A formula for any real number, maybe

• Published in 2026
• Added on 2026-02-05

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.

Links

• https://arxiv.org/abs/2602.02384v1
• https://arxiv.org/pdf/2602.02384v1

Other information

BibTeX entry

@article{Aformulaforanyrealnumbermaybe,
	key = {Aformulaforanyrealnumbermaybe},
	type = {article},
	title = {A formula for any real number, maybe},
	author = {James E. Hanson and Connor Watson},
	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.},
	comment = {},
	date_added = {2026-02-05},
	date_published = {2026-02-09},
	urls = {https://arxiv.org/abs/2602.02384v1,https://arxiv.org/pdf/2602.02384v1},
	collections = {animals,attention-grabbing-titles,fun-maths-facts},
	url = {https://arxiv.org/abs/2602.02384v1 https://arxiv.org/pdf/2602.02384v1},
	urldate = {2026-02-05},
	year = 2026,
	archivePrefix = {arXiv},
	eprint = {2602.02384},
	primaryClass = {math.LO}
}