More forcing notions imply diamond

We prove that the Sacks forcing collapses the continuum onto , answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses then it forces . Received July 4, 1994     

Superstrong and other large cardinals are never Laver indestructible

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σ_n-reflecting cardinals, Σ_n-correct cardinals and Σ_n-extendible cardinals (all for n ≥  3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing \({\mathbb{Q} \in V_\theta}\), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger.     

Projective absoluteness for Sacks forcing

We show that S¹₃{{\bf \Sigma}^1_3}-absoluteness for Sacks forcing is equivalent to the non-existence of a D¹₂{{\bf \Delta}^1_2} Bernstein set. We also show that Sacks forcing is the weakest forcing notion among all of the preorders that add a new real with respect to S¹₃{{\bf \Sigma}^1_3} forcing absoluteness.     

Inner models with large cardinal features usually obtained by forcing

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 κ for 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 ${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$ . 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 δ of <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.     

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω₂. 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.     

Extender-based Radin forcing

We define extender sequences, generalizing measure sequences of Radin forcing.

Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing.

We show that this forcing satisfies a Prikry-like condition, destroys no cardinals, and has a kind of properness.

Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value. It can even blow the power of a cardinal while keeping it regular or measurable.

     

Understanding preservation theorems: chapter VI of Proper and Improper Forcing,I

We present an exposition of Section VI.1 and most of Section VI.2 from Shelah’s book Proper and Improper Forcing. These sections offer proofs of the preservation under countable support iteration of proper forcing of various properties, including proofs that ${\omega^\omega}$ -bounding, the Sacks property, the Laver property, and the P-point property are preserved by countable support iteration of proper forcing.     

Iterated forcing and changing cofinalities

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some interations) are preserved, and it includes some forcing which changes the confinality of a regular cardinal >ℵ₁ to ℵ₀. So, using the right iteractions, we can iterate such forcing without collapsing ℵ₁. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.: (1) for everyS ⊑ ℵ₂,S or ℵ₂-S contains a closed copy of ω₁ (2) there is a normal precipitous filterD on (3) for every is regular inL (δ ∩A)} is statonary. The results can be improved to equi-consistency; this will be discussed in a future paper. The author thanks the United States-Israel Binational Science Foundation for supporting the research by grant 1110.     

Mathias–Prikry and Laver type forcing; summable ideals,coideals, and +-selective filters

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of \({\omega}\)-hitting and \({\omega}\)-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.     

Subcompact cardinals, squares, and stationary reflection

We generalise Jensen’s result on the incompatibility of subcompactness with □. We show that α ⁺-subcompactness of some cardinal less than or equal to α precludes ${\square _\alpha }$ , but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling’s hierarchy of weak squares, showing which cases are precluded by α ⁺-subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold.     

Cichoń’s diagram,regularity properties and {\varvec{\Delta}^1_3} sets of reals

We study regularity properties related to Cohen, random, Laver, Miller and Sacks forcing, for sets of real numbers on the \({\varvec{\Delta}^1_3}\) level of the projective hieararchy. For \({\varvec{\Delta}^1_2}\) and \({\varvec{\Sigma}^1_2}\) sets, the relationships between these properties follows the pattern of the well-known Cichoń diagram for cardinal characteristics of the continuum. It is known that assuming suitable large cardinals, the same relationships lift to higher projective levels, but the questions become more challenging without such assumptions. Consequently, all our results are proved on the basis of ZFC alone or ZFC with an inaccessible cardinal. We also prove partial results concerning \({\varvec{\Sigma}^1_3}\) and \({\varvec{\Delta}^1_4}\) sets.     

Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ?κ   not only does not collapse κ⁺${\kappa }^{+}$ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) [3], Friedman and Halilovi? (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).     

A variant of Mathias forcing that preserves {\mathsf{ACA}_0}

We present and analyze ${F_\sigma}$ -Mathias forcing, which is similar but tamer than Mathias forcing. In particular, we show that this forcing preserves certain weak subsystems of second-order arithmetic such as ${\mathsf{ACA}_0}$ and ${\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2}$ , whereas Mathias forcing does not. We also show that the needed reals for ${F_\sigma}$ -Mathias forcing (in the sense of Blass in Ann Pure Appl Logic 109(1–2):77–88, 2001) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing.     

