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D. Marker, Model Theory:
Corollary 4.3.24. The number of non isomorphic homogeneous models of $T$ of size $\kappa$ is at most $2^{2^{\aleph_0}}$.
Proof.
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.

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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."

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  • $\begingroup$ 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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