2019年度秋季総合分科会(於:金沢大学)

SUMMARY: A very classical subject of study in combinatorics of the real line is Cichon’s diagram, which lists cardinal characteristics (and the provable relations between them) that encompasses combinatorial concepts related to measure and category of the real line, and to compactness of subsets of irrational numbers.

It is known that Cichon’s diagram is complete, in the sense that no other inequality between them can be proved. However, just in the present decade it has been explored consistency results regarding three or more different values considered simultaneously in the diagram.

Is it consistent with ZFC that all the cardinals in Cichon’s diagram are pairwise different? This is the main question of the research mentioned above.

A couple of years ago, Goldstern, Kellner and Shelah used large cardinals to answer this question affirmatively. Just very recently, the speaker joined their work, and they managed to show how to prove this consistency result without using large cardinals.

SUMMARY: In this talk, we reconsider the significance of the completeness theorem, primarily, for classical first-order logic. We argue that the significance of the theorem is more complicated than often discussed, since it ultimately rests on the issue of how we should consider the relationship between pertinent informal (or preformal) logical concepts and formal counterparts thereof. More specifically, we first give a concise historical overview of the theorem. Secondly, we discuss a few conceptual problems concerning the theorem. Thirdly, we pay attention to what is often called ‘Kreisel’s squeezing argument,’ which has been given in order to solve one of the conceptual problems that arise in relation to the significance of the completeness theorem. Finally, we discuss both limitations and repercussions of Kreisel’s argument, focusing on Kreisel’s methodological concept called ‘informal rigour,’ which has been introduced to handle the foregoing issue.

SUMMARY: Legendre applies his theory of elliptic functions not only to the geometry but also to the mechanics in Study of Elliptic functions in 1825. Poisson issues the Study of Mechanics in 1833. Legendre’s mechanics is discussed based on his Theory of Number, however, Poisson’s one is based on the mathematical physics with experiments and the observations. We discuss mechanics to which Legendre applies the elliptic functions.

SUMMARY: In 1826, Legendre issues his integral table based on his theory of elliptic functions in Study of Elliptic functions in 1825, who is then 73 years old. Poisson applies thankfully to his calculation of the capillarity or of Earth temperature. In Construction of Integral Table, Legendre verifies the principle functions and constructs the Table with logarithms.

SUMMARY: In the diagrams of manuscripts of Apollonius’ Conics, all the conic sections are substituted by circle arcs. Modern printed editions do not reproduce them, but provide mathematically correct diagrams. We will examine some of the manuscript diagrams, especially multiple diagrams corresponding to different arrangement of points and lines (partly omitted in modern editions), and try to find the best way to present the complicated diagrams of ancient theory of conic sections to modern readers.

SUMMARY: In ancient China before the invention of extraction of square root, the false position method is used as the approximate calculation. In this talk, we present several problems of the books in ancient China.

SUMMARY: Hrushovski constructed pseudoplanes corresponding to irrational numbers which refute a conjecture by Lachlan. Hrushovski’s construction is valid for any real numbers \(\alpha \) with \(0 < \alpha < 1\). The automorphism groups of the pseudoplanes corresponding to rational numbers \(\alpha \) with \(0 < \alpha < 1\) are simple groups. Also, the theories of these pseudoplanes are model complete.

SUMMARY: In this talk, we show that if \(\alpha \) is irrational then any normal generic structure is not superstable.

SUMMARY: By the Friedman’s theorem, we know that every nonstandard model of PA has a proper initial segment which is isomorphic to itself. One of the author investigated when the intersection of such submodels coincides the standard model in her Master thesis. We report her result and related topics, especially the relation of fixed points of self-embeddings and special kinds of initial segments.

SUMMARY: We present a result that improves the main theorem of:

Akito Tsuboi, On Theories Having a Finite Number of Nonisomorphic Countable Models, J. Symbolic Logic Volume 50, Issue 3 (1985), 806–808.

SUMMARY: In the theory of computability, the concept of reduction is a certain type of pseudo order that compares complexity of two sets of natural numbers, or two real numbers. The aim of our talk is a better understanding of the relationships between reduction and continuity. We observe that Solovay reduction is characterized by the existence of a certain Lipschitz continuous real function. Then we ask whether there is a concept of reducibility that exactly corresponds to Hoelder continuous function (with order less than or equal to 1). We show this is the case. We introduce the concept of quasi Solovay reduction. We separate it from Solovay reduction and from Turing reduction. We investigate the relationships between quasi Solovay complete sets and randomness.

