Remarks in abstract model theory
This rather long paper consists of two distinct parts, the first of which is about the Beth definability property, and the second about compactness. The first half of Part I contains abstract results about Beth closures and preservation properties. At the beginning, a one-step Beth closure \(L^ B\) is defined by adding new formulas for the required explicit definitions. Then the Beth closure \(L^{Beth}\) is defined by iteration. A sufficient condition is given to the compactness of the one-step Beth closure. Then the pair preservation property PPP is related to the reduction property \(URP_ 2\) and the preservation of these under Beth closures is studied. The first half of Part I ends with a proof of the consistency of that there is no definable extension of \(L(Q_ 1)\) with PPP and interpolation. The latter half of Part I contains results about cofinality quantifiers and stationary logic. The quantifier \(Q_ C^{cf}\) says that the cofinality belongs to C. \(L(aa_{\lambda})\) is the analogue of L(aa) for \(\lambda\). The first result is that the pair \((L(Q^{cf}_{\aleph_ 0},L(aa))\) satisfies the interpolation theorem. It is shown that the pair \((L(Q^{cf}_{\leq \lambda},L(aa_{\lambda}))\) has the super \(\aleph_ 0\)-homogeneity property. It says the following. Assume that in a model M to every \(L(aa_{\lambda})\)-formula \(\psi\) corresponds a relation symbol \(R_{\psi}\) whose interpretation is the relation defined by \(\psi\). Then M is \(L(Q^{cf}_{\leq \lambda})\)-equivalent to a model N where any two finite sequences satisfying the same \(L(Q^{cf}_{\leq \lambda})\)- formulas can be mapped onto each other by an automorphism of N. The (full) Beth closure of \(L(Q^{cf}_{\leq \kappa})\) \((\kappa \geq 2^{\aleph_ 0})\) has the following properties: (1) it is compact, (2) the pair \((L(Q^{cf}_{\leq \kappa}),L(aa_{\kappa}))\) has the interpolation and super \(\aleph_ 0\)-homogeneity properties, (3) the Beth closure of \(L(Q^{cf}_{\leq \kappa})\) has (trivially) the Beth property, (4) PPP, and (5) it does not have the interpolation property. This is interesting in abstract model theory for that it shows that the second hypothesis cannot be weakened to PPP in the following theorem of Makowski and Shelah: the Beth and interpolation properties are equivalent for a compact logic satisfying a preservation property for a tree-sum of models. Part II begins with some remarks about logics which are [\(\lambda\)]- compact but not [cf \(\lambda\)]-compact. Here [\(\kappa\)]-compactness means that a set of sentences \(\Sigma \cup \Delta\) has a model whenever \(| \Sigma | \leq \kappa\) and for every \(\Sigma_ 1\subset \Sigma\), \(| \Sigma_ 1| <\kappa\), the set \(\Sigma_ 1\cup \Delta\) has a model. Then certain assumptions about occurrence numbers in a result by Makowski and Shelah relating amalgamation and [\(\lambda\)]-compactness are improved. The paper ends with a sublogic L of \(L_{\lambda\lambda}\) (\(\lambda\) compact) which is [\(\omega\)]-compact and has JEP. In a footnote it is also announced that this logic has the interpolation and Robinson consistency properties and PPP. In L, infinitary conjunctions have been replaced by the operation \(\bigwedge^{{\mathcal D}}_{i<\mu}\psi_ i\) which means that \(\{i| \psi_ i\) holds\(\}\in {\mathcal D}\); \({\mathcal D}\) is a \(\kappa\)-complete ultrafilter \((\kappa<\lambda\) compact).
scientific article; zbMATH DE number 3966047
scientific article