Models Without Indiscernibles

Abstract

For $T$ any completion of Peano Arithmetic and for $n$ any positive integer, there is a model of $T$ of size $\\beth_n$ with no $(n + 1)$-length sequence of indiscernibles. Hence the Hanf number for omitting types over $T, H(T)$, is at least $\\beth_\\omega$. (Now, using an upper bound previously obtained by Julia Knight $H$ (true arithmetic) is exactly $\\beth_\\omega$). If $T \\neq$ true arithmetic, then $H(T) = \\beth_{\\omega1}$. If $\\delta \\not\\rightarrow (\\rho)^{<\\omega}$, then any completion of Peano Arithmetic has a model of size $\\delta$ with no set of indiscernibles of size $\\rho$. There are similar results for theories strongly resembling Peano Arithmetic, e.g., $\\mathrm{ZF} + V = L$.