Infinite Cardinals Research Articles

Rings of formal power series and symmetric real semigroups

Sections 1–6 of this paper deal with the rings ▪ of (generalized) power series over a formally real field F of coefficients and (non-negative) exponents in an arbitrary totally ordered abelian group G. We determine explicitly the nature of the elements constituting the real spectra of these rings, its structure under the partial order of specialization, and the structure of their associated real semigroups (RS). Theorem A is a corollary of these results. Starting with section 7 we introduce a new class of symmetric real semigroups by requiring that their spaces of characters satisfy analogs to some of the basic laws holding in the real spectra of the rings ▪. These spaces have, amongst others, remarkable symmetry properties that justify their name (see 7.16 and 7.17). In §8 we prove that every finite symmetric RS is isomorphic to the RS associated to some ring of formal power series, but that such representation fails, in general, for symmetric RSs of suitably large infinite cardinality. Our study leads to showing, in §10, that the inclusion map G⁎∪{0}↪G of the reduced special group G⁎ of invertible elements of a symmetric RS G has a retract. As a consequence, every symmetric RS, G, is isomorphic to the extension of G⁎ by a 3-semigroup Δ defined in terms of that retract. Extensions of this form are introduced and studied in §9. Finally, in §11 we present a first order axiomatisation of the class of symmetric RSs in the natural language for real semigroups.

A generalisation of Läuchli's lemma

AbstractLäuchli showed in the absence of the Axiom of Choice () that and, consequently, for all infinite cardinals , where and are the cardinalities of the set of finite subsets and the power set, respectively, of a set which is of cardinality . In this article, we give a generalisation of a simple form of Läuchli's lemma from which several results can be obtained. That is, in the latter equation can be replaced by other cardinals which are equal to in but not in , for example, and , the cardinalities of the set of permutations and the set of partitions, respectively, of a set which is of cardinality .

On a cardinal inequality in ZF$\mathsf {ZF}$

AbstractIt is proved in (without the axiom of choice) that for all infinite cardinals and all natural numbers , where is the cardinality of the set of permutations with exactly non‐fixed points of a set which is of cardinality .

The permutations with n non‐fixed points and the subsets with n elements of a set

AbstractWe write and for the cardinalities of the set of permutations with n non‐fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality , where n is a natural number greater than 1. With the Axiom of Choice, and are equal for all infinite cardinals . We show, in ZF, that if is assumed, then for any infinite cardinal . Moreover, the assumption cannot be removed for and the superscript cannot be replaced by n. We also show under that for any infinite cardinal , implies is Dedekind‐infinite.

Peano’s Conception of a Single Infinite Cardinality

Previous articleNext article No AccessPeano's Conception of a Single Infinite CardinalityClaudio Ternullo and ISABELLA FASCITIELLOClaudio Ternullo Search for more articles by this author and ISABELLA FASCITIELLO Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by HOPOS Just Accepted Sponsored by The International Society for the History of Philosophy of Science Article DOIhttps://doi.org/10.1086/726078 HistoryAccepted December 14, 2022 © 2023 The International Society for the History of Philosophy of Science. All Rights reserved.PDF download Crossref reports no articles citing this article.

The power set and the set of permutations with finitely many non‐fixed points of a set

AbstractFor a cardinal , we write for the cardinality of the set of permutations with finitely many non‐fixed points of a set which is of cardinality . We investigate the relationships between and for an arbitrary infinite cardinal in (without the axiom of choice). It is proved in that for all infinite cardinals , and we show that this is the best possible result.

The Löwenheim-Skolem theorem for Gödel logic

We prove the following Löwenheim-Skolem theorems for first-order Gödel logic:(1)For the Gödel logic G[0,1], a sentence ϕ has models of every infinite cardinality if and only if it has a model of cardinality ℶω(=sup⁡{ℵ0,2ℵ0,…}).(2)For an arbitrary Gödel logic GT, a sentence ϕ has models of every infinite cardinality if and only if it has a model of cardinality ℶω1. Moreover, (1) becomes false if ℶω is replaced by a smaller cardinality, and (2) becomes false if ℶω1 is replaced by a smaller cardinality.

On strong chains of sets and functions

AbstractShelah has shown that there are no chains of length ω3 increasing modulo finite in . We improve this result to sets. That is, we show that there are no chains of length ω3 in increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω2 increasing modulo finite in as well as in . More generally, we study the depth of function spaces quotiented by the ideal where are infinite cardinals.

Critical Nash Value Nodes for Control Affine Embedded Graphon Mean Field Games*

Graphon Mean Field Games (GMFGs) (Caines and Huang (2021)) constitute generalizations of Mean Field Games to the case where the agents form subpopulations associated with the nodes of large graphs. The work in (Foguen-Tchuendom et al. (2021), Foguen-Tchuendom et al. (2022a)) analyzed the stationarity of equilibrium Nash values with respect to node location for large populations of non-cooperative agents with linear dynamics on large graphs embedded in Euclidean space together with their limits (termed embedded graphons). That analysis is extended in this investigation to agent systems lying in the class of control affine non-linear systems (see Isidori (1985)). Specifically, control affine GMFG systems are treated where (i) at each node αeV the drift of each generic agent system is affine in the control function, and (ii) the running costs at each node α ϵ V ⊂ Rm are exponentiated negative inverse quadratic (ENIQ) functions of the difference between a generic state and the local graphon weighted mean Zα,µG where µG:= {µβ,β ϵ V ⊂ Rm} is the globally distributed family of mean fields. The infinite cardinality node and edge limits are considered, where it is assumed that the limit embedded graphon g(α, β), (α, β) ϵ V x V, is continuously differentiable. It is shown that the equilibrium Nash value Vα is stationary with respect to the nodal location α ϵ V if and only if the corresponding mean Zα,µG is stationary with respect to nodal location.

