Partition properties of M-ultrafilters and ideals

1987 ◽  
Vol 52 (4) ◽  
pp. 897-907
Author(s):  
Joji Takahashi
Keyword(s):  
Set Theory ◽  
Predicate Symbol ◽  
Infinite Cardinal ◽  
List Type ◽  
Weakly Compact ◽  
Unary Predicate ◽  
Transitive Model ◽  
Basic Set ◽  
Uncountable Cardinal ◽  
The Axiom Of Choice

As is well known, the following are equivalent for any uniform ultrafilter U on an uncountable cardinal:(i) U is selective;(ii) U → ;(iii) U → .In §1 of this paper, we consider this result in terms of M-ultrafilters (Definition 1.1), where M is a transitive model of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). We define the partition properties and for M-ultrafilters (Definition 1.3), and characterize those M-ultrafilters that possess these properties (Theorem 1.5) so that the result mentioned at the beginning is subsumed as the special case that M is V, the universe of all sets. It turns out that the two properties have to be handled separately, and that, besides selectivity, we need to formulate additional conditions (Definition 1.4). The extra conditions become superfluous when M = V because they are then trivially satisfied. One of them is nothing new; it is none other than Kunen's iterability-of-ultrapowers condition.In §2, we obtain characterizations of the partition properties I+ → and I+ → (Definition 2.3) of uniform ideals I on an infinite cardinal κ (Theorem 2.6). This is done by applying the main results of §1 to the canonical -ultrafilter in the Boolean-valued model constructed from the completion of the quotient algebra P(κ)/I. They are related to certain known characterizations of weakly compact and of Ramsey cardinals.Our basic set theory is ZFC. In §1, it has to be supplemented by a new unary predicate symbol M and new nonlogical axioms that make M look like a transitive model of ZFC.

On large cardinals and partition relations

1971 ◽  
Vol 36 (2) ◽  
pp. 305-308 ◽  
Author(s):  
E. M. Kleinberg ◽  
R. A. Shore
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Partition Relation ◽  
Weakly Compact ◽  
Original Result ◽  
Partition Relations ◽  
Uncountable Cardinal

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

Models of set theory with more real numbers than ordinals

1974 ◽  
Vol 39 (3) ◽  
pp. 579-583 ◽  
Author(s):  
Paul E. Cohen
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Infinite Cardinal ◽  
Real Numbers ◽  
Transitive Model ◽  
Lower Case ◽  
The Axiom Of Choice ◽  
Dependent Choice ◽  
Models Of Set Theory

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

On generic extensions without the axiom of choice

1983 ◽  
Vol 48 (1) ◽  
pp. 39-52 ◽  
Author(s):  
G. P. Monro
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Generic Extension ◽  
Transitive Model ◽  
The Axiom Of Choice ◽  
Generic Extensions

AbstractLet ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let M be a countable transitive model of ZF. The method of forcing extends M to another model M[G] of ZF (a “generic extension”). If the axiom of choice holds in M it also holds in M[G], that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive an application to Boolean toposes.

On partitioning the infinite subsets of large cardinals

1984 ◽  
Vol 49 (2) ◽  
pp. 539-541 ◽  
Author(s):  
R. J. Watro
Keyword(s):  
Order Type ◽  
Large Cardinal ◽  
Infinite Cardinal ◽  
Product Topology ◽  
Weakly Compact ◽  
Ramsey Theorem ◽  
Borel Sets ◽  
Analytic Sets ◽  
The Axiom Of Choice ◽  
Usual Product

Let λ be an ordinal less than or equal to an infinite cardinal κ. For S ⊂ κ, [S]λ denotes the collection of all order type λ subsets of S. A set X ⊂ [κ]λ will be called Ramsey iff there exists p ∈ [κ]κ such that either [p]λ ⊂ X or [p]λ ∩ X = ∅. The set p is called homogeneous for X.The infinite Ramsey theorem implies that all subsets of [ω]n are Ramsey for n < ω. Using the axiom of choice, one can define a non-Ramsey subset of [ω]ω. In [GP], Galvin and Prikry showed that all Borel subsets of [ω]ω are Ramsey, where one topologizes [ω]ω as a subspace of Baire space. Silver [S] proved that analytic sets are Ramsey, and observed that this is best possible in ZFC.When κ > ω, the assertion that all subsets of [κ]n are Ramsey is a large cardinal hypothesis equivalent to κ being weakly compact (and strongly inaccessible). Again, is not possible in ZFC to have all subsets of [κ]ω Ramsey. The analogy to the Galvin-Prikry theorem mentioned above was established by Kleinberg, extending work by Kleinberg and Shore in [KS]. The set [κ]ω is given a topology as a subspace of κω, which has the usual product topology, κ taken as discrete. It was shown that all open subsets of [κ]ω are Ramsey iff κ is a Ramsey cardinal (that is, κ → (κ)<ω).In this note we examine the spaces [κ]λ for κ ≥ λ ≥ ω. We show that κ Ramsey implies all open subsets of [κ]λ are Ramsey for λ < κ, and that if κ is measurable, then all open subsets of [κ]κ are Ramsey. Let us remark here that we can with the same methods prove these results with “κ-Borel” in the place of “open”, where the κ-Borel sets are the smallest collection containing the opens and closed under complementation and intersections of length less than κ. Also, although here we consider just subsets of [κ]λ, it is no more difficult to show that partitions of [κ]λ into less than κ many κ-Borel sets have, under the appropriate hypothesis, size κ homogeneous sets.

