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Iteration of Semiproper Forcing Revisited
[article]

2014
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arXiv
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pre-print

We present a method for iterating semiproper forcing which uses side conditions and is inspired by the technique recently introduced by Neeman.

arXiv:1410.5095v1
fatcat:xltsoja7s5ewzi73d4kvkjvoju
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Non-vanishing higher derived limits
[article]

2021
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arXiv
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pre-print

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups 𝐀 indexed by elements of ω^ω. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits lim^n 𝐀 must vanish, for n>0. They also proved that under the Continuum Hypothesis lim^1 𝐀≠ 0. The question whether lim^n 𝐀 vanishes, for n>0, has attracted considerable interest from set theorists. Dow, Simon and

arXiv:2107.03787v1
fatcat:zut5agckuvb55pcnk3ngy63ciy
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... han showed that under PFA lim^1 𝐀 =0. Bergfalk show that it is consistent that lim^2𝐀 does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that lim^n 𝐀 =0, for all n. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Henson. We complete the picture by showing that, for any n>0, it is relatively consistent with ZFC that lim^n 𝐀≠ 0.##
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PFA and precipitousness of the nonstationary ideal
[article]

2015
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arXiv
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pre-print

We apply Neeman's method of forcing with side conditions to show that PFA does not imply the precipitousness of the nonstationary ideal on ω_1.

arXiv:1504.05326v1
fatcat:kr4cqftlhrcrxmgoutchpqe5pe
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Oscillations and their applications in partition calculus
[article]

2011
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arXiv
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pre-print

The following theorem is due to

arXiv:1110.0614v1
fatcat:52ld5hrypjeybkwo52ui5lfype
*Veličković*and Woodin ( [11] ). Theorem 3 ([11]). Let X, Y, Z ⊆ [N] ω be superperfect sets. Then o ′′ [X × Y × Z] ⊇ 2 ω . Proof. ...##
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Ranks of Maharam algebras
[article]

2016
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arXiv
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pre-print

Solving a well-known problem of Maharam, Talagrand [17] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This

arXiv:1608.02468v2
fatcat:aqjoyigoa5en3fh53zvcl5jo6i
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... s rise to uncountably many non isomorphic separable atomless Maharam algebras.##
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Positional strategies in long Ehrenfeucht-Fraissé games
[article]

2013
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arXiv
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pre-print

We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fra\"iss\'e-game of length \omega_1 but there is no \sigma-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.

arXiv:1308.0156v1
fatcat:iqxsarb4lfhrzfrxr7khkglgxm
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Proper forcing remastered
[article]

2013
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arXiv
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pre-print

In these notes we present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: the Friedman-Mitchell poset for adding a club in \omega_2 with finite conditions, Koszmider's forcing construction of a strong chain of length \omega_2 of functions from \omega_1 to \omega_1, and the Baumgartner-Shelah forcing construction of a thin very tall superatomic Boolean algebra.

arXiv:1110.0610v3
fatcat:agafxndjjbhxvnlw7h64urpg2i
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Forcing axioms and stationary sets

1992
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Advances in Mathematics
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*BOBAN*

*VELICKOVIC*LEMMA 2.4. Vy + SPFA. Proof Let 9 in VY be a poset which preserves stationary subsets of w1 and let (D, : a < o, ) be names for dense subsets of 9. ...

*BOBAN*VELICKOVk CLAIM. B * 9 preserves stationary subsets of ml. Let n: 9 *3 + 9 be the natural projection. Set 9*= {n-'(D):DELB}. ...

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A note on Tsirelson type ideals

1999
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Fundamenta Mathematicae
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Using Tsirelson's well-known example of a Banach space which does not contain a copy of c 0 or lp, for p ≥ 1, we construct a simple Borel ideal I T such that the Borel cardinalities of the quotient spaces P(N)/I T and P(N)/I 0 are incomparable, where I 0 is the summable ideal of all sets A ⊆ N such that n∈A 1/(n + 1) < ∞. This disproves a "trichotomy" conjecture for Borel ideals proposed by Kechris and Mazur.

doi:10.4064/fm-159-3-259-268
fatcat:p4ehbejnonc2jfsrhy4gtsbnzu
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Maharam algebras

