Disjoint Amalgamation in Locally Finite Aec

Baldwin, John T.; Koerwien, Martin; Laskowski, M.

doi:10.1017/jsl.2016.25

Cited by 14 publications

(18 citation statements)

References 15 publications

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“…Unfortunately, Hjorth's solution is unsatisfactory. As observed in [2], for every countable ordinal α, Hjorth produces not a single L ω1,ω -sentence, but a whole set S α of L ω1,ω -sentences. 1 In [2], the authors notice that if α is a finite ordinal, then the set S α is finite.…”

Section: Introductionmentioning

confidence: 89%

Non-absoluteness of Hjorth's Cardinal Characterization

Lücke¹,

Souldatos²

2021

Preprint

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In [5], Hjorth proved that for every countable ordinal α, there exists a complete Lω 1 ,ω -sentence φα that has models of all cardinalities less than or equal to ℵα, but no models of cardinality ℵ α+1 . Unfortunately, his solution does not yield a single Lω 1 ,ω -sentence φα, but a set of Lω 1 ,ω -sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property.In the present paper, we prove that this conjecture is true. More specifically, we isolate a diagonalization principle for functions from ω 1 to ω 1 which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorth's solution to characterizing ℵ 2 in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.

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Section: Introductionmentioning

confidence: 89%

Non-absoluteness of Hjorth's Cardinal Characterization

Lücke¹,

Souldatos²

2021

Preprint

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“…Theorem 1.3 then implies that the associated classification problems (Mod ω ( B n ), iso ), for n > 0, are incomparable under * -reductions. The structures in Mod ω ( B n ) are labeled versions of certain families of structures that were introduced and studied in [BKL17] for their interesting behavior with respect to disjoint n-amalgamation.…”

Section: Fig 1 the Combinatorial N-cubementioning

confidence: 99%

“…The BKL n structures were introduced in [BKL17] by Baldwin, Koerwien, and Laskowski, based on a similar construction by Laskowski and Shelah in [LS93]. In [BKL17], the classes B n and B fin n were calledK n−1 and K n−1 0 , respectively.…”

Section: * -Reductions and Becker Graphsmentioning

confidence: 99%

“…As a consequence, there are no irreducible n-cubes in B n all of whose structures lie in Mod ω ( B n ). Most of our remaining work is to prove that, in contrast, Mod ω ( B n ) has many irreducible k-cubes whenever k < n. We begin by observing that B fin n is a Fraïssé class when n ≥ 2 (this was also used in [BKL17]). Then we will find an irreducible k-cube in B n for 1 ≤ k < n such that every structure in the cube is isomorphic to the Fraïssé limit of B fin n .…”

Section: * -Reductions and Becker Graphsmentioning

confidence: 99%

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Higher-dimensional obstructions for star reductions

Kruckman¹,

Panagiotopoulos²

2021

Fund. Math.

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A * -reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a * -reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish G-spaces which is generically preserved under * -reductions. For every natural number n we define a free action of S∞ whose dimension is n on every invariant Baire measurable non-meager set. We also show that the S∞-space which induces the equivalence relation = + of countable sets of reals is ∞-dimensional on every invariant Baire measurable non-meager set. We conclude that the orbit equivalence relations associated to all these actions are pairwise incomparable with respect to * -reductions.

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“…(3) Any AEC with intersections in which the closure operator is locally finite (i.e. the closure of any finite set is finite) is a multiuniversal AEC, see [BKL17] for a discussion of locally finite AECs. (4) The AEC K of algebraically closed fields (ordered by subfield) is a multiuniversal AEC which is not a universal class.…”

Section: Preliminariesmentioning

confidence: 99%

Categoricity in multiuniversal classes

Boney

Vasey

2019

Annals of Pure and Applied Logic

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The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a result of the second author.

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Disjoint Amalgamation in Locally Finite Aec

Cited by 14 publications

References 15 publications

Non-absoluteness of Hjorth's Cardinal Characterization

Non-absoluteness of Hjorth's Cardinal Characterization

Higher-dimensional obstructions for star reductions

Categoricity in multiuniversal classes

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