(Apologies for the TeX below - the formula is too complicated for HN. There is a mouse on page 13 - see the pdf.)
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 \ZFC{} set theory that $r = \log(\sup_{x_0,x_1 \in \Rb} \inf_{x_2 \in \Rb} \sup_{x_3 \in \Rb}\allowbreak\inf_{x_4 \in \Rb}\sup_{m \in \Nb}\inf_{n_0,\dots,n_{A} \in \Nb} x^2_0 [(n_0 - 2)^2 + \allowbreak(n_1-m)^2 + n_2 + (n_B - n_C)^2 + n_3 \sum_{k=0}^4 ( x_k - \frac{n_{k+5}}{1+n_4} + n_4)^2 + \sum_{i,j = 0}^B (n_{9+2^i3^j} - n_i^{n_j})^2])$. 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."
bikenaga•1h ago
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 \ZFC{} set theory that $r = \log(\sup_{x_0,x_1 \in \Rb} \inf_{x_2 \in \Rb} \sup_{x_3 \in \Rb}\allowbreak\inf_{x_4 \in \Rb}\sup_{m \in \Nb}\inf_{n_0,\dots,n_{A} \in \Nb} x^2_0 [(n_0 - 2)^2 + \allowbreak(n_1-m)^2 + n_2 + (n_B - n_C)^2 + n_3 \sum_{k=0}^4 ( x_k - \frac{n_{k+5}}{1+n_4} + n_4)^2 + \sum_{i,j = 0}^B (n_{9+2^i3^j} - n_i^{n_j})^2])$. 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."