Springer Proceedings in Mathematics & statistics, 2021
Strong reflection principles with the reflection cardinal ≤ ℵ 1 or < 2 ℵ 0 imply that the size of... more Strong reflection principles with the reflection cardinal ≤ ℵ 1 or < 2 ℵ 0 imply that the size of the continuum is either ℵ 1 or ℵ 2 or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.
A ccc-generically supercompact cardinal κ can be smaller than or equal to the continuum. On the o... more A ccc-generically supercompact cardinal κ can be smaller than or equal to the continuum. On the other hand, such a cardinal κ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically measurable cardinals (Theorem 4.1). This is in a strong contrast to P-generically supercompact cardinals for the class P of all σ-closed posets, which can be אn for any n > 1.
It is known that the reflection cardinal of countable chromatic number of graphs is fairly large.... more It is known that the reflection cardinal of countable chromatic number of graphs is fairly large. This stands in contrast with the situation of the countable coloring number whose reflection cardinal is less or equal to that of the Fodor-type Reflection Principle and hence can be consistently $\aleph_{2}$. Applying a theorem of Peter Komj\'ath, it can be shown that the reflection of countable list-chromatic number behaves consistently similarly to the reflection of countable chromatic number but it can also behave consistently like the reflection of countable coloring number. Moreover, the Fodor-type Reflection Principle does not decide in which way the reflection of countable list-chromatic number behaves.
We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a ch... more We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a characterization of linear mappings whose range is $\geq$ 2 dimensional, in terms of preservation of lines (and contraction of lines to a point) by the mappings. This characterization and its affine version generalize the Fundamental Theorem of Affine Geometry. While the algebraic characterization of $\mathbb R$-linear mappings as additive functions depend on the axiom of set theory, our results are provable in (the modern version of) Zermelo's axiom system without Axiom of Choice.
We give an algebraic characterization of pre-Hilbert spaces with an orthonormal basis. This chara... more We give an algebraic characterization of pre-Hilbert spaces with an orthonormal basis. This characterization is used to show that there are pre-Hilbert spaces $X$ of dimension and density $\lambda$ for any uncountable $\lambda$ without any orthonormal basis. Let us call a pre-Hilbert space without any orthonormal bases pathological. The pair of the cardinals $\kappa\leq\lambda$ such that there is a pre-Hilbert space of dimension $\kappa$ and density $\lambda$ are known to be characterized by the inequality $\lambda\leq\kappa^{\aleph_0}$. Our result implies that there are pathological pre-Hilbert spaces with dimension $\kappa$ and density $\lambda$ for all combinations of such $\kappa$ and $\lambda$ including the case $\kappa=\lambda$. A Singular Compactness Theorem on pathology of pre-Hilbert spaces is obtained. A reflection theorem asserting that for any pathological pre-Hilbert space $X$ there are stationarily many pathological sub-inner-product-spaces $Y$ of $X$ of smaller densit...
Assuming Fodor-type Reflection Principle, we prove that every $T_{1}$-space with a point countabl... more Assuming Fodor-type Reflection Principle, we prove that every $T_{1}$-space with a point countable base is left-separated if all of its subspaces of cardinality $\leq\aleph_{1}$ are left-separated. This result improves a theorem bv Fleissner [4] who proved the same assertion under Axioin R.
We give two characterizations of graphs with coloring number $\leq\kappa$ in terms of elementary ... more We give two characterizations of graphs with coloring number $\leq\kappa$ in terms of elementary submodels; one under ZFC and another under SSH and the version of very weak square principle of [8]. These characterizations suggest that the graphs with coloring number $\leq\kappa$ behave very much like the Boolean algebras with $\kappa-$Rees-Nation property (see [5], [8]).
The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relati... more The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver's theorem and Bukovský's theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory. In sections 2 and 3 of this note, we give a proof of Bukovsky's theorem in a modern setting (for another proof of this theorem see [4]). In section 4 we check that the multiverse of set-generic extensions can be treated as a collection of countable transitive models in a conservative extension of ZFC. The last section then deals with the problem of the existence of infinitely-many independent buttons, which arose in the modal-theoretic approach to the set-generic multiverse by J. Hamkins and B. Loewe [12].
Proceedings of the American Mathematical Society, 1999
E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointw... more E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.
