Characterizing large cardinals through Neeman's pure side condition forcing

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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Large cardinals and gap-1 morasses

Annals of Pure and Applied Logic ◽

10.1016/j.apal.2008.10.007 ◽

2009 ◽

Vol 159 (1-2) ◽

pp. 71-99 ◽

Cited By ~ 7

Author(s):

Andrew D. Brooke-Taylor ◽

Sy-David Friedman

Keyword(s):

Large Cardinals

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Co-stationarity of the ground model

Journal of Symbolic Logic ◽

10.2178/jsl/1154698589 ◽

2006 ◽

Vol 71 (3) ◽

pp. 1029-1043 ◽

Cited By ~ 3

Author(s):

Natasha Dobrinen ◽

Sy-David Friedman

Keyword(s):

Large Cardinals ◽

Partial Ordering ◽

Proper Class ◽

Ground Model ◽

Covering Theorem ◽

Uncountable Cardinal ◽

Cohen Forcing ◽

Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

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Some results on consecutive large cardinals

Annals of Pure and Applied Logic ◽

10.1016/0168-0072(83)90051-9 ◽

1983 ◽

Vol 25 (1) ◽

pp. 1-17 ◽

Cited By ~ 11

Author(s):

Arthur W. Apter

Keyword(s):

Large Cardinals

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Large Cardinals and Topology: a Short Retrospective and Some New Results

Logic Journal of IGPL ◽

10.1093/jigpal/jzm052 ◽

2007 ◽

Vol 15 (5-6) ◽

pp. 433-443 ◽

Cited By ~ 2

Author(s):

S. G. Da Silva

Keyword(s):

Large Cardinals ◽

And Topology

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Trapping of waves by a submerged elliptical torus

Journal of Fluid Mechanics ◽

10.1017/s0022112001007327 ◽

2002 ◽

Vol 456 ◽

pp. 277-293 ◽

Cited By ~ 13

Author(s):

M. McIVER ◽

R. PORTER

Keyword(s):

Fluid Velocity ◽

Three Dimensional ◽

Mode Frequency ◽

The Body ◽

Necessary Condition ◽

Numerical Evidence ◽

Side Condition ◽

Trapped Mode ◽

Specific Geometry ◽

The Time Domain

An investigation is made into the trapping of surface gravity waves by totally submerged three-dimensional obstacles and strong numerical evidence of the existence of trapped modes is presented. The specific geometry considered is a submerged elliptical torus. The depth of submergence of the torus and the aspect ratio of its cross-section are held fixed and a search for a trapped mode is made in the parameter space formed by varying the radius of the torus and the frequency. A plane wave approximation to the location of the mode in this space is derived and an integral equation and a side condition for the exact trapped mode are obtained. Each of these conditions is satisfied on a different line in the plane and the point at which the lines cross corresponds to a trapped mode. Although it is not possible to locate this point exactly, because of numerical error, existence of the mode may be inferred with confidence as small changes in the numerical results do not alter the fact that the lines cross.If the torus makes small vertical oscillations, it is customary to try to express the fluid velocity as the gradient of the so-called heave potential, which is assumed to have the same time dependence as the body oscillations. A necessary condition for the existence of this potential at the trapped mode frequency is derived and numerical evidence is cited which shows that this condition is not satisfied for an elliptical torus. Calculations of the heave potential for such a torus are made over a range of frequencies, and it is shown that the force coefficients behave in a singular fashion in the vicinity of the trapped mode frequency. An analysis of the time domain problem for a torus which is forced to make small vertical oscillations at the trapped mode frequency shows that the potential contains a term which represents a growing oscillation.

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Homogeneous changes in cofinalities with applications to HOD

Journal of Mathematical Logic ◽

10.1142/s0219061317500076 ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750007 ◽

Cited By ~ 1

Author(s):

Omer Ben-Neria ◽

Spencer Unger

Keyword(s):

Large Cardinals ◽

An AD-like model

Journal of Symbolic Logic ◽

10.2307/2274240 ◽

1985 ◽

Vol 50 (2) ◽

pp. 531-543 ◽

Cited By ~ 3

Author(s):

Arthur W. Apter

Keyword(s):

Measurable Cardinal ◽

Large Cardinals ◽

Strong Sense ◽

Supercompact Cardinals ◽

Singular Cardinals ◽

Partial Orderings ◽

Measurable Cardinals ◽

Jonsson Cardinals

A very fruitful line of research in recent years has been the application of techniques in large cardinals and forcing to the production of models in which certain consequences of the axiom of determinateness (AD) are true or in which certain “AD-like” consequences are true. Numerous results have been published on this subject, among them the papers of Bull and Kleinberg [4], Bull [3], Woodin [15], Mitchell [11], and [1], [2].Another such model will be constructed in this paper. Specifically, the following theorem is proven.Theorem 1. Con(ZFC + There are cardinals κ < δ < λ so that κ is a supercompact limit of supercompact cardinals, λ is a measurable cardinal, and δ is λ supercompact) ⇒ Con(ZF + ℵ1 and ℵ2 are Ramsey cardinals + The ℵn for 3 ≤ n ≤ ω are singular cardinals of cofinality ω each of which carries a Rowbottom filter + ℵω + 1 is a Ramsey cardinal + ℵω + 2 is a measurable cardinal).It is well known that under AD + DC, ℵ2 and ℵ2 are measurable cardinals, the ℵn for 3 ≤ n < ω are singular Jonsson cardinals of cofinality ℵ2, ℵω is a Rowbottom cardinal, and ℵω + 1 and ℵω + 2 are measurable cardinals.The proof of the above theorem will use the existence of normal ultrafilters which satisfy a certain property (*) (to be defined later) and an automorphism argument which draws upon the techniques developed in [9], [2], and [4] but which shows in addition that certain supercompact Prikry partial orderings are in a strong sense “homogeneous”. Before beginning the proof of the theorem, however, we briefly mention some preliminaries.

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