arXiv subjects

Sakaé Fuchino

Publications and source records attributed to Sakaé Fuchino.

At least 19 recordsLinked to original sources

arXiv2025

Generic Absoluteness Revisited

The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom. M. Viale proved that Martin's Maximum$^{++}$ together with the assumption that there are class many Woodin cardinals implies $\mathcal{H}(\aleph_2)^{\mathsf{V}}\prec_{Σ_2}\mathcal{H}(\aleph_2)^{\mathsf{V}[\mathbb{G}]}$ for a generic $\mathbb{G}$ on any stationary preserving $\mathbb{P}$ which also preserves Bounded Martin's Maximum. We show that a similar but more general conclusion follows from each of $(\mathcal{P},\mathcal{H}(κ))_{Σ_2}$-${\sf RcA}^+$ (which is a fragment of a reformulation of the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(κ)$), and the existence of the tightly $\mathcal{P}$-Laver-generically huge cardinal. While under "$\mathcal{P}=$ all stationary preserving posets", our results are not very much more than Viale's Theorem, for other classes of posets, "$\mathcal{P}=$ all proper posets" or "$\mathcal{P}=$ all ccc posets", for example, our theorems are not at all covered by his theorem. The assumptions (and hence also the conclusion) of Viale's Theorem are compatible with the Ground Axiom. In contrast, we show that the assumptions of our theorems (for most of the common settings of $\mathcal{P}$ and with a modification of the large cardinal property involved) imply the negation of the Ground Axiom. This fact is used to show that fragments of Recurrence Axiom $(\mathcal{P},\mathcal{H}(κ))_Γ$-${\sf RcA}^+$ can be different from the corresponding fragments of Maximality Principle ${\sf MP}(\mathcal{P},\mathcal{H}(κ))_Γ$ for $Γ=Π_2$.

math.LO↗

arXiv2025

Extendible cardinals, and Laver-generic large cardinal axioms for extendibility

We introduce (super-$C^{(\infty)}$-)Laver-generic large cardinal axioms for extendibility ((super-$C^{(\infty)}$-)LgLCAs for extendible, for short), and show that most of the previously known consequences of the (super-$C^{(\infty)}$-)LgLCAs for ultrahuge, in particular, general forms of Resurrection Principles, Maximality Principles, and Absoluteness Theorems, already follow from (super-$C^{(\infty)}$\mbox{-)}LgLCAs for extendible. The consistency of LgLCAs for extendible (for transfinitely iterable $Σ_2$-definable classes of posets) follows from an extendible cardinal while the consistency of super-$C^{(\infty)}$-LgLCAs for extendible follows from a model with a strongly super-$C^{(\infty)}$-extendible cardinal. If $μ$ is an almost-huge cardinal, there are cofinally many $κ<μ$ such that\ $V_μ\models$``$κ$ is strongly super-$C^{(\infty)}$ extendible''. Most of the known reflection properties follow already from some of the LgLCAs for supercompact. We give a survey on the related results. We also show the separation between some of the LgLCAs as well as between LgLCAs and their consequences. LgLCAs are generic large cardinal axioms in terms of generic elementary embeddings with the critical point $κ_{\mathfrak{refl}}=\max\{\aleph_2,2^{\aleph_0}\}$. We show that Laver generic large cardinal axioms for all posets\ in terms of generic elementary embeddings with the critical point $2^{\aleph_0}$ is also possible. We abbreviate this type of axiom for the notion of extendibility as the LgLCAA for extendible and examine its consequences.

math.LO↗

arXiv2025

On Recurrence Axioms

The Recurrence Axiom for a class $\mathcal{P}$ of \pos\ and a set $A$ of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from $A$ is forced by a poset in $\mathcal{P}$, then there is a ground containing the parameters and satisfying the statement. The tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge continuum implies the Recurrence Axiom for $\mathcal{P}$ and $\mathcal{H}(2^{\aleph_0})$. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly $\mathcal{P}$-generic hyperhuge cardinal $κ$, and that $κ$ in the bedrock is genuinely hyperhuge, or even super $C^{(\infty)}$ hyperhuge if $κ$ is a tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge definable cardinal. The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For example, double plus version of Martin's Maximum is a consequence of LGM while Cichoń's Maximum is a phenomenon in many grounds of the universe under LGM.

math.LO↗

arXiv2024

Reflection and Recurrence

We examine the Zermelo Fraenkel set theory with Choice (ZFC) enhanced by one of the (structural) reflection principles down to a small cardinal and/or Recurrence Axioms defined below. The strongest forms of reflection principles spotlight the three scenarios in which the size of the continuum is either $\aleph_1$, or $\aleph_2$, or very large, while the maximal setting of Recurrence Axioms points to the set-theoretic universe with the continuum of size $\aleph_2$. We discuss that both the Reflection Principles and Recurrence Axioms can be construed as preferable candidates of the extension of ZFC in terms of the criteria of Gödel's Program. From this view point, the maximal possible (consistent) combination of these principles and axioms, or even some natural strengthening of the combination (which we want to call ``Laver-generic Maximum'' (LGM)) may be considered as the ultimate extension of ZFC (of course ``ultimate'' only for now -- because of the Incompleteness Theorems): LGM resolves the size of the continuum to be $\aleph_2$ and integrates practically all known statements consistent with ZFC in itself either as its consequences (as it is the case with Martin's Maximum$^{++}$) or as theorems holding in many grounds of the universe (as it is the case with Cichoń's Maximum).

math.LO↗

arXiv2023

Maximality Principles and Ressurection Axioms under a Laver-generic large cardinal