Continuing [6], we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary ... more Continuing [6], we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. In [6], it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters SDLS(L ℵ 0 stat , < ℵ 2) down to < ℵ 2 is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak secondorder parameters SDLS − (L ℵ 0 stat , < 2 ℵ 0) down to < 2 ℵ 0 implies that the size of the continuum is ℵ 2. In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to < 2 ℵ 0 under the continuum being of size > ℵ 2. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size < 2 ℵ 0. We also consider a P κ (λ) version of the stationary logic and show that the SDLS for this logic in internal interpretation SDLS int + (L P KL stat , < 2 ℵ 0) for reflection down to < 2 ℵ 0 is consistent under the assumption of the consistency of ZFC + "the existence of a supercompact cardinal" and this SDLS implies that the continuum is (at least) weakly Mahlo. These three "axioms" in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Lavergeneric supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be ℵ 1 or ℵ 2 or very large respectively. We also show that the existence of one of these generic large cardinals implies the "++" version of the corresponding forcing axiom.
Journal of the Japan Association for Philosophy of Science, 2018
Since Gödel's Incompleteness Theorems were published in 1931, not a few mathematicians have been ... more Since Gödel's Incompleteness Theorems were published in 1931, not a few mathematicians have been trying to do mathematics in a framework as weak as possible to remain in a "safe" terrain. While the Incompleteness Theorems do not offer any direct motivations for exploring the terrae incognitae of the alarmingly general and consistency-wise strong settings like the full ZFC or even ZFC with large large cardinals etc., Gödel's Speedup Theorem, a sort of a variant of the Incompleteness Theorems, in contrast, seems to provide positive reasons for studying mathematics in these powerful extended frameworks in spite of the peril called the (in)consistency strength. In this article of purely expository character, we will examine a version of the Speedup Theorem with a detailed proof and discuss the impact of the Speedup Theorem on the whole mathematics.
Annals of the Japan Association for Philosophy of Science, 2012
Platonism and examine the possibility of Platonism viewpoint in ina,thematics in wake of recenL d... more Platonism and examine the possibility of Platonism viewpoint in ina,thematics in wake of recenL devolopments in set theor.v, .
At firstsight it may seem that the theorem is just a special case of Corollary 1 to Theorem 32 in... more At firstsight it may seem that the theorem is just a special case of Corollary 1 to Theorem 32 in [1]. However the /c-categoricityof T is defined there not to be \(k,T) = l but l(/c,T)^kl. So the conclusion of our theorem is stronger than that of the corollary for elementary classes of L^. Unlike in Lma theories, the L^-homogeneity of the models of T in a>i does not simply follow from the a>r categoricity: as proved in [5],there is a countable theory in LWim which is wr categorical but whose models in a)xare not L^-homogeneous. Nevertheless, as far as I know, it seems to be stillan open question, whether Theorem 1 holds without the assumption of homogeneity of the models. With a similar proof to that of Theorem 1 we can also get the following stronger version:
Annals of the Japan Association for Philosophy of Science, 2017
We survey the study of the Fodor-type Reflection Principle (FRP) and discuss the significance of ... more We survey the study of the Fodor-type Reflection Principle (FRP) and discuss the significance of the principle in terms of what we call the reverse-mathematical criterion.
We introduce a principle formulated in terms of the existence of a winning strategy of a game and... more We introduce a principle formulated in terms of the existence of a winning strategy of a game and prove that this principle is placed between the
It is known that the reflection cardinal of countable chromatic number of graphs is fairly large.... more It is known that the reflection cardinal of countable chromatic number of graphs is fairly large. This stands in contrast with the situation of the countable coloring number whose reflection cardinal is less or equal to that of the Fodor-type Reflection Principle and hence can be consistently $\aleph_{2}$. Applying a theorem of Peter Komj\'ath, it can be shown that the reflection of countable list-chromatic number behaves consistently similarly to the reflection of countable chromatic number but it can also behave consistently like the reflection of countable coloring number. Moreover, the Fodor-type Reflection Principle does not decide in which way the reflection of countable list-chromatic number behaves.
We introduce a new reflection principle which we call "Fodor-type Reflection Principle"... more We introduce a new reflection principle which we call "Fodor-type Reflection Principle" (FRP). This principle follows from but strictly weaker than Fleissner's Axiom R. So, for example, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ≤ ℵ2. We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelof if all of its subspaces of cardinality ≤ ℵ1 are meta-Lindelof (Theorem 4.1). It follows from this theorem that, under FRP, every locally countably compact space X is metrizable if all of its subspaces of cardinality ≤ ℵ1 are metriz- able (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.
