An alternative construction of Conway's ordered field No
Let \(\kappa\leq On\) (i.e., the power of all proper classes in NBG). A model A for a theory T in a language L is said to be \(\kappa\)-universally extending iff 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. An \(\aleph_{\alpha}\)-universally extending ordered field of power \(\aleph_{\alpha}\) is constructed in NBG for each regular power \(\aleph_{\alpha}\), where \(0<\alpha \leq On\) and \(\sum_{\beta <\alpha}2^{\aleph_{\beta}}\leq \aleph_{\alpha}\), and it is shown that when \(\aleph_{\alpha}\) is inaccessible the structure is either a (set) model of J. H. Conway's ordered field No or an isomorphic copy of No depending on whether or not \(\aleph_{\alpha}\) is a set. [For Errata, see ibid. 25, 223 (1988).] The author has since provided [Abstr. Am. Math. Soc. Dec. 1988] categorical axiomatizations of the ordered field of reals and shown that by deleting the Archimedean condition one obtains categorical axiomatizations of No; e.g. R is an On-universally extending (On- homogeneous universal) Archimedean ordered field; and R is a d-maximal real-closed Archimedean ordered field. Let T be a theory containing the theory of dense linear orderings and A, B be models of T. If \(A\subset B\) and for some pair (X,Y) of subsets of A there is a \(z\in B\) where \(X<\{z\}<Y\), but for no z'\(\in A\), \(X<\{z'\}<Y\), then B is said to be a d-extension. If A has no d-extension, A is d-maximal.
scientific article; zbMATH DE number 4049627
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