number as a value of a complex formula with three fixed natural numbers, linking set theory and real analysis.

Findings

Any real number can be represented by the formula with fixed A, B, C under ZFC.

The existence of certain large cardinals implies some reals cannot be represented.

The work connects set-theoretic mice with the encoding of real numbers.

Abstract

We discuss how to write down three specific natural numbers $A$, $B$, $C$ such that for any real number $r$ you've probably ever thought of, it is consistent with $\mathsf{ZFC}$ set theory that $$\def\Rb{\mathbb{R}}\def\Nb{\mathbb{N}}r = \log\left(\sup_{x_0,x_1 \in \Rb} \inf_{x_2 \in \Rb} \sup_{x_3 \in \Rb}\inf_{x_4 \in \Rb}\sup_{m \in \Nb}\inf_{n_0,\dots,n_{A} \in \Nb} x^2_0 \begin{bmatrix} \phantom{+}(n_0 - 2)^2 + (n_1-m)^2 \\ + n_2 + (n_B - n_C)^2 \\ + n_3 \sum_{k=0}^4 ( x_k - \frac{n_{k+5}}{1+n_4} +n_4)^2 \\ + \sum_{i,j = 0}^B (n_{9+2^i3^j} - n_i^{n_j})^2 \end{bmatrix} \right).$$ We also discuss why it's possible, assuming the existence of certain large cardinals, for there to be a real number $s$ which cannot be the value of this formula for our particular $A$, $B$, $C$. This involves set-theoretic mice.