A formula for any real number, maybe

TL;DR

The paper constructs a fixed triple $(A,B,C)$ and a Diophantine-encoded expression that, within suitable well-founded models of ${\rm ZFC}$, can realize any nonnegative real in $L$ as the value of a complex supremum–infimum formula. It achieves this by combining a three-step strategy: encode a real into an inner-model lattice via $L[x]$ and $c$-degrees, represent the encoding with a $\Sigma^1_4$ definable set that can be forced to isolate a singleton real, and implement the value with a compact Diophantine computer $f_{A,B,C}$. The approach leverages forcing, descriptive set theory, and inner-model theory (including the notion of mice and large cardinals) to establish a strong form of independence: the realized value can depend on the ambient model of set theory, and, under certain assumptions, not every real is attainable as a value. The results illuminate deep connections between computability, Diophantine encodings, and the multiverse of set-theoretic universes, highlighting both the power and the limitations of definability in high-strength theories.

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.

Figures (3)

• Figure 1: The $n$th bit of $r$ determines whether the corresponding level of the lattice $\mathcal{D}$ has width $1$ or $2$.
• Figure 2: 'Memory layout' of our Diophantine formula. $n_0$ is a fixed reference value of $2$. $n_1$ is the index of the open set being constructed. $n_2$ helps ensure that the maximum value of the function is $1$. $n_3$ is the scale of the paraboloid. $n_4$ sets the denominators of the coordinates of the vertex of the paraboloid and shifts them in the negative direction. $n_5$ through $n_9$ set the numerators of the coordinates of the vertex. Addresses of the form $n_{9+2^i3^j}$ store the value of $n_i^{n_j}$ and the remaining addresses are free variables.
• Figure 3: A mouse bhlpage31059400.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.4
• proof
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 3.1
• proof
• Corollary 3.2