Short recursively saturated models and boundedly saturated models

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Short Recursively Saturated Models

A model is short if there is an element in the model whose Skolem closure is cofinal in the model. That is, M is short iff there is a in M such that M=Sup(Scl(a)). If a model is not short, it is called tall.

A type p(v,b) is bounded if it contains the formula v<b.

A model M is boundedly saturated if for every b in M, every bounded recursive type p(v,b) is realized in M.

A model is short recursively saturated if it is short and boundedly saturated.

Every elementary cut of a recursively saturated model is boundedly saturated.

Countable Short Recursively Saturated Models

If an elementary cut of a countable recursively saturated model is tall, it is recursively saturated (and hence isomorphic to the model). If the elementary cut is short it is bounded recursively saturated, and hence short recursively saturated.

Short Saturated Models