2018年度年会(於:東京大学)

SUMMARY: For a class \(\mathcal {C}\) of structures (with a fixed notion \(\mathcal {N}\) of substructure) and a property \(\mathcal {P}\), the reflection cardinal of \((\mathcal {C},\mathcal {P})\) is the minimal cardinal \(\kappa \) such that, for any \(M\in \mathcal {C}\) of cardinality \(>\kappa \), if \(M\) does not satisfies the property \(P\), then there are stationarily many substructures \(N\) of \(M\) of cardinality \(<\kappa \). If \(kappa\) is the reflection cardinal of \((\mathcal {C},\mathcal {N})\), we shall write \(\kappa =Ref(\mathcal {C}, \mathcal {P})\).

By choosing \(\mathcal {C}\), \(\mathcal {N}\) and \(\mathcal {P}\), we can represent many set-theoretic reflection statements. If, for example \(\mathcal {P}\) is simply a contradiction, and \(\mathcal {N}\) is the elementary submodel relation for some logic \(\mathcal {L}\), then \(\kappa =Ref(\mathcal {C}, \mathcal {P})\) is the strong form of Downward Löwenheim-Skolem Theorem down to \(<\kappa \) for \(\mathcal {L}\).

Of these reflection statements, the cases \(\alpha _2=Ref(\mathcal {C}, \mathcal {P})\) and \(2^{\aleph _0}=Ref(\mathcal {C}, \mathcal {P})\) seems to be of special interest. The former may be interpreted as a pronouncement that the first uncountable cardinal \(\aleph _1\) captures the situation \(\neg \mathcal {P}\) good enough while the latter as the pronouncement that the continuum is large enough in connection with the property \(\mathcal {P}\).

For a class \(\mathcal {C}\) of structures (with a fixed notion \(\mathcal {N}\) of substructure) and a property \(\mathcal {P}\), the reflection cardinal of \((\mathcal {C},\mathcal {P})\) is the minimal cardinal \(\kappa \) such that, for any \(M\in \mathcal {C}\) of cardinality \(>\kappa \), if \(M\) does not satisfies the property \(P\), then there are stationarily many substructures \(N\) of \(M\) of cardinality \(<\kappa \). If \(\kappa \) is the reflection cardinal of \((\mathcal {C},\mathcal {N})\), we shall write \(\kappa =R{e}fl(\mathcal {C}, \mathcal {P})\).

By choosing \(\mathcal {C}\), \(\mathcal {N}\) and \(\mathcal {P}\), we can represent many set-theoretic reflection statements. If, for example \(\mathcal {P}\) is simply a contradiction, and \(\mathcal {N}\) is the elementary submodel relation for some logic \(\mathcal {L}\), then \(\kappa =R{e}fl(\mathcal {C}, \mathcal {P})\) is the strong form of Downward Löwenheim-Skolem Theorem down to \(<\kappa \) for \(\mathcal {L}\).

Of these reflection statements, the cases \(\aleph _2=R{e}fl(\mathcal {C}, \mathcal {P})\) and \(2^{\aleph _0}=R{e}fl(\mathcal {C}, \mathcal {P})\) seems to be of special interest. The former may be interpreted as a pronouncement that the first uncountable cardinal \(\aleph _1\) captures the situation \(\neg \mathcal {P}\) good enough while the latter as the pronouncement that the continuum is large enough in connection with the property \(\mathcal {P}\).

The stronger assertions among \(\aleph _2=R{e}fl(\mathcal {C}, \mathcal {P})\) imply the Continuum Hypothesis while assertions of the form \(2^{\aleph _0}=R{e}fl(\mathcal {C}, \mathcal {P})\) tend to imply that the continuum is extremely large.

Most of the natural assertions of the form \(\aleph _2=R{e}fl(\mathcal {C}, \mathcal {P})\) or \(2^{\aleph _0}=R{e}fl(\mathcal {C}, \mathcal {P})\) involves some kind of countability in the property \(\mathcal {P}\). This is the case with the reflection assertion \(\aleph _2=R{e}fl(\mathcal {C}, \mathcal {P})\) where \(\mathcal {C}\) is the class of all graphs with induced subgraphs as the notion of substructure and \(\mathcal {P}\) is the property “of countable coloring number”. It is shown that this assertion is equivalent to the Fodor-type Reflection Principle (FRP). We can also consider the reflection number for the property obtained from these properties of countable character by replacing the countability by of cardinality \(\kappa \). Recently many interesting results about reflection statements in this vein are obtained.

In this talk we will give a survey on these reflection statements.