2009
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Annals of Pure and Applied Logic
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Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam in [24] in relation to Von Neumann's problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the main open problem in this area for around 60 years. It was finally resolved by Talagrand [31] who provided the first example of a Maharam algebra which is not a measure algebra. In this paper we

doi:10.1016/j.apal.2008.04.006
fatcat:6zpp4hcw3vgsppg4k2pnazhhhm
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... ey some recent work on Maharam algebras in relation to the two conditions proposed by Von Neumann: weak distributivity and the countable chain condition. It turns out that by strengthening either one of these conditions one obtains a ZFC characterization of Maharam algebras. We also present some results on Maharam algebras as forcing notions showing that they share some of the well known properties of measure algebras. 1. B has the countable chain condition (ccc), i.e. if A ⊆ B is such that a∧b = 0, for every a, b ∈ A such that a = b, then A is at most countable. 2. B is weakly distributive, i.e. if {b n,k } n,k is a double sequence of elements of B then the following weak distributivity law holds:##
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Playful Boolean Algebras

1986
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Transactions of the American Mathematical Society
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We show that for an atomless complete Boolean algebra 8 of density < 2N°, the Banach-Mazur, the split and choose, and the Ulam game on S are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff B adds a real; Nonempty wins iff B has a (r-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáá.

doi:10.2307/2000386
fatcat:r6lgmhiqvjb2ppou3j3jwzsmke
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Guessing models and the approachability ideal
[article]

2019
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arXiv
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pre-print

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call GM^+(ω_3,ω_1) holds. This principle implies ISP(ω_2) and ISP(ω_3), and hence the tree property at ω_2 and ω_3, the Singular Cardinal Hypothesis, and the failure of the weak square principle (ω_2,λ), for all regular λ≥ω_2. In addition, it implies that the restriction of the approachability ideal I[ω_2] to the set of ordinals of cofinality ω_1 is the non stationary ideal on

arXiv:1802.10125v2
fatcat:c3nduoznjzcypc4fud5pvdzluu
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... s set. The consistency of this last statement was previously shown by Mitchell.##
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CCC forcing and splitting reals

2005
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Israel Journal of Mathematics
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Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the

doi:10.1007/bf02785365
fatcat:hqvowqwmgrgstfdf6wfb6t6rua
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... roper Forcing Axiom and is proved using the P -ideal dichotomy first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. In the process,##
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The complexity of the reals in inner models of set theory
[article]

1995
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arXiv
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pre-print

The usual definition of the set of constructible reals is Σ ^1_2. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be analytic or even Borel in a nontrivial way. A related problem was posed by K. Prikry: can there exist a nonconstructible perfect set of constructible reals? The main result of this paper is a negative answer to Friedman's question. In fact we prove that if M is an

arXiv:math/9501203v1
fatcat:e44xbwvpzvcgzlmfmytkcjckvm
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... er model of set theory and the set of reals in M is analytic then either all reals are in M or else _1^M is countable. We also extend this result to higher levels of the projective hierarchy under appropriate large cardinal assumptions. Concerning Prikry's problem we show that the answer is negative if "perfect" is replaced by "superperfect" but that it can be positive if "constructible" is replaced by "belonging to some inner model M".##
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Borel partitions of infinite subtrees of a perfect tree
[article]

1993
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arXiv
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pre-print

A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n >= 3 is false. Let us identify the reals with 2^omega ordered by the lexicographical ordering and define for distinct x,y in 2^omega, D(x,y) to be the least n such that x(n) not= y(n). Let the type of an increasing n-tuple x_0, ... x_n-1_< be the ordering <^* on 0,

arXiv:math/9301209v1
fatcat:hj76wbl7brfcdi7jz6qfq5jfeq
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... ,n-2 defined by i<^*j iff D(x_i,x_i+1)< D(x_j,x_j+1). Galvin proved that for any Borel coloring of triples of reals there is a perfect set P such that the color of any triple from P depends only on its type. Blass proved an analogous result is true for any n. As a corollary it follows that if the unordered n-tuples of reals are colored into finitely many Borel classes there is a perfect set P such that the n-tuples from P meet at most (n-1)! classes. We consider extensions of this result to partitions of infinite increasing sequences of reals. We show, that for any Borel or even analytic partition of all increasing sequences of reals there is a perfect set P such that all strongly increasing sequences from P lie in the same class.
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