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"Following Jónnson ([16], p.149), a model A for a theory T in a language L is said to be $\kappa$-universally extending if for any models B and C of T in L where B is a substructure of A, C is an extension of B, and |B|, |C| < $\kappa$, there is a model C’ of T in L that is a substructure of A and an isomorphism from C onto C’ extending the identity map on B. When, as in the case of ordered fields, T is a Jónnson theory, i.e. an inductive first-order theory having the amalgamation and joint embedding properties as well as an infinite model (cf. [18], p.147), the $\kappa$ -universally extending models of T coincide with the models of T that are $\kappa$-homogeneous and $\kappa$-universal with respect to models of T (cf. [24], Corollary 2.5e and the remarks on p. 153 of [18]). For the case at hand, of course, the latter structures are the $\kappa$-saturated models for the theory of real-closed ordered fields, i.e., the real-closed fields that are $\eta_{\alpha}$-orderings for $\alpha$ > 0 (cf. [7], Ch. 5)."

"Following Jónnson ([16], p.149), a model A for a theory T in a language L is said to be $\kappa$-universally extending if for any models B and C of T in L where B is a substructure of A, C is an extension of B, and |B|, |C| < $\kappa$, there is a model C’ of T in L that is a substructure of A and an isomorphism from C onto C’ extending the identity map on B. When, as in the case of ordered fields, T is a Jónnson theory, i.e. an inductive first-order theory having the amalgamation and joint embedding properties as well as an infinite model (cf. [18], p.147), the $\kappa$ -universally extending models of T coincide with the models of T that are $\kappa$-homogeneous and $\kappa$-universal with respect to models of T (cf. [24], Corollary 2.5e and the remarks on p. 153 of [18]). For the case at hand, of course, the latter structures are the $\kappa$-saturated models for the theory of real-closed ordered fields, i.e., the real-closed fields that are $\eta_{\alpha}$-orderings for $\alpha$ > 0 (cf. [7], Ch. 5)."

I then go on to prove as part of Lemma 2: Let 0 < $\alpha$ < or = On. S is an $\aleph_{\alpha}$- universally extending ordered field if and only if S is a real-closed field that is an $\eta_{\alpha}$-ordering.

Victor: Just to be clear, the references to my pre-1999 papers were clearly not intended to supply the proof you sought. On the other hand, contrary to what you say, the general case is addressed in both of those papers. For example, on page 8 of my 1986 paper (with reference to classical works of Jónnson, Morley and Vaught, Chang and Keisler) I write:

Vladimir: Just to be clear, the references to my pre-1999 papers were clearly not intended to supply the proof you sought. On the other hand, contrary to what you say, the general case is addressed in both of those papers. For example, on page 8 of my 1986 paper (with reference to classical works of Jónnson, Morley and Vaught, Chang and Keisler) I write:

Second Emendation

Victor: Just to be clear, the references to my pre-1999 papers were clearly not intended to supply the proof you sought. On the other hand, contrary to what you say, the general case is addressed in both of those papers. For example, on page 8 of my 1986 paper (with reference to classical works of Jónnson, Morley and Vaught, Chang and Keisler) I write:

"Following Jónnson ([16], p.149), a model A for a theory T in a language L is said to be $\kappa$-universally extending if for any models B and C of T in L where B is a substructure of A, C is an extension of B, and |B|, |C| < $\kappa$, there is a model C’ of T in L that is a substructure of A and an isomorphism from C onto C’ extending the identity map on B. When, as in the case of ordered fields, T is a Jónnson theory, i.e. an inductive first-order theory having the amalgamation and joint embedding properties as well as an infinite model (cf. [18], p.147), the $\kappa$ -universally extending models of T coincide with the models of T that are $\kappa$-homogeneous and $\kappa$-universal with respect to models of T (cf. [24], Corollary 2.5e and the remarks on p. 153 of [18]). For the case at hand, of course, the latter structures are the $\kappa$-saturated models for the theory of real-closed ordered fields, i.e., the real-closed fields that are $\eta_{\alpha}$-orderings for $\alpha$ > 0 (cf. [7], Ch. 5)."

[7] Chang, C. C. and Keisler, J. H., Model Theory, North-Holland, 1973.

[16] Jónnson, B. “Homogeneous universal relational systems”, Math. Scand. 8 1960 137–142. [1

[18] Jónnson, B. “Extensions of relational structures” 1965 Theory of Models (Proc. 1963 Internat. Sympos. Berkeley) pp. 146–157 North-Holland, Amsterdam.

[24] Morley, M. and Vaught, R. “Homogeneous universal models”, Math. Scand.11 1962 pp. 37–57.

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