SUMMARY: In a joint work with many people, we have developed a computer system that solves pre-university level math problems written in natural language. The system is comprised of two parts. One is a language processing pipeline, which translates a math problem into a logical formula. The other is a computer algebra system that derives an answer from the translated problem. In the talk, I will mainly talk about the former part. The main obstacle in the translation from a natural language into a logical language is the flexibility of the natural language, which enables us to convey complex meaning in a concise expression but makes the sentences highly ambiguous for a machine. I will explain how we combat with it using both logical and statistical means.

SUMMARY: The fraction \(1/(z^2+1)\) has two singularities \(z=i, -i\). Expand in \(z=2i\). (1) Devide the domain into \(0 \le |z-2i| < 1\), \(1 < |z-2i| < 3\), \(3 < |z-2i|\). (2) Decomposite it into partials: \((i/2) \{1/(z+1)-i/(z-i)\}\). Ex. \(1/(z+1)=1/\{(z-2i)+3i\}\). Let us omit Laurent’s expansion. Wada’s theorem: \(1/(\square - \triangle )^{p+1}=\Sigma (k=1 \rightarrow \infty ) d_p(k) \square ^{-p-k}\triangle ^{k-1}, 0<|\triangle /\square |<1\), where integers \(p \ge 0\); \(k \ge 1\) and \(d_p(k) \equiv (k+p-1)!/p!(k-1)!\), named Suida expansion. If \(1 \div \{(z-2i)-(-3i)\}^{0+1}=\Sigma (k=1 \rightarrow \infty ) d_0(k)(z-2i)^{-0-k} (-3i)^{k-1}\), \(0<|-3i/(z-2i)|<1\). Then \(\Sigma (k=1\rightarrow \infty )\) \((-3i)^{k-1}(z-2i)^{-k}\), \(3<|z-2i|\). If \(1/\{3i-(-(z-2i))\}^{0+1}=\Sigma (k=1 \rightarrow \infty )d_0(k)(3i)^{-0-k}(-(z-2i))^{k-1}\), \(0<|-(z-2i)/3i|<1\). Then \(\Sigma (k=1\rightarrow \infty )d_0(k)(-1)^{k-1}(1/3i)^k(z-2i)^{k-1}\), \(0<|z-2i|<3\).

SUMMARY: Maupertuis treats principle of minimum action and Clairaut discuss effort and principles of equilibrium on earth. Poisson issues Study of Mechanics in 1833, which consists the second book of three books, entitled A Study of Mathematical Physics. He discusses the mathemetical principles from many side of mathematics. Our present to this session shows the points of mathematical scopes in mechanics by Poisson.

SUMMARY: Providing capillary action in the equilibrium, Poisson assures that the rise of the surface of water is due to the abrupt variation of density in the neighborhood of the wall and of the surface. Poisson discusses this problem in 1831, in the rivalry to the paper/book of Laplace 1806–7 and Gauss 1831. We show Poisson’s discussion.

SUMMARY: In the early twentieth century, mathematicians began to eagerly discuss transformations of variables that keep the canonical form of the differential equations, the so-called ^^ ^^ contact transformation” or ^^ ^^ canonical transformation”. This paper examines origins of these two transformations. We also discuss a process how these two transformations were bound and how they were introduced to the Hamilton–Jacobi theory.

SUMMARY: Two ‘Toshoku (doushi)’ problems contained in the book “Shu”, and one of them had been left undeciphered. In this talk, we decipher the problem from the other deciphered problem, and discuss that the method used in Toshoku problems is ‘shaoguang-shu.’

SUMMARY: We try to evaluate the similarity of geometric problems in WASAN using NMF (Non-negative Matrix Factorization). In our study, we calculate characteristic vectors of geometric problems in “SANPOUTENSEISHINAN” written by “AIDA YASUAKI” by NMF. And we try to evaluate the similarity of geometric problems by these characteristic vectors.

SUMMARY: Volumes 8 and 9 of th Taisei Sankei (Great Accomplished Mathematical Treatise, 1710 or 11) are entitled Daily Mathematics and contain 220 problems of mathematics stemmed from daily lives. Because of their elementary character, they have been neglected by historians of Japanese mathematics; for example, in the Meijizen Hihon Sūgakushi (History of Mathematics before the Meiji Restoration), Fujiwara Matsusaburo paid almost no attention to them. Takebe Katahiro, one of three authors of the Taisei Sankei published the Sangaku Keimō Genkai Taisei (Great Colloquial Commentary on the Suanxue Qimeng) in 1690. The Suanxue Qimeng (Introduction to Mathematics, 1299), written by Zhu Shijie of the Yuan dynasty, was a collection of mathematical problems. We shall discuss the relation of the Daily Mathematics with the Suanxue Qimeng.