Springer Proceedings in Mathematics & statistics, 2021
Strong reflection principles with the reflection cardinal ≤ ℵ 1 or < 2 ℵ 0 imply that the size of... more Strong reflection principles with the reflection cardinal ≤ ℵ 1 or < 2 ℵ 0 imply that the size of the continuum is either ℵ 1 or ℵ 2 or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.
A ccc-generically supercompact cardinal κ can be smaller than or equal to the continuum. On the o... more A ccc-generically supercompact cardinal κ can be smaller than or equal to the continuum. On the other hand, such a cardinal κ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically measurable cardinals (Theorem 4.1). This is in a strong contrast to P-generically supercompact cardinals for the class P of all σ-closed posets, which can be אn for any n > 1.
It is known that the reflection cardinal of countable chromatic number of graphs is fairly large.... more It is known that the reflection cardinal of countable chromatic number of graphs is fairly large. This stands in contrast with the situation of the countable coloring number whose reflection cardinal is less or equal to that of the Fodor-type Reflection Principle and hence can be consistently $\aleph_{2}$. Applying a theorem of Peter Komj\'ath, it can be shown that the reflection of countable list-chromatic number behaves consistently similarly to the reflection of countable chromatic number but it can also behave consistently like the reflection of countable coloring number. Moreover, the Fodor-type Reflection Principle does not decide in which way the reflection of countable list-chromatic number behaves.
We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a ch... more We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a characterization of linear mappings whose range is $\geq$ 2 dimensional, in terms of preservation of lines (and contraction of lines to a point) by the mappings. This characterization and its affine version generalize the Fundamental Theorem of Affine Geometry. While the algebraic characterization of $\mathbb R$-linear mappings as additive functions depend on the axiom of set theory, our results are provable in (the modern version of) Zermelo's axiom system without Axiom of Choice.
We give an algebraic characterization of pre-Hilbert spaces with an orthonormal basis. This chara... more We give an algebraic characterization of pre-Hilbert spaces with an orthonormal basis. This characterization is used to show that there are pre-Hilbert spaces $X$ of dimension and density $\lambda$ for any uncountable $\lambda$ without any orthonormal basis. Let us call a pre-Hilbert space without any orthonormal bases pathological. The pair of the cardinals $\kappa\leq\lambda$ such that there is a pre-Hilbert space of dimension $\kappa$ and density $\lambda$ are known to be characterized by the inequality $\lambda\leq\kappa^{\aleph_0}$. Our result implies that there are pathological pre-Hilbert spaces with dimension $\kappa$ and density $\lambda$ for all combinations of such $\kappa$ and $\lambda$ including the case $\kappa=\lambda$. A Singular Compactness Theorem on pathology of pre-Hilbert spaces is obtained. A reflection theorem asserting that for any pathological pre-Hilbert space $X$ there are stationarily many pathological sub-inner-product-spaces $Y$ of $X$ of smaller densit...
Assuming Fodor-type Reflection Principle, we prove that every $T_{1}$-space with a point countabl... more Assuming Fodor-type Reflection Principle, we prove that every $T_{1}$-space with a point countable base is left-separated if all of its subspaces of cardinality $\leq\aleph_{1}$ are left-separated. This result improves a theorem bv Fleissner [4] who proved the same assertion under Axioin R.
We give two characterizations of graphs with coloring number $\leq\kappa$ in terms of elementary ... more We give two characterizations of graphs with coloring number $\leq\kappa$ in terms of elementary submodels; one under ZFC and another under SSH and the version of very weak square principle of [8]. These characterizations suggest that the graphs with coloring number $\leq\kappa$ behave very much like the Boolean algebras with $\kappa-$Rees-Nation property (see [5], [8]).
The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relati... more The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver's theorem and Bukovský's theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory. In sections 2 and 3 of this note, we give a proof of Bukovsky's theorem in a modern setting (for another proof of this theorem see [4]). In section 4 we check that the multiverse of set-generic extensions can be treated as a collection of countable transitive models in a conservative extension of ZFC. The last section then deals with the problem of the existence of infinitely-many independent buttons, which arose in the modal-theoretic approach to the set-generic multiverse by J. Hamkins and B. Loewe [12].
Proceedings of the American Mathematical Society, 1999
E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointw... more E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.
