%0 Journal Article
%T A Partial Order Where All Monotone Maps Are Definable
%A Martin Goldstern
%A Saharon Shelah
%J Mathematics
%D 1997
%I arXiv
%X It is consistent that there is a partial order (P,<) of size aleph_1 such that every monotone (unary) function from P to P is first order definable in (P,<). The partial order is constructed in an extension obtained by finite support iteration of Cohen forcing. The main points is that (1) all monotone functions from P to P will (essentially) have countable range (this uses a Delta-system argument) and (2) that all countable subsets of P will be first order definable, so we have to code these countable sets into the partial order. Amalgamation of finite structures plays an essential role.
%U http://arxiv.org/abs/math/9707202v1