An axiomatics for nonstandard set theory, based on von Neumann–Bernays–Gödel Theory

2001 ◽  
Vol 66 (3) ◽  
pp. 1321-1341 ◽  
Author(s):  
P. V. Andreev ◽  
E. I. Gordon
Keyword(s):  
Set Theory ◽  
Nonstandard Analysis ◽  
Predicate Symbol ◽  
Unary Predicate ◽  
Von Neumann ◽  
Class Theory ◽  
Bounded Set ◽  
Nonstandard Set Theory ◽  
Alternative Set ◽  
Transfer Principles

AbstractWe present an axiomatic framework for nonstandard analysis—the Nonstandard Class Theory (NCT) which extends von Neumann–Gödel–Bernays Set Theory (NBG) by adding a unary predicate symbol St to the language of NBG (St(X) means that the class X is standard) and axioms—related to it—analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory by V. Kanovei and M. Reeken. In many aspects NCT resembles the Alternative Set Theory by P. Vopenka. For example there exist semisets (proper subclasses of sets) in NCT and it can be proved that a set has a standard finite cardinality iff it does not contain any proper subsemiset. Semisets can be considered as external classes in NCT. Thus the saturation principle can be formalized in NCT.

Inner Models and Large Cardinals

1995 ◽  
Vol 1 (4) ◽  
pp. 393-407 ◽  
Author(s):  
Ronald Jensen
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Ordinal Number ◽  
Infinite Cardinal ◽  
Recursive Definition ◽  
Von Neumann ◽  
Ordinal Numbers ◽  
Kurt Gödel ◽  
Uncountable Cardinal ◽  
Image Position

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture: The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x); Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.

Indescribable cardinals and elementary embeddings

1991 ◽  
Vol 56 (2) ◽  
pp. 439-457 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Set Theory ◽  
Predicate Symbol ◽  
Large Cardinal ◽  
Unary Predicate ◽  
Finite Sequences ◽  
Elementary Embeddings ◽  
Reflection Principles ◽  
The One ◽  
Image Position ◽  
The Universe

Indescribability is closely related to the reflection principles of Zermelo-Fränkel set theory. In this axiomatic setting the universe of all sets stratifies into a natural cumulative hierarchy (Vα: α ϵ On) such that any formula of the language for set theory that holds in the universe already holds in the restricted universe of all sets obtained by some stage.The axioms of ZF prove the existence of many ordinals α such that this reflection scheme holds in the world Vα. Hanf and Scott noticed that one arrives at a large cardinal notion if the reflecting formulas are allowed to contain second order free variables to which one assigns subsets of Vα. For a given collection Ω of formulas in the ϵ language of set theory with higher type variables and a unary predicate symbol they define an ordinal α to be Ω indescribable if for all sentences Φ in Ω and A ⊆ VαSince a sufficient coding apparatus is available, this definition is (for the classes of formulas that we are going to consider) equivalent to the one that one obtains by allowing finite sequences of relations over Vα, some of which are possibly k-ary. We will be interested mainly in certain standardized classes of formulas: Let (, respectively) denote the class of all formulas in the language introduced above whose prenex normal form has n alternating blocks of quantifiers of type m (i.e. (m + 1)th order) starting with ∃ (∀, respectively) and no quantifiers of type greater than m. In Hanf and Scott [1961] it is shown that in ZFC, indescribability is equivalent to inaccessibility and indescribability coincides with weak compactness.

Weakly compact cardinals in models of set theory

1985 ◽  
Vol 50 (2) ◽  
pp. 476-486
Author(s):  
Ali Enayat
Keyword(s):  
Set Theory ◽  
Countable Model ◽  
Weakly Compact ◽  
Uncountable Cardinal ◽  
Power Set ◽  
Central Notion ◽  
Countable Compactness ◽  
Omitting Types ◽  
Models Of Set Theory ◽  
Generic Ultrapowers

The central notion of this paper is that of a κ-elementary end extension of a model of set theory. A model is said to be a κ-elementary end extension of a model of set theory if > and κ, which is a cardinal of , is end extended in the passage from to , i.e., enlarges κ without enlarging any of its members (see §0 for more detail). This notion was implicitly introduced by Scott in [Sco] and further studied by Keisler and Morley in [KM], Hutchinson in [H] and recently by the author in [E]. It is not hard to see that if has a κ-elementary end extension then κ must be regular in . Keisler and Morley [KM] noticed that this has a converse if is countable, i.e., if κ is a regular cardinal of a countable model then has a κ-elementary end extension. Later Hutchinson [H] refined this result by constructing κ-elementary end extensions 1 and 2 of an arbitrary countable model in which κ is a regular uncountable cardinal, such that 1 adds a least new element to κ while 2 adds no least new ordinal to κ. It is a folklore fact of model theory that the Keisler-Morley result gives soft and short proofs of countable compactness and abstract completeness (i.e. recursive enumera-bility of validities) of the logic L(Q), studied extensively in Keisler's [K2]; and Hutchinson's refinement does the same for stationary logic L(aa), studied by Barwise et al. in [BKM]. The proof of Keisler-Morley and that of Hutchinson make essential use of the countability of since they both rely on the Henkin-Orey omitting types theorem. As pointed out in [E, Theorem 2.12], one can prove these theorems using “generic” ultrapowers just utilizing the assumption of countability of the -power set of κ. The following result, appearing as Theorem 2.14 in [E], links the notion of κ-elementary end extension to that of measurability of κ. The proof using (b) is due to Matti Rubin.

The Axiom of Choice Machine

2018 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Set Theory ◽  
Laws Of Nature ◽  
Axiom Of Choice ◽  
Dutch Book ◽  
The Axiom Of Choice

This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of nature) way to implement the Axiom of Choice—and, for our purposes, it is ideal if that involves infinite causal histories, so the causal finitist can reject it. Such a construction is offered. Moreover, other paradoxes involving the Axiom of Choice are given, including two Dutch Book paradoxes connected with the Banach–Tarski paradox. Again, all this is argued to provide evidence for causal finitism.

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

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