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Abstract: Abstract This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a \(\Pi _2\) -property formalized in an appropriate language for second order number theory is forcible from some \(T\supseteq \mathsf {ZFC}+\) large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of \(H_{\omega _1}\) is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for \(\Delta _0\) -properties and for all universally Baire sets of reals. We will extend these results also to the theory of \(H_{\aleph _2}\) in a follow up of this paper. PubDate: 2023-02-01
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Abstract: Abstract An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ( \({{\mathbf {IUL}}^{fp}}\) ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ( \({\mathbf {IUL}}\) , posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL \(_e\) -chains which have finitely many positive idempotent elements. PubDate: 2023-02-01
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Abstract: Abstract We have provided a model-theoretic proof for the decidability of the additive structure of integers together with the function f mapping x to \(\lfloor \varphi x\rfloor \) where \(\varphi \) is the golden ratio. PubDate: 2023-02-01
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Abstract: Abstract We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality \(\kappa \) , where \(\kappa \) is a regular cardinal. The corresponding new notion is called \(\kappa \) -filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different \(\kappa \) -filter pairs give rise to a fixed logic of cardinality \(\kappa \) . To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality \(\kappa \) . Along the way we use \(\kappa \) -filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair. PubDate: 2023-02-01
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Abstract: Abstract 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 negation-less 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. PubDate: 2023-02-01
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Abstract: Abstract We construct a 2-equivalence \(\mathfrak {CohTheory}^{op }\simeq \mathfrak {TypeSpaceFunc}\) . Here \(\mathfrak {CohTheory}\) is the 2-category of positive theories and \(\mathfrak {TypeSpaceFunc}\) is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in \(\mathfrak {CohTheory}\) . The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is ‘the same’ as the collection of its type spaces (i.e. its type space functor). In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories. PubDate: 2023-02-01
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Abstract: Abstract We investigate several ideal versions of the pseudointersection number \(\mathfrak {p}\) , ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant \(\mathtt {cov}^*({\mathcal I})\) has a crucial influence on the studied notions. For an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal J})\) introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J})\) introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have $$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathtt {cov}^*({\mathcal I})\}=\mathfrak {p},\qquad \min \{\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}\le \mathtt {cov}^*({\mathcal I}), \end{aligned}$$ respectively. In addition to the first inequality, for a slalom invariant \(\mathfrak {sl_e}({\mathcal I},{\mathcal J})\) introduced in Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that $$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathfrak {sl_e}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}=\mathfrak {p}. \end{aligned}$$ Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished. PubDate: 2023-02-01
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Abstract: Abstract For a non-zero natural number n, we work with finitary \(\Sigma ^0_n\) -formulas \(\psi (x)\) without parameters. We consider computable structures \({\mathcal {S}}\) such that the domain of \({\mathcal {S}}\) has infinitely many \(\Sigma ^0_n\) -definable subsets. Following Goncharov and Kogabaev, we say that an infinite list of \(\Sigma ^0_n\) -formulas is a \(\Sigma ^0_n\) -classification for \({\mathcal {S}}\) if the list enumerates all \(\Sigma ^0_n\) -definable subsets of \({\mathcal {S}}\) without repetitions. We show that an arbitrary computable \({\mathcal {S}}\) always has a \({{\mathbf {0}}}^{(n)}\) -computable \(\Sigma ^0_n\) -classification. On the other hand, we prove that this bound is sharp: we build a computable structure with no \({{\mathbf {0}}}^{(n-1)}\) -computable \(\Sigma ^0_n\) -classifications. PubDate: 2023-02-01
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Abstract: Abstract We prove that every quasi-Hopfian finitely presented structure A has a d- \(\Sigma _2\) Scott sentence, and that if in addition A is computable and Aut(A) satisfies a natural computable condition, then A has a computable d- \(\Sigma _2\) Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable d- \(\Sigma _2\) Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable d- \(\Sigma _2\) Scott sentence, as well as any strongly rigid Coxeter group of finite rank. Finally, we show that the free projective plane of rank 4 has a computable d- \(\Sigma _2\) Scott sentence, thus exhibiting a natural example where the assumption of quasi-Hopfianity is used (since this structure is not Hopfian). PubDate: 2023-02-01
