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The continuum hypothesis was advanced by Georg Cantor in 1878, [1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being ...
the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four independence results are also due to Gödel/Cohen.);
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.
The continuum hypothesis and the generalized continuum hypothesis; The Suslin conjecture; The following statements (none of which have been proved false) cannot be proved in ZFC (the Zermelo–Fraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.
For example, the number 111 0 626 0 112 0 262. translates to "= ∀ + x", which is not well-formed. Because each natural number can be obtained by applying the successor operation S to 0 a finite number of times, every natural number has its own Gödel number. For example, the Gödel number corresponding to 4, SSSS0, is: 123 0 123 0 123 0 123 0 ...
The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between and , means that =. [2] The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).
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 ...
1 Examples. 2 Model-theoretic ... More generally, if is a set of formulas in the common language of and , then is ... with the continuum hypothesis is a ...