It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from [3]. If (...) A0, A1,..., An,… are sets of reals, then there are disjoint sets B0, B1,..., Bn,… such that BnAn and μ*=μ* for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. Let be a σ-centered forcing notion and An ¦n<ω subsets of P witnessing this. If P, An's and the relation of compatibility are Borel, then P adds a Cohen real. The forcing with a κ-complete ideal over a set X, ¦X¦κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ. (shrink)

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa $, the existence of a strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is a theorem of $\textsf{ZFC}$. Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring (...) $c:[\kappa ]^2 \rightarrow \theta $ is independent of $\textsf{ZFC}$. We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of $\kappa $ -Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is equivalent to a certain weak indexed square principle $\boxminus ^{\operatorname {\mathrm {ind}}}(\kappa, \theta )$. We conclude the paper with an application to the failure of the infinite productivity of $\kappa $ -stationarily layered posets, answering a question of Cox. (shrink)

If [Formula: see text] is regular and [Formula: see text], then the existence of a weakly presaturated ideal on [Formula: see text] implies [Formula: see text]. This partially answers a question of Foreman and Magidor about the approachability ideal on [Formula: see text]. As a corollary, we show that if there is a presaturated ideal [Formula: see text] on [Formula: see text] such that [Formula: see text] is semiproper, then CH holds. We also show some barriers to getting the tree (...) property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods. (shrink)

We show that it is independent whether club $\kappa $ -Miller forcing preserves $\kappa ^{++}$. We show that under $\kappa ^{ \kappa $, club $\kappa $ -Miller forcing collapses $\kappa ^{<\kappa }$ to $\kappa $. Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter $\kappa $ -Miller forcing does not have the Laver property.

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call GM+ holds. This principle implies ISP and ISP, and hence th...

We present a new method of constructing a model of \SCH+\AP.

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...) a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$. (shrink)

James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied (...) his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. (shrink)

We deal with several pcf problems: we characterize another version of exponentiation: maximal number ofk-branches in a tree withλnodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for eachλwith a pcf restriction and investigate furtherTD{f). The sections can be read independently, although there are some minor dependencies.

We consider mainly the following version of set theory: “ZF+DC and for every λ,λℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda, \lambda^{\aleph_0}}$$\end{document} is well ordered”, our thesis is that this is a reasonable set theory, e.g. on the one hand it is much weaker than full choice, and on the other hand much can be said or at least this is what the present work tries to indicate. In particular, we prove that for a sequence δ¯=⟨δs:s∈Y⟩,cf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\delta} = \langle\delta_{s}: s \in Y\rangle, {\rm cf}}$$\end{document} large enough compared to Y, we can prove the pcf theorem with minor changes.We then deduce the existence of covering numbers and define and prove existence of a class of true successor cardinals. Using this we give some diagonalization arguments on Abelian groups, chosen as a characteristic case.We end by showing that some such consequences hold even in ZF above. (shrink)

We introduce the weak partial square principles View the MathML source and View the MathML source, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial square principles fail. The main result of the paper is that □λ,κ does not imply View the MathML source.

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...) some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

We strengthen nonstructure theorems for almost free Abelian groups by studying long Ehrenfeucht–Fraı̈ssé games between a fixed group of cardinality λ and a free Abelian group. A group is called ε -game-free if the isomorphism player has a winning strategy in the game of length ε ∈ λ . We prove for a large set of successor cardinals λ = μ + the existence of nonfree -game-free groups of cardinality λ . We concentrate on successors of singular cardinals.

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it holds on a stationary (...) subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

We define , a square principle in the context of , and prove its consistency relative to ZFC by a directed-closed forcing and hence that it is consistent to have hold when κ is supercompact, whereas □κ is known to fail under this condition. The new principle is then extended to produce a principle with a non-reflection property. Another variation on is also considered, this one based on a family of club subsets of . Finally, a new square principle for (...) cardinals, denoted , is introduced. This principle is proved consistent with κ being supercompact. It is shown to yield a non-reflection result similar to that given by □κ. (shrink)

We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality λ when certain cardinal arithmetic assumptions about λ implying the failure of GCH hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to (...) non-universality, as was raised by the earlier work of Shelah. One of our examples is the theory for which non-universality results similar to the ones we obtain are already known; hence we may view our results as an abstraction of the known results from a concrete theory to a class of theories. We show that no theory with the oak property is simple. (shrink)

We strengthen the revised GCH theorem by showing, e.g., that for , for all but finitely many regular κ ω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many .We strengthen previous results on the black box and the middle diamond: previously it was established that these principles hold on for sufficiently large n; here we succeed in replacing a sufficiently large n with a sufficiently large n.The main theorem, concerning the accessibility (...) of λ on cofinality κ, Theorem 3.1, implies as a special case that for every regular λ>ω, for some κ<ω, we can find a sequence such that , , and we can fix a finite set of “exceptional” regular cardinals θ<ω so that if Aλ satisfies A<ω, there is a pair-coloring so that for every -monochromatic BA with no last element, letting δ:=supB it holds that —provided that is not one of the finitely many “exceptional” members of. (shrink)

We give some applications of mixed support forcing iterations to the topics of disjoint stationary sequences and internally approachable sets. In the first half of the paper we study the combinatorial content of the idea of a disjoint stationary sequence, including its relation to adding clubs by forcing, the approachability ideal, canonical structure, the proper forcing axiom, and properties related to internal approachability. In the second half of the paper we present some consistency results related to these ideas. We construct (...) a model in which a disjoint stationary sequence exists at the successor of an arbitrary regular uncountable cardinal. We also construct models in which the properties of being internally stationary, internally club, and internally approachable are distinct. (shrink)

We construct a model in which for all 1 ≤ n < ω, there is no stationary subset of ${\aleph_{n+1} \cap {\rm cof}(\aleph_n)}$ which carries a partial square.

It is shown that the power set of $\kappa$ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on $2^\kappa$ under Borel reducibility. Here $\kappa$ is an uncountable regular cardinal with $\kappa^{<\kappa}=\kappa$.

Under certain cardinal arithmetic assumptions, we prove that for every large enough regular λ cardinal, for many regular κ < λ, many stationary subsets of λ concentrating on cofinality κ has the “middle diamond”. In particular, we have the middle diamond on {δ < λ: cf(δ) = κ}. This is a strong negation of uniformization.

We prove that if μ is a regular cardinal and ℙ is a μ-centered forcing poset, then ℙ forces that $(I[\mu ^{ + + } ])^V $ generates I[µ⁺⁺] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences.

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof carries a partial square.

We present a question about Suslin trees and the weak square hierarchy which was contributed to the list of open problems of the BIRS workshop.

We use a $\kappa^{+}-Mahlo$ cardinal to give a forcing construction of a model in which there is no sequence $\langle A_{\beta} : \beta \textless \omega_{2} \rangle$ of sets of cardinality $\omega_{1}$ such that $\{\lambda \textless \omega_{2} : \existsc \subset \lambda & (\bigcupc = \lambda otp(c) = \omega_{1} & \forall \beta \textless \lambda (c \cap \beta \in A_{\beta}))\}$ is stationary.

We deal with several pcf problems: we characterize another version of exponentiation: maximal number of κ-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further T D (f). The sections can be read independently, although there are some minor dependencies.

We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.