A guide to “Coding the universe” by Beller, Jensen, Welch

1985 ◽  
Vol 50 (4) ◽  
pp. 1002-1019 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Affirmative Answer ◽  
Partial Ordering ◽  
Proper Class ◽  
Covering Lemma ◽  
Singular Cardinals ◽  
The Universe

In the wake of Silver's breakthrough on the Singular Cardinals Problem (Silver [74]) followed one of the landmark results in set theory, Jensen's Covering Lemma (Devlin-Jensen [74]): If 0# does not exist then for every uncountable x ⊆ ORD there exists a constructible Y ⊇ X, card(Y) = card(X). Thus it is fair to say that in the absence of large cardinals, V is “close to L”.It is natural to ask, as did Solovay, if we can fairly interpret the phrase “close to L” to mean “generic over L”. For example, if V = L[a], a ⊆ ω and if 0# does not exist then is V-generic over L for some partial ordering ∈ L? Notice that an affirmative answer implies that in the absence of 0#, no real can “code” a proper class of information.Jensen's Coding Theorem provides a negative answer to Solovay's question, in a striking way: Any class can be “coded” by a real without introducing 0#. More precisely, if A ⊆ ORD then there is a forcing definable over 〈L[A], A〉 such that ⊩ V = L[a], a ⊆ ω, A is definable from a. Moreover if 0# ∉ L[A] then ⊩ 0# does not exist. Now as any M ⊨ ZFC can be generically extended to a model of the form L[A] (without introducing 0#) we obtain: For any 〈M, A〉 ⊨ ZFC (that is, M ⊨ ZFC and M obeys Replacement for formulas mentioning A as a predicate) there is an 〈M, A〉-definable forcing such that ⊩ V = L[a], a ⊆ ω, 〈M, A〉 is definable from a. Moreover if 0# ∉ M then ⊩ 0# does not exist.

Co-stationarity of the ground model

2006 ◽  
Vol 71 (3) ◽  
pp. 1029-1043 ◽  
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.

scholarly journals Large cardinals and definable well-orders on the universe

2009 ◽  
Vol 74 (2) ◽  
pp. 641-654 ◽  
Author(s):  
Andrew D. Brooke-Taylor
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Proper Class ◽  
The Universe

AbstractWe use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

scholarly journals IDEAL PROJECTIONS AND FORCING PROJECTIONS

2014 ◽  
Vol 79 (4) ◽  
pp. 1247-1285 ◽  
Author(s):  
SEAN COX ◽  
MARTIN ZEMAN
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Normal Ideal ◽  
List Type ◽  
Type Number ◽  
Inner Model ◽  
Image Position ◽  
Open Question ◽  
Woodin Cardinal

AbstractIt is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1)If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;(2)For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

0# and some forcing principles

1986 ◽  
Vol 51 (1) ◽  
pp. 39-46 ◽  
Author(s):  
Matthew Foreman ◽  
Menachem Magidor ◽  
Saharon Shelah
Keyword(s):  
Real Number ◽  
Large Cardinals ◽  
Singular Limit ◽  
Partial Ordering ◽  
The Other ◽  
List Type ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Partial Orderings ◽  
The Universe

It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the axiom of determinacy holding in L(R). This consistency has been held to be evidence for the truth of these properties. It is with this in mind that the first author suggested the following:Maximality Principle If P is a partial ordering and G ⊆ P is a V-generic ultrafilter then eithera) there is a real number r ∈ V [G] with r ∉ V, orb) there is an ordinal α such that α is a cardinal in V but not in V[G].This maximality principle applied to garden variety partial orderings has startling results for the structure of V.For example, if for some , then P = 〈{p: p ⊆ κ, ∣p∣ < κ}, ⊆〉 neither adds a real nor collapses a cardinal. Thus from the maximality principle we can deduce that the G. C. H. fails everywhere and there are no inaccessible cardinals. (Hence this principle contradicts large cardinals.) Similarly one can show that there are no Suslin trees on any cardinal κ. These consequences help justify the title “maximality principle”.Since the maximality principle implies that the G. C. H. fails at strong singular limit cardinals it has consistency strength at least that of “many large cardinals”. (See [M].) On the other hand it is not known to be consistent, relative to any assumptions.

Singular Cardinals and the PCF Theory

1995 ◽  
Vol 1 (4) ◽  
pp. 408-424 ◽  
Author(s):  
Thomas Jech
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Natural Generalization ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Quarter Century ◽  
Pcf Theory ◽  
Cardinal Numbers ◽  
Singular Cardinals ◽  
The Rich

§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory (“pcf” stands for “possible cofinalities”). The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4. In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory. §2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P(κ), the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ.

scholarly journals Internal Consistency and the Inner Model Hypothesis

2006 ◽  
Vol 12 (4) ◽  
pp. 591-600 ◽  
Author(s):  
Sy-David Friedman
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Weak Sense ◽  
Large Cardinals ◽  
Model Method ◽  
Inner Models ◽  
Inner Model ◽  
The Relationship ◽  
The Universe ◽  
The Way

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening of Cohen's result:Theorem 1 (Easton's Theorem). There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals.

On elementary embeddings from an inner model to the universe

2001 ◽  
Vol 66 (3) ◽  
pp. 1090-1116 ◽  
Author(s):  
J. Vickers ◽  
P. D. Welch
Keyword(s):  
Measurable Cardinal ◽  
Negative Answer ◽  
Core Model ◽  
Elementary Embedding ◽  
Proper Class ◽  
The Core ◽  
Inner Model ◽  
Elementary Embeddings ◽  
The Universe ◽  
Closure Assumption

AbstractWe consider the following question of Kunen:Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M → V)imply Con(ZFC + ∃ a measurable cardinal)?We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j. M are definable.We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.

The κ-closed unbounded filter and supercpmpact cardinals

1981 ◽  
Vol 46 (1) ◽  
pp. 31-40
Author(s):  
Mitchell Spector
Keyword(s):  
Set Theory ◽  
A Priori ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Convincing Evidence ◽  
The Other ◽  
Other Hand ◽  
Time Period ◽  
The One ◽  
The Universe

The consistency of the Axiom of Determinateness (AD) poses a somewhat problematic question for set theorists. On the one hand, many mathematicians have studied AD, and none has yet derived a contradiction. Moreover, the consequences of AD which have been proven form an extensive and beautiful theory. (See [5] and [6], for example.) On the other hand, many extremely weird propositions follow from AD; these results indicate that AD is not an axiom which we can justify as intuitively true, a priori or by reason of its consequences, and we thus cannot add it to our set theory (as an accepted axiom, evidently true in the cumulative hierarchy of sets). Moreover, these results place doubt on the very consistency of AD. The failure of set theorists to show AD inconsistent over as short a time period as fifteen years can only be regarded as inconclusive, although encouraging, evidence.On the contrary, there is a great deal of rather convincing evidence that the existence of various large cardinals is not only consistent but actually true in the universe of all sets. Thus it becomes of interest to see which consequences of AD can be proven consistent relative to the consistency of ZFC + the existence of some large cardinal. Earlier theorems with this motivation are those of Bull and Kleinberg [2] and Spector ([14]; see also [12], [13]).

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§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.

Homogeneous changes in cofinalities with applications to HOD

2017 ◽  
Vol 17 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Omer Ben-Neria ◽  
Spencer Unger
Keyword(s):  
Large Cardinals ◽  
Related Question ◽  
New Technique ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
A New Technique

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.

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