Mathematics

This paper shows how to transform explosive many-valued systems into paraconsistent logics. We investigate especially the case of three-valued systems showing how paraconsistent three-valued logics can be obtained from them.

Developing a system of parallel non-linear iterations, we establish the consistency of b<s<d<c where b,d,c are arbitrary subject to the known ZFC restrictions and s is regular. By evaluating other invariants we achieve also the constellations b<r<d<c , b<e<d<c and b<u<d<c .

This paper presents a general framework for unifying functional interpretations. It is based on families of parameters allowing for different degrees of freedom on the design of the interpretation. In this way we are able to generalise previous work on unifying functional interpretations, by including in the unification the more recent bounded and Herbrandized functional interpretations.

We study the topological and set-theoretical nature of Paretian social welfare relations in a setting with infinite time horizon. Specifically, we answer questions posed in \citet{mathias2020} about the interplay between total welfare relations satisfying Pareto and anonymity principles with subsets of real numbers not satisfying the Baire property.

This paper answers several open questions around structures with o-minimal open core. We construct an expansion of an o-minimal structure R by a unary predicate such that its open core is a proper o-minimal expansion of R . We give an example of a structure that has an o-minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o-minimal open core and definable Skolem functions, but is not o-minimal.

We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels V α of the cumulative hierarchy are defined via iterated application of the power set operation, starting from V 0 =∅ , and taking unions at limit stages. Assuming that j: V α+1 → V α+1 is a (non-trivial) elementary embedding, we show that the structure of V α is fundamentally different to that of V α+1 . We show that j is definable from parameters over V α+1 iff α+1 is an odd ordinal. Moreover, if α+1 is odd then j is definable over V α+1 from the parameter j‘‘ V α ={j(x) ∣ ∣ x∈ V α } , and uniformly so. This parameter is optimal in that j is not definable from any parameter which is an element of V α . In the case that α=β+1 , we also give a characterization of such j in terms of ultrapower maps via certain ultrafilters. Assuming λ is a limit ordinal, we prove that if j: V λ → V λ is Σ 1 -elementary, then j is not definable over V λ from parameters, and if β<λ and j: V β → V λ is fully elementary and ∈ -cofinal, then j is likewise not definable; note that this last result is relevant to embeddings of much lower consistency strength than rank-to-rank. If there is a Reinhardt cardinal, then for all sufficiently large ordinals α , there is indeed an elementary j: V α → V α , and therefore the cumulative hierarchy is eventually periodic (with period 2).

The paper is in the field of Region Based Theory of Space (RBTS), sometimes called mereotopology. RBTS is a kind of point-free theory of space based on the notion of region. Its origin goes back to some ideas of Whitehead, De Laguna and Tarski to build the theory of space without the use of the notion of point. More information on RBTS and mereotopology can be found, for instance, in \cite{Vak2007}. Contact algebras present an algebraic formulation of RBTS and in fact give axiomatizations of the Boolean algebras of regular closed sets of some classes of topological spaces with an additional relation of contact. An exhaustive study of this theory is given in \cite{DiVak2006}. Dynamic contact algebra (DCA) \cite{Vak2014} (see also \cite{Vak2010,Vak2012}) introduced by the present author, is a generalization of contact algebra studying regions changing in time and presents a formal explication of Whitehead's ideas of integrated point-free theory of space and time. DCA is an abstraction of a special \emph{dynamic model of space}, called also \emph{snapshot} or \emph{cinematographic} model and the paper \cite{Vak2014} contains the expected representation theorem with respect to such models. In the present paper we introduce a new version of DCA which is a simplified version of the definition from \cite{Vak2014} and similar to that of \cite{Vak2012}. The aim is to use this version as a representative example of a DCA and to develop for this example not only the snapshot models but also topological models and the expected topological duality theory, generalizing in a certain sense the well known Stone duality for Boolean algebras. Abstract topological models of DCAs present a new view on the nature of space and time and show what happens if we are abstracting from their metric properties.

We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.

Lindström's Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negationless logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

We investigate predicative aspects of order theory in constructive univalent foundations. By predicative and constructive, we respectively mean that we do not assume Voevodsky's propositional resizing axioms or excluded middle. Our work complements existing work on predicative mathematics by exploring what cannot be done predicatively in univalent foundations. Our first main result is that nontrivial (directed or bounded) complete posets are necessarily large. That is, if such a nontrivial poset is small, then weak propositional resizing holds. It is possible to derive full propositional resizing if we strengthen nontriviality to positivity. The distinction between nontriviality and positivity is analogous to the distinction between nonemptiness and inhabitedness. We prove our results for a general class of posets, which includes directed complete posets, bounded complete posets and sup-lattices, using a technical notion of a δ V -complete poset. We also show that nontrivial locally small δ V -complete posets necessarily lack decidable equality. Specifically, we derive weak excluded middle from assuming a nontrivial locally small δ V -complete poset with decidable equality. Moreover, if we assume positivity instead of nontriviality, then we can derive full excluded middle. Secondly, we show that each of Zorn's lemma, Tarski's greatest fixed point theorem and Pataraia's lemma implies propositional resizing. Hence, these principles are inherently impredicative and a predicative development of order theory must therefore do without them. Finally, we clarify, in our predicative setting, the relation between the traditional definition of sup-lattice that requires suprema for all subsets and our definition that asks for suprema of all small families.