SUMMARY: The notion of probability plays a crucial role in quantum mechanics. It appears as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In our former works, based on the toolkit of algorithmic randomness, we presented an alternative rule to the Born rule, called the principle of typicality, for specifying the property of results of measurements in an operational way. In this talk, we reformulate the quantum operations formalism in terms of the principle of typicality, in order to demonstrate how properly our framework works in practical problems in quantum mechanics.

SUMMARY: In classical set theory, De Morgan’s laws (i) \(\neg (\forall x\in a)~ A(x) \Leftrightarrow (\exists x\in a) ~\neg A(x)\) and (ii) \(\neg (\exists x\in a) ~A(x) \Leftrightarrow (\forall x\in a) ~\neg A(x)\) hold for bounded quantifications. However, these laws do not hold in Takeuti’s quantum set theory. Here, we show that Takeuti’s quantum set theory can be improved so that these De Morgan’s laws hold, while maintaining the results obtained so far, such as the transfer principle for theorems of ZFC set theory. As a result, duality is established between conjunction and disjunction, universal quantification and existential quantification in quantum set theory just as in classical set theory, and more powerful developments will be expected in quantum set theory and its applications.

SUMMARY: For a space \(X\), \(e(X)\) denotes the extent of \(X\). We give a characterization of rectangular (equivalently, countably paracompact) products of two subspaces of an ordinal in terms of the equality of extents, under the non-existence of weakly inaccessible cardinal. We also give a simple example that the characterization does not hold under the existence of a weakly inaccessible cardinal.

SUMMARY: Archipelago groups \(\mathcal {A}(G)\) are defined for groups \(G\), i.e. the quotient group of the free \(\sigma \)-product of countable copies of \(G\) by the narmal closure of the free product of countable copies of \(G\). Though theses groups appear as the fundamental groups of natural spaces, the fundamental questions are open.

SUMMARY: Uniformization of ladder system colorings has been introduced by analysis of a proof of the Shelah’s solution of Whitehead problem. Here, for a subset \(\mathcal {S}\) of \(\omega _1\cap \textsf {Lim}\), \(\textsf {U}(\mathcal {S})\) is the assertion that, for any ladder system coloring \(\langle d_\alpha : \alpha \in \omega _1 \cap \textsf {Lim} \rangle \), there exists \(S\in \mathcal {S}\) such that the restricted coloring \(\langle d_\alpha : \alpha \in S \rangle \) can be uniformized. Shelah’s proof can be separated into the following two theorems: \(\textsf {MA}_{\aleph _1}\) implies \(\textsf {U}(\{\omega _1\cap \textsf {Lim} \} )\), and \(\textsf {U}(\{\omega _1\cap \textsf {Lim} \} )\) implies the existence of a non-free Whitehead group. It is proved that the assertion \(\mathcal {K}_3\), which is one of Todorcevic’s fragments of Martin’s Axiom, implies that \(\textsf {U}(\textsf {stat})\) holds, where \(\textsf {stat}\) stands for the set of stationary subsets of \(\omega _1\cap \textsf {Lim}\).

SUMMARY: \(P\)-points over \(\mathcal {P}_{\kappa }\lambda \) are defined using functions. We present an equivalent condition for an ideal \(I\) to be a \(P\)-point using a family of sets belonging to \(I\).

SUMMARY: We investigate variants of so-called Hat Guessing Games; played by infinitely many agents, finitely or infinitely many colors, restricted visibility, non-simultaneous guesses, and so on. We show, assuming the axiom of choice, that in several typical cases the agents have optimal strategies.

SUMMARY: It is known that many concrete forcings such as Cohen forcing destroy AD. In this talk, we show that one cannot preserve AD via forcings as long as the forcing increases \(\Theta \) and V satisfies \(\mathrm {AD}^+\) and \(\mathrm {V} = \mathrm {L}(\mathcal {P}\bigl (\mathbb {R})\bigr )\). We also provide an example of forcings which preserve AD while increasing \(\Theta \) when V is not of the form \(\mathrm {L}\bigl (\mathcal {P}(\mathbb {R})\bigr )\). This is joint work with Nam Trang.

SUMMARY: Zero and infinity are, of course, have long histories over mathematics. From the division by zero, we found some basic meanings of zero and infinity, as mathematics. We would like to talk some global viewpoints on zero and infinity. We will give their definitions first clearly as mathematics. We will introduce a surprising relationship of zero and infinity, clearly in this talk based on the cited references.

SUMMARY: In the previous talk (MSJ Spring Meeting 2018), we reported that there exists a continuum of intermediate predicate logics that have the disjunction property but lack the existence property. In this talk, we report that the logics constructed in this way are all incomplete with respect to Kripke-frame semantics and algebraic frame semantics. Note that these logics have the same propositional part as intuitionistic logic.

