We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κfor which 2^κ=κ^+, another for which 2^κ=κ^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that H_{κ^+}^V\subseteq HOD^W. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δof <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.
Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as well as giving a template for preserving Woodin cardinals through forcing. Using this template, we form an indestructibility result under certain Easton iterations.
After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.
The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof is by interpolation and uses the Mapping Reflection Principle.
Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal $κ$ becomes superdestructible---any further ${<}κ$-closed forcing which adds a subset to $κ$ will destroy the measurability, even the weak compactness, of $κ$. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.
We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $ω_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties. Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings.
We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.
Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles hold for $ω_2$. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.
A ccc-generically supercompact cardinal $κ$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $κ$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically measurable cardinals (Theorem 4.1). This is in a strong contrast to $\cal P$-generically supercompact cardinals for the class $\cal P$ of all $σ$-closed posets, which can be $\aleph_n$ for any n>1.
This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering properties are also applied to investigate models of strongly compact cardinals or of strong forcing axioms like MM or PFA.
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal kappa, for example, becomes fully indestructible by kappa-directed closed forcing; a strong cardinal kappa becomes indestructible by less-than-or-equal-kappa-strategically closed forcing; and a strongly compact cardinal kappa becomes indestructible …
We define extender sequences, generelizing measure sequences from Radin Forcing. Using the extender sequences we combine Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing. This forcing satisfies Prikry like condition, destroys no cardinals, and has a kind of properness. Depending on the large cardinal we start with, this forcing can blow the power of a cardinal together with changing its' cofinality to a prescribe value. If this prescribed value is the cardinal itself then it remains regular and even can remain measurable.
Let $D$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $D$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $D$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $D$ is subcomplete above $μ$, where $μ$ is any regular cardinal below the first limit point of $D$.
Let $\mathcal{E}$ be the $σ$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.
W.H. Woodin showed that if $κ_1 < \cdots < κ_n$ are strong cardinals then two-step ${\bfΣ}^1_{n+3}$ generic absoluteness holds after collapsing $2^{2^{κ_n}}$ to be countable. We show that this number can be reduced to $2^{κ_n}$, and to $κ_n^+$ in the case $n = 1$, but cannot be further reduced to $κ_n$.
The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.
We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness under proper forcings. In particular, said absoluteness does not imply Pi^1_1 determinacy.
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
Suppose $κ$ is a singular strong limit cardinal of countable cofinality and let $\langle κ_{n}: n<ω\rangle$ be an incrasing sequence of regular cardinals cofinal in $κ$. We show that if $cf(2^κ)= κ^+$, then forcing with the full product $\prod_{n<ω}Add(κ_n,1)$ collapses $2^κ$ into $κ^+$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length $κ^+,$ then forcing with $\prod_{n<ω}Add(κ_n,1)$ adds a generic filter for $Add(κ^+, 1)$, and indeed \[ \prod_{n<ω}Add(κ_n,1)…
We investigate the problem of when $\leqλ$--support iterations of $<λ$--complete notions of forcing preserve $λ^+$. We isolate a property -- {\em properness over diamonds} -- that implies $λ^+$ is preserved and show that this property is preserved by $λ$--support iterations. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{δ<λ^+:\cf(δ)=λ\}$ that complements a theorem of Shelah.
This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $κ$, and ordinal $α$, if $κ$ is $α$-inaccerssible, then there is a $\mathbb{P}$ forcing that $κ$ which preserves that $α$-inaccessible but destorys that $κ$ is $(α+1)$-inaccessible. I also consider Mahlo cardinals and d…
It is known that the set of possible cofinalities $\mathrm{pcf}(A)$ has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that $\mathrm{pcf}(A)$ has no good properties in the presense of a measurable cardinal, or in generic extensions by Prikry-type forcing.
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…
Answering two questions of D. Fremlin [Real-valued measurable cardinals, in Set Theory of the Reals, H. Judah ed. 1993, 151-305 ] we show the following: (1) If c is real-valued measurable then the Maharam type of (c,P(c),sigma) is 2^c. (2) It is consistent to have k real-valued measurable but for every submodel V_1 with k measurable in it there are no k reals which are random over V_1.
Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as P*Q, where P has size less than delta and Q is forced to be delta-strategically closed. In this paper, generalizing the Levy-Solovay theorem, I show that after such forcing, every embedding j:V[G]-->M[j(G)] in the extension which satisfies a mild closure condition is the lift of an embedding j:V-->M in the ground model. In particular, every ultrapower embedding in the extension lifts an embedding from the ground m…
We present a forcing for blowing up 2^lambda and making ``many positive polarized partition relations'' (in a sense made precise in (c) of our main theorem) hold in the interval [lambda, 2^lambda]. This generalizes results of [276], Section 1, and the forcing is a ``many cardinals'' version of the forcing there.
Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.
In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some "damage" to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of "nice" properties, when changing cofinalities to be uncountable.
We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. …
We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a set of ordinals in $V$ cannot be added to $L({\mathbb R})$ by small forcing. The large cardinal needed corresponds to the consistency strength of $AD^{L({\mathbb R})}$; roughly $ω$ Woodin cardinals.
We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the indestructibility properties of remarkable cardinals. We show that if $κ$ is remarkable, then there is a forcing extension in which the remarkability of $κ$ becomes indestructible by all $\ltκ$-closed $\leqκ$-distributive forcing and all two-step iterations of the form ${\rm Add}(κ,θ)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\ltκ$-closed and $\leqκ$-di…
Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_κ$ consisting of perfect trees of height $λ$ and width $κ$ where $κ$ is a singular $ω$-strong limit of cofinality $λ$. They showed that if $κ$ is singular of countable cofinality, then $\mathbb{P}_κ$ is minimal for $ω$-sequences assuming that $κ$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_κ$ is $(ω,ν)$-distributive for all $ν<κ$ given a singular $ω$-strong limit cardinal $κ$ of countable cofinality, and Dobrinen et al$.$ asked whether …
The bounded proper forcing axiom BPFA is the statement that for any family of aleph_1 many maximal antichains of a proper forcing notion, each of size aleph_1, there is a directed set meeting all these antichains. A regular cardinal kappa is called {Sigma}_1-reflecting, if for any regular cardinal chi, for all formulas phi, ``H(chi) models `phi ' '' implies ``exists delta < kappa, H(delta) models `phi ' '' We show that BPFA is equivalent to the statement that two nonisomorphic models of size aleph_1 cannot be made isomorphic by a proper forcing notion, and we show that the consistency strength…
We introduce bounded category forcing axioms for well-behaved classes $Γ$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{λ_Γ^+}$ modulo forcing in $Γ$, for some cardinal $λ_Γ$ naturally associated to $Γ$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $λ_Γ=ω$--to classes $Γ$ with $λ_Γ>ω$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions o…
$Σ^1_3$-absoluteness for ccc forcing means that for any ccc forcing $P$, ${H_{ω_1}}^V \prec_{Σ_2}{H_{ω_1}}^{V^P}$. "$ω_1$ inaccessible to reals" means that for any real $r$, ${ω_1}^{L[r]}<ω_1$. To measure the exact consistency strength of "$Σ^1_3$-absoluteness for ccc forcing and $ω_1$ is inaccessible to reals", we introduce a weak version of a weakly compact cardinal, namely, a (lightface) $Σ^1_2$-indescribable cardinal; $κ$ has this property exactly if it is inaccessible and $H_κ\prec_{Σ_2} H_{κ^+}$.
Motivated by the goal of constructing a model in which there are no $κ$-Aronszajn trees for any regular $κ>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.
We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.
We show: There are pairs of universes V_1 subseteq V_2 and there is a notion of forcing P in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P-generic filter G over V_2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.
We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong (0<n<omega+1), hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existence is equi…
In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[ω_1]^2\rightarrowω$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of posi…
We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncoun…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.
Given a Woodin cardinal $δ$, I show that if $F$ is any Easton function with $F"δ\subseteqδ$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $λ$-supercompact cardinal $κ$ such that $\GCH$…
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.
We give an application of our extender based Radin forcing to cardinal arithmetic. Using a preparation forcing and interleaving of Cohen and Levy forcings in the normal Radin sequence we get a model with a power function having a fixed finite gap on all cardinals.
