Mathematics

I prove several independence results in the choiceless ZF+DC theory which separate algebraic and non-algebraic consequences of the axiom of choice. As an example, let E be an equivalence relation resulting from a turbulent Polish group action, let X be a Polish field, and let F be a countable subfield. It is consistent with the choiceless theory ZF+DC that X has a transcendence basis over F and E does not have a selector.

In this work we study the decidability of the global modal logic arising from Kripke frames evaluated on certain residuated lattices (including all BL algebras), known in the literature as crisp modal many-valued logics. We exhibit a large family of these modal logics that are undecidable, in opposition to classical modal logic and to the propositional logics defined over the same classes of algebras. These include the global modal logics arising from the standard Lukasiewicz and Product algebras. Furthermore, it is shown that global modal Lukasiewicz and Product logics are not recursively axiomatizable. We conclude the paper by solving negatively the open question of whether a global modal logic coincides with the local modal logic closed under the unrestricted necessitation rule.

We show that generalized eventually narrow sequences on a strongly inaccessible cardinal κ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on κ . Moreover assuming GCH, κ <κ =κ , we show that: (1) if κ is strongly unfoldable, κ + ≤β=cf(β)≤cf(δ)≤δ≤μ and cf(μ)>κ ,then there is a cardinal preserving generic extension in which s(κ)= κ + ≤b(κ)=β≤d(κ)=δ≤ 2 κ =μ. (2) if κ is strongly inaccessible, λ> κ + , then in the generic extension obtained as the <κ -support iteration of κ -Hechler forcing of length λ there are no κ -towers of length λ .

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics.

Questions concerning the proof-theoretic strength of classical versus non-classical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment -- a logic recently studied by Hannes Leitgeb under the label `HYPE'. We show in particular that, by formulating the theory PKF over HYPE one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension -- in the sense of Feferman -- is as strong as the schematic extension of KF, thus matching the strength of predicative analysis.

Let K be an arbitrary Kripke model of Heyting Arithmetic, HA . For every node k in K , we can view the classical structure of k , M k as a model of some classical theory of arithmetic. Let T be a classical theory in the language of arithmetic. We say K is locally T , iff for every k in K , M k ⊨T . One of the most important problems in the model theory of HA is the following question: {\it Is every Kripke model of HA locally PA ?} We answer this question negatively. We introduce two new Kripke model constructions to this end. The first construction actually characterizes the arithmetical structures that can be the root of a Kripke model K⊩HA+EC T 0 ( EC T 0 stands for Extended Church Thesis). The characterization says that for every arithmetical structure M , there exists a rooted Kripke model K⊩HA+EC T 0 with the root r such that M r =M iff M⊨ Th Π 2 (PA) . One of the consequences of this characterization is that there is a rooted Kripke model K⊩HA+EC T 0 with the root r such that M r ⊭I Δ 1 and hence K is not even locally I Δ 1 . The second Kripke model construction is an implicit way of doing the first construction which works for any reasonable consistent intuitionistic arithmetical theory T with a recursively enumerable set of axioms that has the existence property. We get a sufficient condition from this construction that describes when for an arithmetical structure M , there exists a rooted Kripke model K⊩T with the root r such that M r =M .

We introduce trace definability, a weak notion of interpretability, and trace equivalence, a weak notion of equivalence for first order structures and theories. In particular we get an interesting weak equivalence notion for NIP theories. We describe a close connection to indiscernible collapse. We also show that if Q is a divisible subgroup of (R;+) and Q is a dp-rank one expansion of (Q;+,<) then exactly one of the following holds: Th(Q) trace defines RCF or Q is trace equivalent to a reduct of an ordered vector space.

Diproche ("Didactical Proof Checking") is an automatic system for supporting the acquistion of elementary proving skills in the initial phase of university education in mathematics. A key feature of Diproche - which is designed by the example of the Naproche system developed by M. Cramer and others - is an automated proof checker for proofs written in a controlled fragment of natural language specifically designed to capture the language of beginners' proving exercises in mathematics. Both the accepted language and proof methods depend on the didactical and mathematical context and vary with the level of education and the topic proposed. An overall presentation of the system in general was given in Carl and Krapf 2019. Here, we briefly recall the basic architecture of Diproche and then focus on explaining key features and the working principles of Diproche in the sample topics of elementary number theory and axiomatic geometry.

A permutation group G on a set A is κ -homogeneous iff for all X,Y∈[A ] κ with |A∖X|=|A∖Y|=|A| there is a g∈G with g[X]=Y . G is κ -transitive iff for any injective function f with dom(f)∪ran(f)∈[A ] ≤κ and |A∖dom(f)|=|A∖ran(f)|=|A| there is a g∈G with f⊂g . Giving a partial answer to a question of P. M. Neumann we show that there is an ω -homogeneous but not ω -transitive permutation group on a cardinal λ provided (i) λ< ω ω , or (ii) 2 ω <λ , and μ ω = μ + and □ μ hold for each μ≤λ with ω=cf(μ)<μ , or (iii) our model was obtained by adding ω 1 many Cohen generic reals to some ground model. For κ>ω we give a method to construct large κ -homogeneous, but not κ -transitive permutation groups. Using this method we show that there exists κ + -homogeneous, but not κ + -transitive permutation groups on κ +n for each infinite cardinal κ and natural number n≥1 provided V=L .

In this paper, we shall give another proof of the faithfulness of Blass translation (for short, B -translation) of the propositional fragment L 1 of Leśniewski's ontology in the modal logic K \it by means of Hintikka formula\rm . And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a result of observing the proofs we shall give general theorems on the faithfulness of B -translation with respect to normal modal logics complete to certain sets of well-known accessibility relations with a restriction that transitivity and symmetry are not set at the same time. As an application of the theorems, for example, B -translation is faithful for the provability logic PrL (= GL ), that is, K + □(□ϕ⊃ϕ)⊃□ϕ . The faithfulness also holds for normal modal logics, e.g., KD , K4 , KD4 , KB . We shall conclude this paper with the section of some open problems and conjectures.