The double density spectrum of a topological space

It is an interesting, maybe surprising, fact that different dense subspaces of even “nice” topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space X that we call the double density spectrum of X and denote by dd(X).We improve a result from [1] by showing that dd(X) is always ω-closed (i.e., countably closed) if X is Hausdorff.We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows.Let S be a non-empty set of infinite cardinals. Then (1) S = dd(X) holds for a Hausdorff space X iff S is ω-closed and \(\sup \,S \le {2^{{2^{\min S}}}}\) (2) S = dd(X) holds for a regular space X iff S is ω-closed and sup S ≤ 2min S. We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance: (i) If κ = cf(κ)embeds in \({\cal P}\left( \omega \right)/{\rm{fin}}\) and S is any set of uncountable regular cardinals < κ with ∣S∣ < min S, then there is a compactum C such that {ω, κ}∪ S ⊂ dd(C), moreover λ ∉ dd(C) whenever ∣S∣ + ω < cf(λ) < κ and cf(λ) ∉ S. (ii) It is consistent to have a separable compactum C such that dd(C) is not ω1-closed.

Infinite Cardinals and Combinatorial Principles

Infinite Cardinals and Combinatorial Principles

Characterising k-connected sets in infinite graphs

A k-connected set in an infinite graph, where k>0 is an integer, is a set of vertices such that any two of its subsets of the same size ℓ≤k can be connected by ℓ disjoint paths in the whole graph.We characterise the existence of k-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. We also prove a duality theorem for the existence of such k-connected sets: if a graph contains no such k-connected set, then it has a tree-decomposition which, whenever it exists, precludes the existence of such a k-connected set.

The secret life of μ-clubs

Let μ<κ<λ be three infinite cardinals, the first two being regular. We compare five versions for Pκ(λ) of the ideal NSκ|Eμκ (the restriction of the nonstationary ideal on κ to the set of all limit ordinals less than κ of cofinality μ): NSκ,λ|Eμκ,λ (the restriction of the nonstationary ideal on Pκ(λ) to the set of all a in Pκ(λ) of uniform cofinality μ), NSμ,κ,λ (the smallest (μ,κ)-normal ideal on Pκ(λ)), J(μ,κ,λ) (the smallest projection on Pκ(λ) of a restriction of the nonstationary ideal on some Pκ(π) to the set of all x in Pκ(π) such that x∩λ can be reconstructed from a subset of x of size μ (and any of its subsets of size μ)), the ideal Nμ-Sκ,λ dual to the μ-club filter on Pκ(λ) and the game ideal NGκ,λμ. We show that if λ<κ+ω, then the first four ideals (and even all five ideals in case ρ<μ<κ for any cardinal ρ<κ) coincide. Our main result asserts that if there are no large cardinals in an inner model, then Nμ-Sκ,λ=J(μ,κ,λ). This throws some light on the so far rather mysterious μ-club filter.

KNASTER AND FRIENDS III: SUBADDITIVE COLORINGS

AbstractWe continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta &lt; \kappa $ , the existence of a strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is a theorem of $\textsf{ZFC}$ . Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is independent of $\textsf{ZFC}$ . We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of $\kappa $ -Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is equivalent to a certain weak indexed square principle $\boxminus ^{\operatorname {\mathrm {ind}}}(\kappa , \theta )$ . We conclude the paper with an application to the failure of the infinite productivity of $\kappa $ -stationarily layered posets, answering a question of Cox.

Attaining strong diameter two property for infinite cardinals

We extend the (attaining of) strong diameter two property to infinite cardinals. In particular, a Banach space has the 1-norming attaining strong diameter two property with respect to ω (1-ASD2Pω for short) if every convex series of slices of the unit ball intersects the unit sphere. We characterize the C(K) spaces and the L1(μ) spaces having the 1-ASD2Pω. We establish dual implications between the 1-ASD2Pω, ω-octahedral norms and Banach spaces failing the (−1)-ball-covering property. The stability of these new properties under direct sums and tensor products is also investigated.

Group topologies making every continuous homomorphic image to a compact group connected

A topological group is minimally almost periodic (MinAP) if the only continuous homomorphism to any compact group is trivial. Dikranjan and Shakhmatov proved that if an abelian group can be equipped with a MinAP group topology, then for every m∈N the subgroup mG of G is either the trivial group or has infinite cardinality. In this paper we prove the following: if an abelian group G can be equipped with a group topology making all of its continuous homomorphic images to a compact group connected, then it admits a MinAP group topology. This condition becomes sufficient as well, as every MinAP topological group only has trivial continuous homomorphic images in compact groups.

Multiplier Algebras of Normed Spaces of Continuous Functions

In this article, we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF’s which only admit constant multipliers. To do that, using a method from Mashreghi and Ransford (Anal Math Phys 9(2):899–905, 2019), we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space of infinite cardinality. On the other hand, we give a sufficient condition for non-separability of a multiplier algebra.

Normal subgroups in the group of column-finite infinite matrices

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

A Choice-Free Cardinal Equality

For a cardinal a, let fin(a) be the cardinality of the set of all finite subsets of a set which is of cardinality a. It is proved without the aid of the axiom of choice that, for all infinite cardinals a and all natural numbers n, 2fin(a)n=2[fin(a)]n. On the other hand, it is proved consistent with ZF that there exists an infinite cardinal a such that 2fin(a)<2fin(a)2<2fin(a)3<⋯<2fin(fin(a)).

Existence and Exactness of Exponential Riesz Sequences and Frames for Fractal Measures

We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ℝd generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the Rϵ-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermore, when the measure is the middle-third Cantor measure, or more generally for self-similar measures with no-overlap condition, there are always exponential Riesz sequences of maximal possible Beurling dimension.