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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.
The continuum hypothesis is equivalent to =. The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., = for all ordinals .
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]
Actually, even this complex formula has been simplified from what the instructions given in the first paragraph would yield. But the point remains, there is a formula of set theory that is true only for the desired constructible set and that contains parameters only for ordinals.
If the continuum hypothesis holds, all real closed fields with cardinality of the continuum and having the η 1 property are order isomorphic. This unique field Ϝ can be defined by means of an ultrapower , as R N / M {\displaystyle \mathbb {R} ^{\mathbb {N} }/\mathbf {M} } , where M is a maximal ideal not leading to a field order-isomorphic to ...