The ultrapower theorem states that every two elementary equivalent models have isomorphic ultrapowers (Shelah, Israel Journal of Mathematics, 1971, Vol. 10).
Are there other proofs than Shelah's?
2026-04-09 18:14:27.1775758467
Ultrapower theorem: reference request
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I'm not sure if this is what you're looking for, since their proof is essentially the same as Shelah's, but you may enjoy reading the discussion in Chang & Keisler's Model Theory, section 6.1. They show first how to prove the theorem using GCH, then how to prove a stronger version of the theorem also assuming GCH, and then how to prove Shelah's result. Keisler's result is also discussed in Bell & Slomson's Models and Ultraproducts, section 7.2.