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Strong \(\Delta_1\)-definability of a model in an admissible set. (English. Russian original) Zbl 0914.03044
Sib. Math. J. 39, No. 1, 168-175 (1998); translation from Sib. Mat. Zh. 39, No. 1, 191-200 (1998).
A model \(\langle M, P_1^{n_1},\ldots,P_m^{n_m}\rangle \) is called strongly \(\Delta_1\)-definable over an admissible set \(A\) provided there exist \(\Delta_1\)-formulas \(\varphi(x,\overline{y})\), \(\psi_1(x_1,\ldots,x_{n_1},\overline{y})\), \(\ldots\), \(\psi_m(x_1,\ldots,x_{n_m},\overline{y})\) and a tuple of elements \(\overline{a}\) in \(A\) such that \(M\) is isomorphic to the model \(\langle X, \overline{P}_1,\ldots,\overline{P}_m\rangle \), where \(X=\{x\in A\mid A\models\varphi(x,\overline{a})\}\) and \(\overline{P}_i= \{ \langle x_1,\ldots,x_{n_i}\rangle \mid A \models \psi_i (x_1,\ldots,x_{n_i},\overline{a}) \}\).
The author calls a model \(M\) a \(B\)-model if, for every finite nonempty subset \(M_0\subseteq M\), there is a finite \(M^\ast_0\subseteq M\), \(M_0\subseteq M^\ast_0\) such that, for each finite subset \(M_1{\not\subseteq}M_0^\ast\) of \(M\), there exists an automorphism \(f\) of \(M\) such that \(f|{}M^\ast_0=\text{id}_{M_0^\ast}\) and \(fM_1\neq M_1\).
The author proves that, if a rigid model \(C\) is definable over a Cartesian product \(M\times N\), where \(N\) is isomorphic to a recursive model and \(M\) is a \(B\)-model, then \(C\) is isomorphic to a recursive model. Then he uses this assertion for proving that every ordinal is \(\Delta_1\)-definable over at most countable Abelian groups and Boolean algebras if and only if it is recursive. This gives examples of Abelian groups and Boolean algebras whose respective Ulm types and ordinal types are not \(\Delta_1\)-definable in the superstructure of hereditarily finite sets over these algebras.

03C57 Computable structure theory, computable model theory
03D60 Computability and recursion theory on ordinals, admissible sets, etc.
03D45 Theory of numerations, effectively presented structures
Full Text: DOI
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