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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 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 .
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.
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.
The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel set theory without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis ...
The continuum hypothesis asserts that is also the second aleph number, . [2] In other words, the continuum hypothesis states that there is no set A {\displaystyle A} whose cardinality lies strictly between ℵ 0 {\displaystyle \aleph _{0}} and c {\displaystyle {\mathfrak {c}}}
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.