Minami–Sakai (Arch Math Logic 55(7–8):883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all $$F_\sigma $$ ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following: In the course of the proof of the latter result, we also prove that for any analytic ideal $$\mathcal {I}$$ there is a Borel ideal $$\mathcal {J}$$ with $$\mathcal {I} \subseteq \mathcal {J}$$ .

In this paper, we study an algebraically closed field $$\Omega $$ expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup $$\Gamma $$ . This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple $$(\Omega , k, \Gamma )$$ . This enables us to characterize the interpretable groups when $$\Gamma $$ is divisible. Every interpretable group H in $$(\Omega ,k, \Gamma )$$ is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in $$\Gamma $$ by an interpretable group N, which is the quotient of an algebraic group by a subgroup $$N_1$$ , which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in $$\Gamma $$ .

We introduce a new typed combinatory calculus with a type constructor that, to each type $$\sigma $$ , associates the star type $$\sigma ^*$$ of the nonempty finite subsets of elements of type $$\sigma $$ . We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.

Assuming $$\text {ZF}+\text {DC}$$ZF+DC, we prove that if there exists a strong partition cardinal greater than $$\varTheta $$ź, then (1) there is an inner model of $$\text {ZF}+\text {AD}+\text {DC}+ {{{\mathbb {R}}} }^{{\#}}$$ZF+AD+DC+R# exists, and (2) there is an inner model of $$\text {ZF}+\text {AD}+\text {DC}+ (\exists \kappa >\varTheta )\,(\kappa $$ZF+AD+DC+(źź>ź)(ź is measurable). Here $$\varTheta $$ź is the supremum of the ordinals which are the surjective image of the set of reals $${{{\mathbb {R}}} }$$R.

We show that Shelah cardinals are preserved under the canonical $${{\mathrm{GCH}}}$$GCH forcing notion. We also show that if $${{\mathrm{GCH}}}$$GCH holds and $$F:{{\mathrm{REG}}}\rightarrow {{\mathrm{CARD}}}$$F:REGźCARD is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $$\forall \kappa \in {{\mathrm{REG}}},~ 2^{\kappa }=F(\kappa )$$źźźREG,2ź=F(ź). This gives a partial answer to a question asked by Cody (Arch Math Logic 52(5---6):569---591, 2013) and independently by Honzik (Acta Univ Carol 1:55---72, 2015). We also prove an indestructibility result for Shelah cardinals.

Suppose A is a linear order, possibly with countably many unary predicates added. We classify the isomorphism relation for countable models of $$\text {Th}(A)$$Th(A) up to Borel bi-reducibility, showing there are exactly five possibilities and characterizing exactly when each can occur in simple model-theoretic terms. We show that if the language is finite (in particular, if there are no unary predicates), then the theory is $$\aleph _0$$ź0-categorical or Borel complete; this generalizes a theorem due to Schirmann (Theories des ordres totaux et relations dequivalence. Master's thesis, Universite de Paris VII, 1997).

In this paper we investigate the consistency and consequences of the downward Löwenheim–Skolem–Tarski theorem for extension of the first order logic by the Magidor–Malitz quantifier. We derive some combinatorial results and improve the known upper bound for the consistency of Chang’s conjecture at successor of singular cardinals.

In Ramsey's Theorem and Recursion Theory, Theorem 4.2, Jockusch proved that for any computable k-coloring of pairs of integers, there is an infinite $$\Pi ^0_2$$ź20 homogeneous set. The proof used a countable collection of $$\Pi ^0_2$$ź20 sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch stated without proof that it can be shown that there is no computable way to prove this result with a finite number of $$\Pi ^0_2$$ź20 sets. We provide a proof of this claim, showing that there is no computable way to take an index for an arbitrary computable coloring and produce a finite number of indices of $$\Pi ^0_2$$ź20 sets with the property that one of those sets will be homogeneous for that coloring. While proving this result, we introduce n-trains as objects with useful combinatorial properties which can be used as approximations to infinite $$\Pi ^0_2$$ź20 sets.

We show that every $$\varPi _1$$ź1-elementary epimorphism between models of $$\mathrm{ZF}$$ZF is an isomorphism and hence, trivial. On the other hand, nonisomorphic $$\varSigma _1$$Σ1-elementary epimorphisms between models of $$\mathrm{ZF}$$ZF can be constructed, as can fully elementary epimorphisms between models of $$\mathrm{ZFC}^-$$ZFC- (a formulation of ZFC without powerset). We construct examples of such elementary epimorphisms. We also construct an inverse-directed system of elementary epimorphisms between models of $$\mathrm{ZFC}^-$$ZFC- and determine the inverse limit of this system. Elementary epimorphisms were introduced by Rothmaler (J Symb Log 70(2):473---488, 2005). An elementary epimorphism is somewhat like an elementary embedding taken in reverse. To be precise, a surjective homomorphism $$f{: } M \rightarrow N$$f:MźN between two model-theoretic structures is an elementary epimorphism if and only if every formula with parameters satisfied by N is satisfied in M using a preimage of those parameters. Given a class of formulas $$\varGamma $$Γ, a $$\varGamma $$Γ-elementary epimorphism is defined by restricting the above definition to this class of formulas.

We investigate the definable subsets of Stone algebras, introduce the notion of pseudo o-minimality of partially ordered structures, and prove that the completion of the theory of Stone algebras is pseudo o-minimal. We also give the decomposition theorem of Stone algebras.