A Partial Order Where All Monotone Maps Are Definable

Details A Partial Order Where All Monotone Maps Are Definable A Partial Order Where All Monotone Maps Are Definable

1997-07-16

arxiv.org/abs/math/9707202v1

Mathematics

Logic

This is paper number GoSh:554 in Shelah's list. The paper is to appear in Fundamenta Mathematicae.

Scientific paper

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.

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