Non-existence of Universal Orders in Many Cardinals

Details Non-existence of Universal Orders in Many Cardinals Non-existence of Universal Orders in Many Cardinals

1992-09-03

arxiv.org/abs/math/9209201v1

Mathematics

Logic

Scientific paper

Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<\l<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $\l$ and its existence is not a result of a trivial cardinal arithmetical reason, then $\l$ ``resembles'' $\aleph_1$ --- a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the non-existence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

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