A formula for any real number, maybe

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 $$r=\log \left(\underset{x_0,x_1\in \mathbb{R}}{\sup }\underset{x_2\in \mathbb{R}}{\inf }\underset{x_3\in \mathbb{R}}{\sup }\underset{x_4\in \mathbb{R}}{\inf }\underset{m\in \mathbb{N}}{\sup }\underset{n_0,\dots ,n_A\in \mathbb{N}}{\inf }{x}_0^2\left[\begin{array}{c}(n_0-2{)}^2+(n_1-m{)}^2\\ +n_2+(n_B-n_C{)}^2\\ +n_3{\sum }_{k=0}^4(x_k-\frac{n_{k+5}}{1+n_4}+n_4{)}^2\\ +{\sum }_{i,j=0}^B(n_{9+2^i{3}^j}-n_i^{n_j}{)}^2\end{array}\right]\right).$$ 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.

17 pages, 3 figures