SUMMARY: Standard sequent calculi for modal logics S5 and K4B do not enjoy the cut-elimination property. Takano showed that the application of (cut) in these calculi can be restricted to analytic cut. For some modal logics, he investigated the relationships between the inference rules and semantical properties by introducing analytically saturated sequents. By extending his method, we found that we can improve the restriction of (cut) in S5 and K4B. We can prove finite model property of these logics simultaneously.

SUMMARY: The modal propositonal logic \(\mathbf {KD4Z_{14}}\) is an irreflexive counter part of \(\mathbf {S4.1.4}\). We treat\(\mathbf {KD4Z_{14}}\) and a little weaker logic \(\mathbf {K4Z_{14}^+}\).

1. Both logics have cut-free systems.
2. Both logics have f.m.p. and Craig interpolation property, and these properties can be proved simultaneously.

SUMMARY: In non-associative substructural logics, we can consider not only the weakening rules in the usual sense but also restricted weakening rules associated with associativity. In this talk, we introduce a cut-free Gentzen-style formulation for some non-associative substructural logics with restricted weakening rules, and consider some related topics.

SUMMARY: We introduce a formal system of reduction paths as an extension of a monoid-like structure. Our motivation comes from a quantitative analysis of reduction systems based on the perspective of computational cost and orbit. From the perspective, we define a formal system of reduction paths for parallel reduction, wherein paths are generated from a quiver by means of three path-operators. Next, we introduce an equational theory and reduction rules for paths, and show that the rules on paths are terminating and confluent so that normal paths are obtained. Following normal paths, a graphical representation of reduction paths is provided. Then we show that the reduction graph is a plane graph, and unique path and universal common-reduct properties are established.

SUMMARY: We consider how to define adequate semantics for the simple type assignment system for lambda calculus such that \(M:\tau \) is provable if and only if \(M:\tau \) is valid in the semantics.

SUMMARY: A sentence \(\varphi \) is said to be \(\Gamma \)-conservative over \(T\) if for every \(\Gamma \) sentence \(\theta \), if \(T+\varphi \vdash \theta \), then \(T \vdash \theta \). For \(\Gamma = \Sigma _{n}\) (resp. \(\Pi _{n}\)), let \(\Gamma ^{d} = \Pi _{n}\) (resp. \(\Sigma _{n}\)). In 1979, Guaspari proved that for any reasonable theory \(T\), there is an independent \(\Gamma ^{d}\) sentence \(\varphi \) which is \(\Gamma \)-conservative over \(T\). Also Guaspari asked the following question: For any reasonable theories \(T_{0}\) and \(T_{1}\), is there a \(\Gamma ^{d}\) sentence which is simultaneously independet and \(\Gamma \)-conservative over both \(T_{0}\) and \(T_{1}\)? For \(\Gamma = \Sigma _n\), this problem was solved negatively by Bennet, however, for \(\Gamma = \Pi _n\), it has not been settled yet.

First, we introduce a new sufficient condition about the existence of simultaneously independent \(\Pi _n\)-conservative \(\Sigma _n\) sentences. Secondly, we investigate the situation of the existence of such sentences in the case of finite sequences of theories.

SUMMARY: It is known that the propositional provability logic \(\mathbf {GL}\) satisfies the fixed-point property. However, Montagna showed that the predicate modal logic \(\mathbf {QGL}\) loses the fixed-point property. In this talk, we prove that several extensions of \(\mathbf {QGL}\) including Tanaka’s system \(\mathbf {NQGL}\) do not have the fixed-point property. Secondly, we prove that the fixed-point theorem for \(\mathbf {QK} + \Box ^{n+1}\bot \). As a consequence, we obtain that the class \(\mathcal {BL}\) of Kripke frames which are transitive and of bounded length satisfies the fixed-point property locally. We also obtain that \(\mathbf {NQGL}\) does not satisfy the Craig interpolation property. Finally, we investigate a sufficient condition for formulas to have a fixed-point in \(\mathbf {QGL}\).

SUMMARY: We investigate relationships between Gödel’s second incompleteness theorem and derivability conditions for provability predicates. First, we exhibit some new sets of derivability conditions which are sufficient for unprovability of the consistency statement \(\forall x({\rm Pr}_T(x) \to \neg {\rm Pr}_T(\dot {\neg }x))\). Secondly, we show that Hilbert–Bernays’ conditions and Löb’s conditions are mutually incomparable. Thirdly, we show that both of Hilbert–Bernays’ conditions and Löb’s conditions do not accomplish Gödel’s original statement of the second incompleteness theorem. At last, we improve Buchholz’s proof of uniform version of provable \(\Sigma _1\)-completeness.