Mathias–Prikry and Laver–Prikry type forcing

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martin?s number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.     

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.     

Sets in Prikry and Magidor generic extensions

We continue [4] and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.     

The short extenders gap three forcing using a morass

We show how to construct Gitik??s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.     

Supercompact extender based Prikry forcing

The extender based Prikry forcing notion is being generalized to super compact extenders.     

The short extenders gap two forcing is of Prikry type

We show that Gitik’s short extender gap-2 forcing is of Prikry type.     

Changing cofinalities and the nonstationary ideal

Aκ-c.c. iteration of a Prikry type forcing notion is defined. Applications to the nonstationary ideal are given.     

Prikry on extenders,revisited

The extender based forcing of Gitik and Magidor is generalized to yield, given any extender j: V å M with critical point κ, a cardinal preserving generic extension with no new bounded subset of κ in which cf(κ) = ω and \(\kappa ^\omega = |j(\kappa )|\).Assuming a superstrong cardinal exists, the forcing notion is used to construct a model in which the added Prikry sequences are a scale in the normal Prikry sequence.In addition, several ways to produce generic filter over an iterated ultrapower are presented.     

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.

     

A new proof of a theorem of Magidor

We give a new proof using iterated Prikry forcing of Magidor's theorem that it is consistent to assume that the least strongly compact cardinal is the least supercompact cardinal. Received: 8 December 1997 / Revised version: 12 November 1998     

Restrictions on forcings that change cofinalities

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.     

Destroying stationary sets

We present a forcing poset for destroying the stationarity of certain subsets ofP _kk⁺. Using this poset along with Prikry forcing techniques we establish some consistency results concerning saturated ideals andS(k, k ⁺). This paper forms a part of the author’s Ph.D. dissertation written under the supervision of Professor Cummings at Carnegie Mellon University.     

CCC forcing and splitting reals

The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.     

Generalized Prikry forcing and iteration of generic ultrapowers

It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈M_n, j_m,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j_0,n(κ) | n ∈ ω〉 is a Prikry generic sequence over M_ω. In this paper we generalize this for normal precipitous filters. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Characterization of Generalized Příkrý Sequences

A generalization of Příkry's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkry generic sequences reminiscent of Mathias' criterion for Příkry genericity is provided, together with a maximality theorem which states that a generalized Příkry sequence almost contains every other one lying in the same extension. This forcing can be used to falsify the covering lemma for a higher core model if there is an inner model with infinitely many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ. During the research for this paper the author was supported by DFG-Project Je209/1-2.     

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     

Normality and Related Properties of Forcing Algebras

The authors present a sufficient condition for irreducibility of forcing algebras and study the (non)reducedness phenomenon. Furthermore, the authors prove a criterion for normality of forcing algebras over a polynomial base ring with coefficients in a perfect field. This gives over algebraically closed fields a geometrical normality criterion for algebraic (forcing) varieties. Besides, the authors examine in detail a specific (enlightening) example with several forcing equations. Finally, the authors compute explicitly the normalization of a particular forcing algebra by means of finding explicitly the generators of the ideal defining it as an affine ring.     

Disassociated indiscernibles

This work addresses a basic question by Kunen: how many normal measures can there be on the least measurable cardinal? Starting with a measurable cardinal κ of Mitchell order less than two () we define a Prikry type forcing which turns the number of normal measures over κ to any while making κ the first measurable.     

Chaos and the shadowing property

We present, as a simpler alternative for the results of [P. Ko?cielniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl. 310 (2005) 188-196; P. Ko?cielniak, M. Mazur, On C⁰ genericity of various shadowing properties, Discrete Contin. Dynam. Syst. 12 (2005) 523-530], an elementary proof of C⁰ genericity of the periodic shadowing property. We also characterize chaotic behavior (in the sense of being semiconjugated to a shift map) of shadowing systems.     

Study of a Generalized Nonlinear Euler-Poisson-Darboux System: Numerical and Bessel Based Solutions

In this paper a nonlinear Euler-Poisson-Darboux system is considered. In a first part, we proved the genericity of the hypergeometric functions in the development of exact solutions for such a systemin some special cases leading to Bessel type differential equations. Next, a finite difference scheme in two-dimensional case has been developed. The continuous system is transformed into an algebraic quasi linear discrete one leading to generalized Lyapunov-Sylvester operators. The discrete algebraic system is proved to be uniquely solvable, stable and convergent based on Lyapunov criterion of stability and Lax-Richtmyer equivalence theorem for the convergence. A numerical example has been provided at the end to illustrate the efficiency of the numerical scheme developed in section 3. The present method is thus proved to be more accurate than existing ones and lead to faster algorithms.