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1
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Consistency of the Silver dichotomy in generalised Baire space

100%
Friedman S.-D.
Fundamenta Mathematicae
|
2014
|
tom 227
|
nr 2
179-186
EN
Silver's fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $κ^{κ}$ for a regular uncountable κ fails in Gödel's L, even for κ-Borel equivalence relations. We show here that Silver's dichotomy for κ-Borel equivalence relations in $κ^{κ}$ for uncountable regular κ is however consistent (with GCH), assuming the existence of $0^{#}$.
2
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The enriched stable core and the relative rigidity of HOD

100%
Friedman S.-D.
Fundamenta Mathematicae
|
2016
|
tom 235
|
nr 1
1-12
EN
In the author's 2012 paper, the V-definable Stable Core 𝕊 = (L[S],S) was introduced. It was shown that V is generic over 𝕊 (for 𝕊-definable dense classes), each V-definable club contains an 𝕊-definable club, and the same holds with 𝕊 replaced by (HOD,S), where HOD denotes Gödel's inner model of hereditarily ordinal-definable sets. In the present article we extend this to models of class theory by introducing the V-definable Enriched Stable Core 𝕊* = (L[S*],S*). As an application we obtain the rigidity of 𝕊* for all embeddings which are "constructible from V". Moreover, any "V-constructible" club contains an "𝕊*-constructible" club. This also applies to the model (HOD,S*), and therefore we conclude that, relative to a V-definable predicate, HOD is rigid for V-constructible embeddings.
3
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Condensation and large cardinals

64%
Friedman S.-D., Holy P.
Fundamenta Mathematicae
|
2011
|
tom 215
|
nr 2
133-166
EN
We introduce two generalized condensation principles: Local Club Condensation and Stationary Condensation. We show that while Strong Condensation (a generalized condensation principle introduced by Hugh Woodin) is inconsistent with an ω₁-Erdős cardinal, Stationary Condensation and Local Club Condensation (which should be thought of as weakenings of Strong Condensation) are both consistent with ω-superstrong cardinals.
4
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The Tree Property at ω₂ and Bounded Forcing Axioms

64%
Friedman S.-D., Torres-Pérez V.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
tom 63
|
nr 3
207-216
EN
We prove that the Tree Property at ω₂ together with BPFA is equiconsistent with the existence of a weakly compact reflecting cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a weakly compact cardinal. Similarly, we show that the Special Tree Property for ω₂ together with BPFA is equiconsistent with the existence of a reflecting Mahlo cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a Mahlo cardinal.
5
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Supercompactness and failures of GCH

64%
Friedman S.-D., Honzik R.
Fundamenta Mathematicae
|
2012
|
tom 219
|
nr 1
15-36
EN
Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver's methods show that this can be done starting with κ being λ⁺⁺-supercompact (note that Silver's result above is the special case when κ = λ). One can ask if there is an analogue of Woodin's result for λ-supercompactness. We answer this question in the following strong sense: starting with the GCH and κ being λ-supercompact and λ⁺⁺-tall, we preserve λ-supercompactness of κ and kill the GCH at λ by directly manipulating the size of $2^{λ}$ (i.e. we do not force the failure of GCH at λ as a consequence of having $2^{κ}$ large enough). The direct manipulation of $2^{λ}$, where λ can be a successor cardinal, is the first step toward understanding which Easton functions can be realized as the continuum function on regular cardinals while preserving instances of λ-supercompactness.
6
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Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH

64%
Friedman S.-D., Golshani M.
Fundamenta Mathematicae
|
2013
|
tom 223
|
nr 2
171-193
EN
Starting from large cardinals we construct a pair V₁⊆ V₂ of models of ZFC with the same cardinals and cofinalities such that GCH holds in V₁ and fails everywhere in V₂.
7
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The tree property at the double successor of a measurable cardinal κ with $2^{κ}$ large

64%
Friedman S.-D., Halilović A.
Fundamenta Mathematicae
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2013
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tom 223
|
nr 1
55-64
EN
Assuming the existence of a λ⁺-hypermeasurable cardinal κ, where λ is the first weakly compact cardinal above κ, we prove that, in some forcing extension, κ is still measurable, κ⁺⁺ has the tree property and $2^{κ} = κ⁺⁺⁺$. If the assumption is strengthened to the existence of a θ -hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ) then the proof can be generalized to get $2^{κ} = θ $.
8
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The consistency strength of the tree property at the double successor of a measurable cardina

64%
Dobrinen N., Friedman S.-D.
Fundamenta Mathematicae
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2010
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tom 208
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nr 2
123-153
EN
The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal. If $0^{#}$ exists, then we can construct an inner model in which the tree property holds at the double successor of each strongly inaccessible cardinal. We also find upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double successor of a measurable cardinal. The upper and lower bounds differ only by 1 in the Mitchell order.
9
Content available remote

Measurable cardinals and the cofinality of the symmetric group

64%
Friedman S.-D., Zdomskyy L.
Fundamenta Mathematicae
|
2010
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tom 207
|
nr 2
101-122
EN
Assuming the existence of a P₂κ-hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinal κ such that $2^{κ} = κ⁺⁺$ and the group Sym(κ) of all permutations of κ cannot be written as the union of a chain of proper subgroups of length < κ⁺⁺. The proof involves iteration of a suitably defined uncountable version of the Miller forcing poset as well as the "tuning fork" argument introduced by the first author and K. Thompson [J. Symbolic Logic 73 (2008)].
10
Content available remote

Easton functions and supercompactness

52%
Cody B., Friedman S.-D., Honzik R.
Fundamenta Mathematicae
|
2014
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tom 226
|
nr 3
279-296
EN
Suppose that κ is λ-supercompact witnessed by an elementary embedding j: V → M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) ∀α α < cf(F(α)), and (2) α < β ⇒ F(α) ≤ F(β). We address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while preserving the λ-supercompactness of κ ? We show that, assuming GCH, if F is any function as above, and in addition for some regular cardinal λ > κ there is an elementary embedding j: V → M with critical point κ such that κ is closed under F, the model M is closed under λ-sequences, H(F(λ)) ⊆ M, and for each regular cardinal γ ≤ λ one has $(|j(F)(γ)| = F(γ))^{V}$, then there is a cardinal-preserving forcing extension in which $2^{δ} = F(δ)$ for every regular cardinal δ and κ remains λ-supercompact. This answers a question of [CM14].
11
Content available remote

Δ₁-Definability of the non-stationary ideal at successor cardinals

52%
Friedman S.-D., Wu L., Zdomskyy L.
Fundamenta Mathematicae
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2015
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tom 229
|
nr 3
231-254
EN
Assuming V = L, for every successor cardinal κ we construct a GCH and cardinal preserving forcing poset ℙ ∈ L such that in $L^{ℙ}$ the ideal of all non-stationary subsets of κ is Δ₁-definable over H(κ⁺).
12
Content available remote

On Borel reducibility in generalized Baire space

52%
Friedman S.-D., Hyttinen T., Kulikov V.
Fundamenta Mathematicae
|
2015
|
tom 231
|
nr 3
285-298
EN
We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{<κ} = κ$. The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.
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