Countably decomposable admissible sets
This paper is an important contribution to model and definability theory for infinitary languages unifying work on this subject done by various authors. An admissible structure \({\mathcal A}\) is defined to be countably decomposable if its domain A is the union of countably many of its elements and \({\mathcal A}\) satisfies a certain decomposability property DP. The main result of Part I states that for structures with the DP and theories \(T\subseteq {\mathcal L}_ A\) such that T is \({\mathcal A}\)-r.e. and a countable union of members of A, (a) the set of consequences of T in A is \({\mathcal A}\)-r.e., (b) if every \(T_ 0\subseteq T\), \(T_ 0\in A\) has a model then T has a model and (c) the least ordinal not pinned down by T is a member of A. This result includes all the known theorems on \(\Sigma_ 1\)-completeness, \(\Sigma_ 1\)-compactness, non-definability of well-orderings, etc. for various admissible structures. (The proof uses the method of open games.) In Part II, it is shown that under the assumption \(V=L\) the authors' result is the best possible in a certain sense. The notion of \(\Sigma_ 1\)-compactness is strengthened to stable \(\Sigma_ 1\)-compactness, and in Part III it is proved that \(L_{\alpha}\) is stably \(\Sigma_ 1\)-compact if and only if it satisfies the right decomposability conditions. The paper contains an appendix by \textit{L. Harrington}: ''Admissible ordinals are not immortal''.
scientific article; zbMATH DE number 3889512
scientific article