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Abstract: Abstract Second-order arithmetic and class theory are second-order theories of mathematical subjects of foundational importance, namely, arithmetic and set theory. Despite the similarity in appearance, there turned out to be significant mathematical dissimilarities between them. The present paper studies various principles in class theory, from such a comparative perspective between second-order arithmetic and class theory, and presents a few new dissimilarities between them. PubDate: 2023-02-01
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Abstract: In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination. PubDate: 2023-02-01
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Abstract: Abstract For \(n\in \omega \) , the weak choice principle \(\textrm{RC}_n\) is defined as follows: For every infinite set X there is an infinite subset \(Y\subseteq X\) with a choice function on \([Y]^n:=\{z\subseteq Y: z =n\}\) . The choice principle \(\textrm{C}_n^-\) states the following: For every infinite family of n-element sets, there is an infinite subfamily \({\mathcal {G}}\subseteq {\mathcal {F}}\) with a choice function. The choice principles \(\textrm{LOC}_n^-\) and \(\textrm{WOC}_n^-\) are the same as \(\textrm{C}_n^-\) , but we assume that the family \({\mathcal {F}}\) is linearly orderable (for \(\textrm{LOC}_n^-\) ) or well-orderable (for \(\textrm{WOC}_n^-\) ). In the first part of this paper, for \(m,n\in \omega \) we will give a full characterization of when the implication \(\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-\) holds in \({\textsf {ZF}}\) . We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that \(\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-\) and that \(\textrm{RC}_6\Rightarrow \textrm{C}_3^-\) , answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that \(\textrm{RC}_6\Rightarrow \textrm{C}_9^-\) and that \(\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-\) . PubDate: 2022-12-16
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Abstract: Abstract In \(\textbf{ZF}\) , the well-known Fichtenholz–Kantorovich–Hausdorff theorem concerning the existence of independent families of X of size \( {\mathcal {P}} (X) \) is equivalent to the following portion of the equally well-known Hewitt–Marczewski–Pondiczery theorem concerning the density of product spaces: “The product \({\textbf{2}}^{{\mathcal {P}}(X)}\) has a dense subset of size X ”. However, the latter statement turns out to be strictly weaker than \(\textbf{AC}\) while the full Hewitt–Marczewski–Pondiczery theorem is equivalent to \(\textbf{AC}\) . We study the relative strengths in \(\textbf{ZF}\) between the statement “X has no independent family of size \( {\mathcal {P}}(X) \) ” and some of the definitions of “X is finite” studied in Levy’s classic paper, observing that the former statement implies one such definition, is implied by another, and incomparable with some others. PubDate: 2022-12-14
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Abstract: Abstract A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation based on a notion of embeddability of derivations. Additionally, conservativity for classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes. PubDate: 2022-12-08
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Abstract: Abstract We study \(\kappa \) -maximal cofinitary groups for \(\kappa \) regular uncountable, \(\kappa = \kappa ^{<\kappa }\) . Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that: Any \(\kappa \) -maximal cofinitary group has \({<}\kappa \) many orbits under the natural group action of \(S(\kappa )\) on \(\kappa \) . If \(\mathfrak {p}(\kappa ) = 2^\kappa \) then any partition of \(\kappa \) into less than \(\kappa \) many sets can be realized as the orbits of a \(\kappa \) -maximal cofinitary group. For any regular \(\lambda > \kappa \) it is consistent that there is a \(\kappa \) -maximal cofinitary group which is universal for groups of size \({<}2^\kappa = \lambda \) . If we only require the group to be universal for groups of size \(\kappa \) then this follows from \(\mathfrak {p}(\kappa ) = 2^\kappa \) . PubDate: 2022-12-04
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Abstract: Abstract A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set A in this ultrafilter such that the restriction of the function to A is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\) . (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) such that every filter base of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\) . (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\) . PubDate: 2022-11-07 DOI: 10.1007/s00153-022-00853-3
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Abstract: Abstract An appropriate framework is put forward for the construction of \(\lambda \) -models with \(\infty \) -groupoid structure, which we call homotopic \(\lambda \) -models, through the use of an \(\infty \) -category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and \(\lambda \) -calculus, in the sense that the concept of proof (path) of equality of \(\lambda \) -terms is raised to higher proof (homotopy). PubDate: 2022-11-04 DOI: 10.1007/s00153-022-00856-0
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Abstract: Abstract Answering in negative a question of M. Hrušák, we construct a Borel ideal not extendable to any \(F_\sigma \) ideal and such that it is not Katětov above the ideal \(\mathrm {conv}\) . PubDate: 2022-11-01 DOI: 10.1007/s00153-022-00822-w
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Abstract: Abstract We systematically study the interrelations between all possible variations of \(\Delta ^0_1\) variants of the law of excluded middle and related principles in the context of intuitionistic arithmetic and analysis. PubDate: 2022-11-01 DOI: 10.1007/s00153-022-00827-5
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