SUMMARY: Aida Yasuaki (1747–1817) wrote the 6 volume Sanpou Kokon Tsuuran in 1797, which incisively criticized 19 already published books of mathematics. I have discussed the first volume of Aida’s work, which contained comments on 7 books from the Sanpou Kongen Ki (1666) to the Katsuyou Sanpou (1712). In this instance, I examined the second volume, which contained comments on 8 books from the Kagaku Sanpou (1715) to the Meigen Sanpou (1764). Though some comments were the same as those found in volume 1, some were new. I will comment on the history of difficult problems, and on the relationship between solutions using equations and solutions using the abacus.

SUMMARY: In recent years, construction of image databases from digitized historical documents of Japanese mathematics (wasan) has been progressing. Since previous studies, the authors have proposed ideas for automatic tagging in these image databases of wasan based on geometric elements (triangles, squares, circles and so on) and those relationship (tangency of circles, number of elements and so on). In this study, the authors implement the program that tags images of geometric problems in wasan automatically and verify the effectiveness of our proposal for geometric problems included in actual documents of wasan. As a result, it is found that automatic tagging succeeds for geometric problems of more than 80 percent in “Sanpo tensei-ho shinan” by Yasuaki Aida.

SUMMARY: Liu and Tanaka (2007) asserted that among independent distributions on a uniform binary AND-OR tree, the minimum cost (achieved by an algorithm) is maximized only by an independent and identical distribution. In this decade, the assertion has been justified under a hypothesis that only depth-first algorithms are taken into consideration. The uniform binary tree case was shown by S. and Niida (2015). Balanced multi-branching tree case was shown (with a certain hypothesis) by Peng et al. (2017). We extend the results of S.–Niida and Peng et al. to the case where non-depth-first algorithms are taken into consideration.

SUMMARY: In 2007, Liu and Tanaka characterized the eigen-distributions that achieve the distributional complexity for AND-OR trees, and among others, they proved the uniqueness of eigen-distribution for a uniform binary tree. Later, Suzuki and Nakamura showed that the uniqueness fails if only directional algorithms are considered. In this talk, we introduce the weighted trees, namely, trees with weighted cost depending on the value of a leaf. Using such models, we prove that for balanced multi-branching trees, the uniqueness of eigen-distribution holds w.r.t. all deterministic algorithms, but fails w.r.t. only directional algorithms.

SUMMARY: Solovay reducibility is a well-known and inmoprtant notion in theory of randomness. We defined pseudo Solovay reducibility to generalize Solovay reducibility. We have some results of pseudo Solovay reducibility.

SUMMARY: The indicator argument is a model-theoretic framework to obtain independence and conservation results in the study of first-order arithmetic. In the talk, we will consider new formulation of indicator arguments with the idea of generic cuts and forcing. With this method, we will analyze the conservation results for bounding principle and Ramsey’s theorem for pairs.

SUMMARY: The notion of probability plays a crucial role in quantum mechanics. It appears as the Born rule. In this talk we reveal that every event with probability one occurs certainly in quantum measurement.

SUMMARY: It is proved that every \(C^*\)-embedded subset in \(\Bbb N^{\omega _1}\) is \(C\)-embedded in \(\Bbb N^{\omega _1}\) under a certain set-theoretic assumption, where \(\Bbb N^{\omega _1}\) denotes the product of \(\omega _1\) copies of natural numbers \(\Bbb N\). As a consequence, it is independent of ZFC that there is a (closed) \(C^*\)-embedded subset in \(\Bbb N^{\omega _1}\).

SUMMARY: For a topological space \(X\), the Lindelöf degree of \(X\) is the minimal cardinal \(\kappa \) such that every open cover of \(X\) has a subcover of size \(\le \kappa \). If \(S\) is the Sorgenfrey line, then its product \(S \times S\) has the Lindelöf degree \(2^\omega \). On the other hand, it is unknown whether there are Lindelöf spaces \(X\) and \(Y\) with \(L(X \times Y)>2^\omega \). In this talk, we prove that, in the Cohen forcing extension, the Lindelöf degree of the product of two regular Lindelöf spaces can be arbitrary large up to the least \(\omega _1\)-strongly compact cardinal. We also show that if there is no such Lindelöf spaces, then \(\omega _2\) is weakly compact in \(L\).

SUMMARY: The forcing notion which adds an uncountable antichain through an Aronszajn tree has two similar combinatorial properties, which are called the rectangle refining property and the property \(\mathsf {R}_{1,\aleph _1}\). By a viewpoint of specialization of an Aronszajn tree, we can conclude that two properties are different.

