2026/02/02 by James E. Hanson, Connor Watson · 2 voices
Mathematics · #math.LO #math.HO
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 \def\Rbℝ\def\Nbℕr = log(supx0,x1 ∈ \Rb infx2 ∈ \Rb supx3 ∈ \Rbinfx4 ∈ \Rbsupm ∈ \Nbinf_n0,…,nA ∈ \Nb x20 \beginbmatrix \phantom+(n0 - 2)2 + (n1-m)2
+ n2 + (nB - nC)2
+ n3 ∑k=04 ( xk - \fracnk+51+n4 +n4)2
+ ∑i,j = 0B (n9+2i3j - ninj)2 \endbmatrix ). 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.