A parallel to the null ideal for inaccessible $$\lambda $$: Part I

It is well known how 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 naturally, to call it a generalization we require it to be \(({<}\lambda )\)-complete and \(\lambda ^+\)-c.c. and more. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of \(\lambda \)-Borel sets modulo the ideal. Common wisdom have said that there is no such thing because we have no parallel of Lebesgue integral, but here surprisingly first we get a positive \(=\) existence answer for a generalization of the null ideal for a “mild” large cardinal \(\lambda \)—a weakly compact one. Second, we try to show that this together with the meagre ideal (for \(\lambda \)) behaves as in the countable case. In particular, we consider the classical Cichoń diagram, which compares several cardinal characterizations of those ideals. We shall deal with other cardinals, and with more properties of related forcing notions in subsequent papers (Shelah in The null ideal for uncountable cardinals; Iterations adding no \(\lambda \)-Cohen; Random \(\lambda \)-reals for inaccessible continued; Creature iteration for inaccesibles. Preprint; Bounding forcing with chain conditions for uncountable cardinals) and Cohen and Shelah (On a parallel of random real forcing for inaccessible cardinals. arXiv:1603.08362 [math.LO]) and a joint work with Baumhauer and Goldstern.     

Potential continuity of colorings

We say that a coloring is continuous if it is continuous with respect to some second countable topology on κ. A coloring c is potentially continuous if it is continuous in some -preserving extension of the set-theoretic universe. Given an arbitrary coloring , we define a forcing notion that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that is c.c.c. if and only if c is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding Cohen reals to any model of set theory introduces a coloring which is potentially continuous but not continuous. has no uncountable c-homogeneous subset in the Cohen extension, but such a set can be introduced by forcing. The potential continuity of c can be destroyed by some c.c.c. forcing. The research for this paper was supported by G.I.F. Research Grant No. I-802-195.6/2003. The author would like to thank Uri Abraham for very fruitful discussions on the subject of this article.     

Robinson forcing is not absolute

Robinson (or infinite model theoretic) forcing is studied in the context of set theory. The major result is that infinite forcing, genericity, and related notions are not absolute relative to ZFC. This answers a question of G. Sacks and provides a non-trivial example of a non-absolute notion of model theory. This non-absoluteness phenomenon is shown to be intrinsic to the concept of infinite forcing in the sense that any ZFC-definable set theory, relative to which forcing is absolute, has the flavor of asserting self-inconsistency. More precisely: IfT is a ZFC-definable set theory such that the existence of a standard model ofT is consistent withT, then forcing is not absolute relative toT. For example, if it is consistent that ZFC+ “there is a measureable cardinal” has a standard model then forcing is not absolute relative to ZFC+ “there is a measureable cardinal.” Some consequences: 1) The resultants for infinite forcing may not be chosen “effectively” in general. This answers a question of A. Robinson. 2) If ZFC is consistent then it is consistent that the class of constructible division rings is disjoint from the class of generic division rings. 3) If ZFC is consistent then the generics may not be axiomatized by a single sentence ofL _w/w. In Memoriam: Abraham Robinson     

Special subsets of the reals and tree forcing notions

We study relationships between classes of special subsets of the reals (e.g. meager-additive sets, -sets, -sets, -sets) and the ideals related to the forcing notions of Laver, Mathias, Miller and Silver.

     

Distributive proper forcing axiom and cardinal invariants

In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.     

The two-cardinals transfer property and resurrection of supercompactness

We show that the transfer property for singular does not imply (even) the existence of a non-reflecting stationary subset of . The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of ``resurrection of supercompactness'. Our forcing extension destroys the supercompactness of some cardinals; to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension.

     

Martin’s maximum revisited

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM⁺⁺ makes the \({\Pi_2}\)-fragment of the theory of \({H_{\aleph_2}}\) invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to \({H_{\aleph_2}}\) of Woodin’s absoluteness results for \({L(\mathbb{R})}\). In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis.