D. Marker, Model Theory:
Corollary 4.3.24. The number of non isomorphic homogeneous models of $T$ of size $\kappa$ is at most $2^{2^{\aleph_0}}$.
Homogeneous models of cardinality $\kappa$ are determined by the set of types realized. Because $|S_n(T)| ≤ 2^{\aleph_0}$, the number of possible sets of types realized in a model is at most $2^{2^{\aleph_0}}$.

I can't understand why the sentence which I've made it bold in the proof is true. Would be grateful for your help.


This is Theorem 4.3.23, which appears directly before Corollary 4.3.24.

Marker probably should have written "Homogeneous models of cardinality $\kappa$ are determined up to isomorphism by the set of types realized."

  • $\begingroup$ Yes, that's right. It's true to be written "up to isomorphism". $\endgroup$
    – Aref
    Jan 4 '17 at 20:33

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