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The continuum hypothesis remains an active topic of research; see Woodin [9] [10] and Peter Koellner [11] for an overview of the current research status. The continuum hypothesis and the axiom of choice were among the first genuinely mathematical statements shown to be independent of ZF set theory.
The following four independence results are also due to Gödel/Cohen.); the generalized continuum hypothesis (GCH); a related independent statement is that if a set x has fewer elements than y, then x also has fewer subsets than y. In particular, this statement fails when the cardinalities of the power sets of x and y coincide;
The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press. 1947. "What is Cantor's continuum problem?" The American Mathematical Monthly 54: 515–25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of Mathematics: Selected ...
This phenomenon has been labeled the independence of the continuum hypothesis. [30] Both the hypothesis and its negation are thought to be consistent with the axioms of ZFC. [31] Many noteworthy discoveries have preceded the establishment of the continuum hypothesis.
The second continuum hypothesis is independent of Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC): its truth is consistent with ZFC since it is true in Cohen's model of ZFC with the negation of the Continuum Hypothesis; [5] [6]: 109–110 its falsity is also consistent since it is contradicted by the Continuum Hypothesis, which ...
Suslin hypothesis; Remarks: The consistency of V=L is provable by inner models but not forcing: every model of ZF can be trimmed to become a model of ZFC + V=L. The diamond principle implies the continuum hypothesis and the negation of the Suslin hypothesis. Martin's axiom plus the negation of the continuum hypothesis implies the Suslin hypothesis.
In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory. For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967. [12]
The axiom of constructibility and the generalized continuum hypothesis each imply the axiom of choice and so are strictly stronger than it. In class theories such as Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory , there is an axiom called the axiom of global choice that is stronger than the axiom of choice for sets ...