Set-theoretic axioms formulated in terms of existence of a Laver-generic large cardinal were introduced in [16] and studied further in [17], [18], [20]. These axioms, let us call them Laver-genericity axioms, claim the existence of a $\mathcal{P}$-Laver generic large cardinal for various classes $\mathcal{P}$-of proper or semi-proper posets, and they still vary depending on the notions of large cardinal involved, and a modification (tightness) of the definition of Laver-genericity. Laver-genericity axioms we consider here are divided into three groups depending on whether they imply that the Laver generic large cardinal $κ$ is either $\aleph_2=(2^{\aleph_0})^+$, or it is $\aleph_2 = 2^{\aleph_0}$, or else it is very large and $= 2^{\aleph_0}$ (see the Trichotomy Theorem (Theorem 3.5)). Many set-theoretic axioms and principles considered in the recent development of set theory follow from a Laver-genericity axiom in one of these three groups, and by this, they are placed uniformly in a global context (see Figure 3). In spite of this very strong unifying feature of the Laver genericity axioms, we show that Maximality Principle (MP) without parameters is independent over ZFC with any of the Laver-genericity axioms we consider in our present context (Theorem 4.8, Theorem 5.11). Similar independence is also shown for parameterized versions of Maximality Principles (Theorem 6.1, Theorem 6.5). In contrast to these independence results, we can show that local versions of Maximality Principle as well as versions of Resurrection Axioms including the Unbounded Resurrection Axioms of Tsaprounis follow from the existence of a tightly Laver-generic large cardinal for a strong enough notion of large cardinal (Theorem 6.6, Theorem 7.1, Theorem 7.2).

math.LO↗

arXiv2023

Weakly extendible cardinals and compactness of extended logics

We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V=L, a cardinal $κ$ is the weak compactness number of ${\cal L}^{\aleph_0,II}_{stat,κ,ω}$ if and only if it is the weak compactness number of ${\cal L}^{II}_{κ,ω}$. The latter condition is equivalent to the condition that $κ$ is weakly extendible by the characterization mentioned above (this equivalence holds without the assumption of V=L).

math.LO↗

arXiv2022

Generically supercompact cardinals by forcing with chain conditions

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 $\cal P$-generically supercompact cardinals for the class $\cal P$ of all $σ$-closed posets, which can be $\aleph_n$ for any n>1.

math.LO↗

arXiv2021

Strong downward Löwenheim-Skolem theorems for stationary logics, III -- mixed support iteration

Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]], we further study reflection principles in connection with the Löwenheim-Skolem Theorems of stationary logics. In this paper, we mainly analyze the situations in the models obtained by mixed support iteration of a supercompact length and then collapsing another supercompact cardinal to make it $(2^{\aleph_0})^+$. We show, among other things, that the reflection down to $< 2^{\aleph_0}$ of the non-metrizability of topological spaces with small character is independent from the reflection properties studied in [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]].

math.LO↗

arXiv2021

The first-order definability of generic large cardinals

We show that the notions of generic and Laver-generic supercompactness are first-order definable in the language of ZFC. This also holds for generic and Laver-generic (almost) hugeness as well as for generic versions of other large cardinals.

math.LO↗

arXiv2020

Strong downward Löwenheim-Skolem theorems for stationary logics, II -- reflection down to the continuum

Continuing the previous paper, we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. It has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters down to $<\aleph_2$ 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 second-order parameters down to $<2^{\aleph_0}$ implies that the size of the continuum is $\aleph_2$. In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to $<2^{\aleph_0}$ under the continuum being of size $>\aleph_2$. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size $<2^{\aleph_0}$. We also consider a ${\cal P}_κλ$ version of the stationary logic and show that the SDLS for this logic in internal interpretation for reflection down to $<2^{\aleph_0}$ 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 Laver-generic supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be $\aleph_1$ or $\aleph_2$ 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.

math.LO↗

arXiv2020

Reflection principles, generic large cardinals, and the Continuum Problem

Strong reflection principles with the reflection cardinal $\leq\aleph_1$ or $<2^{\aleph_0}$ imply that the size of the continuum is either $\aleph_1$ or $\aleph_2$ 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.

math.LO↗

arXiv2020

On geometrical characterizations of $\mathbb R$-linear mappings

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.

math.GM↗

arXiv2017

How to drive our families mad

Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$, and $a^+(κ)=\sup\{a^+(F): |F| \leq κ\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent. We also give a several constructions of mad families with some additional properties.

math.LO↗

arXiv2016

On the set-generic multiverse

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 Bukovský's theorem in a modern setting (for another proof of this theorem see Bukovský [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].

math.LO↗

arXiv2016

Pre-Hilbert spaces without orthonormal bases

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 $λ$ for any uncountable $λ$ without any orthonormal basis. Let us call a pre-Hilbert space without any orthonormal bases pathological. The pair of the cardinals $κ\leqλ$ such that there is a pre-Hilbert space of dimension $κ$ and density $λ$ are known to be characterized by the inequality $λ\leqκ^{\aleph_0}$. Our result implies that there are pathological pre-Hilbert spaces with dimension $κ$ and density $λ$ for all combinations of such $κ$ and $λ$ including the case $κ=λ$. 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 density is shown to be equivalent with Fodor-type Reflection Principle (FRP).

math.LO↗

arXiv2007

Coloring ordinals by reals

We study combinatorial principles we call Homogeneity Principle HP(κ) and Injectivity Principle IP(κ,λ) for regular κ>\aleph_1 and λ\leqκwhich are formulated in terms of coloring the ordinals <κby reals.

math.LO↗

arXiv2006

Models of real-valued measurability

Solovay's random-real forcing (1971) is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay's model that do not follow from the existence of real-valued measurability.

math.LO↗

arXiv1999

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_{omega+1},aleph_omega)-->(aleph_1,aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding aleph_omega Cohen reals destroys the weak Freese-Nation property of (P(omega),subseteq)

math.LO↗