Internal cohen extensions

Martin, Donald A.; Solovay, Robert M.

doi:10.1016/0003-4843(70)90009-4

Cited by 394 publications

(149 citation statements)

References 8 publications

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“…We have been unable to eliminate it entirely from the argument, but we can get by with certain weaker hypotheses. Readers familiar with Martin's axiom [6] will have no difficulty carrying out essentially the same argument (or the argument outlined in 4.1) on the basis of that assumption. The weakest hypothesis from which we have deduced the conclusion of Theorem 1 is There exist two nonprincipal ultrafilters D¡ on N such that /(/>,) =£f(D2) for all finite-to-one functions /: N -» N.…”

Section: Define G(«) = «(Next(£¿ «))mentioning

confidence: 97%

A characterization and sum decomposition for operator ideals

Blass¹,

Weiss²

1978

Trans. Amer. Math. Soc.

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Abstract.Let L(H) be the ring of bounded operators on a separable Hubert space. Assuming the continuum hypothesis, we prove that in L(H) every two-sided ideal that contains an operator of infinite rank is the sum of two smaller two-sided ideals. The proof involves a new combinatorial description of ideals of L(H). This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice.

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Section: Define G(«) = «(Next(£¿ «))mentioning

confidence: 97%

A characterization and sum decomposition for operator ideals

Blass¹,

Weiss²

1978

Trans. Amer. Math. Soc.

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“…Their original result was an iteration of the specific forcing destroying Souslin lines. However, Martin realised that their technique could be extended to prove Theorem 2.1 and introduced the Martin's Axiom (see below), which appears in Section 6 of [31] and the paper [22] by Martin and Solovay. The point is that iterated forcing is quite a complex technique and there is no reason to expect that a mathematician not working in set theory but interested in the possible independence of some concrete statement in mathematics should be learning the technique of iterated forcing, or of course that a set theorist will be able to work in any given part of mathematics to answer the question-although many examples of both the former and the latter are known in the literature.…”

Section: Iterated Forcing and Martin's Axiommentioning

confidence: 99%

Forcing Axioms, Finite Conditions and Some More

Džamonja

2013

Logic and Its Applications

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Abstract. We survey some classical and some recent results in the theory of forcing axioms, aiming to present recent breakthroughs and interest the reader in further developing the theory. The article is written for an audience of logicians and mathematicians not necessarily familiar with set theory. IntroductionWe shall work within the axioms of the Zermelo-Fraenkel set theory with Choice (ZFC). These axioms were introduced basically starting from 1908 and improving to a final version in the 1920s as an attempt to axiomatize the foundations of mathematics. There have been other such attempts at about the same time and later, but it is fair to say that for the purposes of much of modern mathematics the axioms of ZFC represent the accepted foundation (see [13] for a detailed discussion of foundational issues in set theory). Gödel's Incompleteness theorems [16] prove that for any consistent theory T which implies the Peano Axioms and whose axioms are presentable as a recursively enumerable set of sentences, so for any reasonable theory one would say, there is a sentence ϕ in the language of T such that T does not prove or disprove ϕ. In some sense the discussion of which axioms to use is made less interesting by these theorems, which can be interpreted as saying that a perfect choice of axioms does not exists. We therefore do like the most, we concentrate on the axioms that correctly model most of mathematics, and for the rest, we try to understand the limits and how we can improve them. For us ZFC is a basis for a foundation which in some circumstances can be extended to a larger set of axioms which provide an insight into various parts of mathematics. In here we concentrate on the forcing axioms (and their negations).

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“…In [2, Theorem 3.2] there is constructed, under (ii), such a decomposition of a Baire metric space of weight 2 ℵ2 , which is ℵ 3 by (i) (see [13]). …”

Section: Lemma 6 (Under A)mentioning

confidence: 99%

On subclasses of weak Asplund spaces

Kalenda

Kunen

2004

Proc. Amer. Math. Soc.

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Abstract. Assuming the consistency of the existence of a measurable cardinal, it is consistent to have two Banach spaces, X, Y , where X is a weak Asplund space such that X * (in the weak* topology) in not in Stegall's class, whereas Y * is in Stegall's class but is not weak* fragmentable.A Banach space X is weak Asplund if every convex continuous real function on X is Gâteaux differentiable at all points of a dense G δ set. This is a large class of spaces; it contains for example all Asplund spaces and all weakly compactly generated spaces. A detailed study of weak Asplund spaces and their subclasses is given in [1]. The largest known subclass of Asplund spaces with reasonable stability properties was introduced by C. Stegall [16].A topological space T belongs to Stegall's class if for any Baire space B and any upper-semicontinuous nonempty compact-valued mapping ϕ : B → T which is minimal with respect to inclusion, ϕ(b) is a singleton for all b ∈ B except for a meager set. If (X * , w * ) is in Stegall's class, then X is weak Asplund. The converse does not hold by [7].A further subclass is the class of those X such that (X * , w * ) is fragmentable. Recall that a topological space T is fragmentable if there exists a metric ρ on the set T such that every nonempty subset of T has nonempty relatively open subsets of arbitrarily small ρ-diameter. Every fragmentable topological space is easily seen to belong to Stegall's class. An example of a Banach space X such that (X * , w * ) is in Stegall's class but not fragmentable is given in [9].Both results of [9] and [7] use some additional axioms beyond ZFC. As remarked in [7], these two sets of axioms cannot hold at the same time. In the present paper, we construct a model of ZFC in which all three classes -weak Asplund spaces, spaces with dual in Stegall's class, and spaces with weak* fragmentable dual -are mutually different. The main result is contained in the following theorem.

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Internal cohen extensions

Cited by 394 publications

References 8 publications

A characterization and sum decomposition for operator ideals

A characterization and sum decomposition for operator ideals

Forcing Axioms, Finite Conditions and Some More

On subclasses of weak Asplund spaces

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