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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.);
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.
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 ...
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 ...
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 effect of Yates's correction is to prevent overestimation of statistical significance for small data. This formula is chiefly used when at least one cell of the table has an expected count smaller than 5. = = The following is Yates's corrected version of Pearson's chi-squared statistics:
The model-theoretic viewpoint has been useful in set theory; for example in Kurt Gödel's work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set ...