Ultrapowers and Elementary Embeddings

[Dragonfall]

First some definitions:

Given similarity structures N and M, we say that "M is elementarily equivalent to N" if for any sentence S in the lanauge of M and N, "M models S iff N models S".

An "elementary embedding" f:|N| -> |M| between the underlying sets of N and M is such that for any formula F(x) and some tuple a of M, "M models F(a) implies N models F(f(a))".

So, assume that M is elementarily equivalent to N, prove that there is an ultrafilter U=(I,U) and an elementary embedding g: N->M^U, where M^U is an ultrapower.

I don't know how to do this. I don't even fully comprehend what it says.

 

[daon]

You haven't defined: "Similarity Structure", "Sentence", "Ultrapower", "Ultrafilter", "Formula", "Models" and the notion of raising a set to a set... M^U. What does |M| mean, cardinality? Is U an interval? Is an "underlying set" of N just a subset of N?

Looks like there is a lot of prerequisite knowledge required. Hopefully someone here knows what you're asking.

Sounds like a ultra-fun class though. :wink: