ON C(n)-EXTENDIBLE CARDINALS

2018 ◽  
Vol 83 (3) ◽  
pp. 1112-1131 ◽  
Author(s):  
KONSTANTINOS TSAPROUNIS
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Homotopy Theory ◽  
Continuum Hypothesis ◽  
Exact Relation ◽  
Generalized Continuum ◽  
Alternative Characterization ◽  
Desirable Feature ◽  
Extendible Cardinals

AbstractThe hierarchies of C(n)-cardinals were introduced by Bagaria in [1] and were further studied and extended by the author in [18] and in [20]. The case of C(n)-extendible cardinals, and of their C(n)+-extendibility variant, is of particular interest since such cardinals have found applications in the areas of category theory, of homotopy theory, and of model theory (see [2], [3], and [4], respectively). However, the exact relation between these two notions had been left unclarified. Moreover, the question of whether the Generalized Continuum Hypothesis (GCH) can be forced while preserving C(n)-extendible cardinals (for n1) also remained open. In this note, we first establish results in the direction of exactly controlling the targets of C(n)-extendibility embeddings. As a corollary, we show that every C(n)-extendible cardinal is in fact C(n)+-extendible; this, in turn, clarifies the assumption needed in some applications obtained in [3]. At the same time, we underline the applicability of our arguments in the context of C(n)-ultrahuge cardinals as well, as these were introduced in [20]. Subsequently, we show that C(n)-extendible cardinals carry their own Laver functions, making them the first known example of C(n)-cardinals that have this desirable feature. Finally, we obtain an alternative characterization of C(n)-extendibility, which we use to answer the question regarding forcing the GCH affirmatively.

COMPLEMENTATION AND EMBEDDINGS OF c0(I) IN BANACH SPACES

2002 ◽  
Vol 85 (3) ◽  
pp. 742-768 ◽  
Author(s):  
SPIROS A. ARGYROS ◽  
JESÚS F. CASTILLO ◽  
ANTONIO S. GRANERO ◽  
MAR JIMÉNEZ ◽  
JOSÉ P. MORENO
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Compact Subset ◽  
Continuum Hypothesis ◽  
Finite Partition ◽  
Weakly Compact ◽  
Complemented Subspaces ◽  
Generalized Continuum ◽  
Compactly Generated

We investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $\mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, \alpha ])$, where $\alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $\mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:(i) in a Banach space $X \in \mathcal{V}$ every copy of $c_0(I)$ with $\# I < \aleph _{\omega}$ is complemented;(ii) if $\alpha _0 = \aleph _0$, $\alpha _{n+1} = 2^{\alpha _n}$, $n \geq 0$, and $\alpha = \sup \{\alpha _n : n \geq 0\}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(\alpha )$.So, under the generalized continuum hypothesis (GCH), $\aleph _{\omega}$ is the greatest cardinal $\tau$ such that every copy of $c_0(I)$ with $\# I < \tau$ is complemented in the class $\mathcal{V}$. If $T : c_0(I) \to C([1,\alpha ])$ is an isomorphism into its image, we prove that:(i) $c_0(I)$ is complemented, whenever $\| T \| ,\| T^{-1} \| < (3/2)^{\frac 12}$;(ii) there is a finite partition $\{I_1, \dots , I_k\}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.Concerning the class $\mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X \in \mathcal{C}$, the following are equivalent:(i) there is a weakly compact subset $W \subset X$ with ${\rm Dens}(W) = \tau$;(ii) $c_0(\tau )$ is isomorphically embedded into $X$.This yields a new characterization of a class of injective Banach spaces.2000 Mathematical Subject Classification: 46B20, 46B26.

A new proof of a theorem of Shelah

1972 ◽  
Vol 37 (1) ◽  
pp. 133-134 ◽  
Author(s):  
John W. Rosenthal
Keyword(s):  
Model Theory ◽  
Continuum Hypothesis ◽  
Order Theory ◽  
Transfinite Induction ◽  
Original Proof ◽  
First Order ◽  
Elementary Chain ◽  
First Order Theory ◽  
Definable Subset ◽  
Generalized Continuum

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

Model Theory: Geometrical and Set-Theoretic Aspects and Prospects

2003 ◽  
Vol 9 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Theoretic Model ◽  
Topos Theory ◽  
Sheaf Theory ◽  
Special Cases ◽  
Algebraic Spaces ◽  
The Subject ◽  
Near Future ◽  
Modern Mathematics

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

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