Large cardinal axioms from tameness in AECs

Boney, Will; Unger, Spencer

doi:10.1090/proc/13555

Cited by 19 publications

(19 citation statements)

References 11 publications

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“…As we proceed by a careful analysis of the central proof of [BTR16], we also recall the cardinal notions at play there. Notice, though, that we refer to these notions using the terminology of [BM14, 4.6], as it is more in keeping with Definition 2.1 (and, of course, [BU17]).…”

Section: Preliminariesmentioning

confidence: 99%

A category-theoretic characterization of almost measurable cardinals

Lieberman

2020

Proc. Amer. Math. Soc.

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Through careful analysis of an argument of [BTR16], we show that the powerful image of any accessible functor is closed under colimits of κ-chains, κ a sufficiently large almost measurable cardinal. This condition on powerful images, by methods resembling those of [LR16], implies κ-locality of Galois types. As this, in turn, implies sufficient measurability of κ, via [BU17], we obtain an equivalence: a purely category-theoretic characterization of almost measurable cardinals.

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Section: Preliminariesmentioning

confidence: 99%

A category-theoretic characterization of almost measurable cardinals

Lieberman

2020

Proc. Amer. Math. Soc.

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“…Also, due to monotonicity results for tameness, the Boney results show that the Hanf number for < λ-tameness is at most the first almost strongly compact above λ (if such a thing exists). The results [BU,Theorem 4.9] put a large restriction on the structure of the tameness spectrum for any ZFC Hanf number. In particular, the following This means that a ZFC (i. e., not a large cardinal) Hanf number for < κtameness would consistently have to avoid cardinals of the form σ (λ <κ ) (under GCH, all cardinals are of this form except for singular cardinals and successors of singulars of cofinality less than κ).…”

Section: The Big Gapmentioning

confidence: 99%

Hanf numbers and presentation theorems in AECs

Boney¹,

Baldwin²

2017

Beyond First Order Model Theory

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This paper addresses a number of fundamental problems in logic and the philosophy of mathematics by considering some more technical problems in model theory and set theory. The interplay between syntax and semantics is usually considered the hallmark of model theory. At first sight, Shelah's notion of abstract elementary class shatters that icon. As in the beginnings of the modern theory of structures ([Cor92]) Shelah studies certain classes of models and relations among them, providing an axiomatization in the Bourbaki ([Bou50]) as opposed to the Gödel or Tarski sense: mathematical requirements, not sentences in a formal language. This formalism-free approach ([Ken13]) was designed to circumvent confusion arising from the syntactical schemes of infinitary logic; if a logic is closed under infinite conjunctions, what is the sense of studying types? However, Shelah's presentation theorem and more strongly Boney's use [Bon] of aec's as theories of L κ,ω (for κ strongly compact) reintroduce syntactical arguments. The issues addressed in this paper trace to the failure of infinitary logics to satisfy the upward Löwenheim-Skolem theorem or more specifically the compactness theorem. The compactness theorem allows such basic algebraic notions as amalgamation and joint embedding to be easily encoded in first order logic. Thus, all complete first order theories have amalgamation and joint embedding in all cardinalities. In contrast *

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“…Our notation itself has a long history (see for example [10,11,19]), and has the benefit of descriptiveness: κ is L µ,ω -compact if and only if for every set T of sentences in the language L µ,ω (or indeed L µ,µ ), if every subset of T of cardinality less than κ is satisfiable, then T is satisfiable. Our notation is perhaps cumbersome when λ is specified, and for this general case Boney and Unger's proposal "(µ, λ)-strong compactness" [8] might be a better solution, but since we shall never need to specify λ, our "L µ,ω -compact" seems a more elegant choice than their "(µ, ∞)strongly compact".…”

Section: Preliminariesmentioning

confidence: 99%

“…Will Boney and Spencer Unger have independently obtained similar results about tameness of AECs from similarly reduced large cardinal assumptions, and moreover can derive large cardinal strength back from tameness assumptions. Specifically, they show in a forthcoming paper [8] the equivalence of "every AEC K with LS(K) < κ is < κ tame" with κ being almost strongly compact, that is, L µ,ω -compact for every µ < κ. Note that the existence of an L µ,ω -compact cardinal for every regular µ is equivalent to the existence of a proper class of almost strongly compact cardinals -see Proposition 2.4 below.…”

Section: Introductionmentioning

confidence: 99%

Accessible images revisited

Brooke-Taylor

Rosický

2016

Proc. Amer. Math. Soc.

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Abstract. We extend and improve the result of Makkai and Paré [15] that the powerful image of any accessible functor F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of L µ,ω -compact cardinals for sufficiently large µ, and also show that under this assumption the λ-pure powerful image of F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of employing the large cardinal assumption to prove each result -one by a direct ultraproduct construction and one using the machinery of elementary embeddings of the settheoretic universe.

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Large cardinal axioms from tameness in AECs

Cited by 19 publications

References 11 publications

A category-theoretic characterization of almost measurable cardinals

A category-theoretic characterization of almost measurable cardinals

Hanf numbers and presentation theorems in AECs

Accessible images revisited

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