Continuing [6], we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary ... more Continuing [6], we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. In [6], it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters SDLS(L ℵ 0 stat , < ℵ 2) down to < ℵ 2 is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak secondorder parameters SDLS − (L ℵ 0 stat , < 2 ℵ 0) down to < 2 ℵ 0 implies that the size of the continuum is ℵ 2. In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to < 2 ℵ 0 under the continuum being of size > ℵ 2. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size < 2 ℵ 0. We also consider a P κ (λ) version of the stationary logic and show that the SDLS for this logic in internal interpretation SDLS int + (L P KL stat , < 2 ℵ 0) for reflection down to < 2 ℵ 0 is consistent under the assumption of the consistency of ZFC + "the existence of a supercompact cardinal" and this SDLS implies that the continuum is (at least) weakly Mahlo. These three "axioms" in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Lavergeneric supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be ℵ 1 or ℵ 2 or very large respectively. We also show that the existence of one of these generic large cardinals implies the "++" version of the corresponding forcing axiom.
Journal of the Japan Association for Philosophy of Science, 2018
Since Gödel's Incompleteness Theorems were published in 1931, not a few mathematicians have been ... more Since Gödel's Incompleteness Theorems were published in 1931, not a few mathematicians have been trying to do mathematics in a framework as weak as possible to remain in a "safe" terrain. While the Incompleteness Theorems do not offer any direct motivations for exploring the terrae incognitae of the alarmingly general and consistency-wise strong settings like the full ZFC or even ZFC with large large cardinals etc., Gödel's Speedup Theorem, a sort of a variant of the Incompleteness Theorems, in contrast, seems to provide positive reasons for studying mathematics in these powerful extended frameworks in spite of the peril called the (in)consistency strength. In this article of purely expository character, we will examine a version of the Speedup Theorem with a detailed proof and discuss the impact of the Speedup Theorem on the whole mathematics.
Annals of the Japan Association for Philosophy of Science, 2012
Platonism and examine the possibility of Platonism viewpoint in ina,thematics in wake of recenL d... more Platonism and examine the possibility of Platonism viewpoint in ina,thematics in wake of recenL devolopments in set theor.v, .
At firstsight it may seem that the theorem is just a special case of Corollary 1 to Theorem 32 in... more At firstsight it may seem that the theorem is just a special case of Corollary 1 to Theorem 32 in [1]. However the /c-categoricityof T is defined there not to be \(k,T) = l but l(/c,T)^kl. So the conclusion of our theorem is stronger than that of the corollary for elementary classes of L^. Unlike in Lma theories, the L^-homogeneity of the models of T in a>i does not simply follow from the a>r categoricity: as proved in [5],there is a countable theory in LWim which is wr categorical but whose models in a)xare not L^-homogeneous. Nevertheless, as far as I know, it seems to be stillan open question, whether Theorem 1 holds without the assumption of homogeneity of the models. With a similar proof to that of Theorem 1 we can also get the following stronger version:
Annals of the Japan Association for Philosophy of Science, 2017
We survey the study of the Fodor-type Reflection Principle (FRP) and discuss the significance of ... more We survey the study of the Fodor-type Reflection Principle (FRP) and discuss the significance of the principle in terms of what we call the reverse-mathematical criterion.
We introduce a principle formulated in terms of the existence of a winning strategy of a game and... more We introduce a principle formulated in terms of the existence of a winning strategy of a game and prove that this principle is placed between the
It is known that the reflection cardinal of countable chromatic number of graphs is fairly large.... more It is known that the reflection cardinal of countable chromatic number of graphs is fairly large. This stands in contrast with the situation of the countable coloring number whose reflection cardinal is less or equal to that of the Fodor-type Reflection Principle and hence can be consistently $\aleph_{2}$. Applying a theorem of Peter Komj\'ath, it can be shown that the reflection of countable list-chromatic number behaves consistently similarly to the reflection of countable chromatic number but it can also behave consistently like the reflection of countable coloring number. Moreover, the Fodor-type Reflection Principle does not decide in which way the reflection of countable list-chromatic number behaves.
We introduce a new reflection principle which we call "Fodor-type Reflection Principle"... more We introduce a new reflection principle which we call "Fodor-type Reflection Principle" (FRP). This principle follows from but strictly weaker than Fleissner's Axiom R. So, for example, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ≤ ℵ2. We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelof if all of its subspaces of cardinality ≤ ℵ1 are meta-Lindelof (Theorem 4.1). It follows from this theorem that, under FRP, every locally countably compact space X is metrizable if all of its subspaces of cardinality ≤ ℵ1 are metriz- able (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.
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Papers by Sakae Fuchino