SUMMARY: We present recent developments in forcing theory that have been motivated by problems on combinatorics of the real line. Concretely, the construction of tree-dimensional arrangements of forcing generic extensions (joint work with Fischer, Friedman and Montoya), and the incorporation of ultrafilter limits in two-dimensional arrangements of generic extensions (joint with Brendle and Cardona). These techniques work to construct models where the cardinals in Cichoń’s diagram (classical diagram of cardinal numbers associated with combinatorial properties of the real line) can be divided into 7 different values, which is the maximum number known modulo ZFC alone.

SUMMARY: We discuss what kinds of uncountable linearly ordered sets are embeddable into models generated by uncountable indiscernible sequences.

SUMMARY: In this paper, we give the definition of sets and prove its existence theorem by using the axiomatic method. The system of axioms I used here is the system of axioms ZFC and the axiom of ordinary numbers and the axiom of transfinite induction.

SUMMARY: In this talk, we introduce a Kripke complete predicate extension of the logic of provability, that is, the propositional modal logic defined by the Löb formula.

The proof system for the logic of the talk is a modal extension of Gentzen-style sequent calculus for predicate logic. It has a standard derivation rule for necessitation, but does not include the Löb formula as an axiom schema. Instead, it has a non-compact inference rule. We show the logic is complete with respect to the class of Kripke frames of bounded length.

SUMMARY: The disjunction and existence properties in intermediate predicate logics were revealed to be independent in our previous paper. In that paper, we constructed a continuum of intermediate predicate logics having existence property but lacking disjunction property. In this talk, we report the existence of a continuum of intermediate predicate logics having disjunction property but lacking existence property.

SUMMARY: We analyze quantitative properties of witnesses of the Church–Rosser Theorem for beta-equality in terms of Takahashi translation and by using the notion of parallel reduction. We show that the proof method developed here can be applied to other reduction systems such as lambda-calculus with beta-eta-reduction, Girard’s system F, and Gödel’s system T as well.

SUMMARY: In Routley–Meyer semantics, relevant logics or relevant modal logics are characterized by unreduced frames in general. In this talk, we consider modal extensions of Slaney’s reduced modeling theorem for relevant logics without WI.

SUMMARY: We proved that if \(L\) is one of the modal logics \(\mathsf {GL}_\alpha \), \(\mathsf {D}_\beta \), \(\mathsf {S}_\beta \) and \(\mathsf {GL}_\beta ^-\) where \(\alpha \subseteq \omega \) is \(\Sigma _1\) and \(\beta \subseteq \omega \) is cofinite, then for any \(\Sigma _1\)-definable consistent extension \(U\) of Peano Arithmetic \(\mathsf {PA}\), there exists a \(\Sigma _1\) definition \(\tau (v)\) of some extension of \(I\Sigma _1\) such that the provability logic \(\mathsf {PL}_\tau (U)\) of \(\tau (v)\) relative to \(U\) is exactly \(L\). We proved this theorem by using Jeroslow’s method of decomposing theories.

SUMMARY: We associate quantum 2-tori \(T_\theta \) with the structure over \({\mathbb C}_\theta = ({\mathbb C}, +, \cdot , y = x^\theta ),\) where \(\theta \in {\mathbb R} \setminus {\mathbb Q}\), and introduce the notion of geometric isomorphisms between such quantum 2-tori.

We show that this notion is closely connected with the fundamental notion of Morita equivalence of non-commutative geometry. Namely, we prove a model theoretic version of Rieffel’s theorem of quantum 2-tori.

SUMMARY: The automorphism group on the random graph is simple. Moreover, for all non identical element \(g\) of the automorphism group, every element can be denoted by a product of at most three conjugates of \(g\) or \(g^{-1}\), by Truss in 2003. We call the number of the product the width of the automorphism group. It’s known that the similar fact holds in generic structures analogues of the random graph. That is, if the generic structure has FAP (Free Amalgamation Property) and its automorphism group is transitive, then the group is simple and the width is less than or equal to 32. In this talk, we proof we can improve the width to 12.

SUMMARY: Hrushovski defined an amalgamation class \(K_f\) by defining a concave increasing function \(f\) referring to an irrational number \(\alpha \) with \(1/2 < \alpha < 2/3\). His construction works for any real number \(\alpha \) with \(0 < \alpha < 1\). We obtained the following: (1) \(f\) is concave and strictly increasing. (2) \(K_f\) is an amalgamation class for any \(\alpha \). (3) \(f\) is unbounded if \(\alpha \) is rational. (4) If \(\alpha = m/d\) then \(f(2x) \leq f(x) + 1/d\). Therefore, the generic model of \(K_f\) has a model complete theory. (5) There is \(\alpha \) (e.g. \(\alpha = 1/\sqrt {2}\)) such that \(f\) is bounded. (6) If \(f\) is bounded then the generic model of \(K_f\) has no model complete theory. (7) We can give an \(\alpha \) in a form of continued fractions where \(f\) is unbounded.