The continuum hypothesis and the axiom of choice were among the first mathematical statements shown to be independent of ZF set theory.
42.
The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does not collapse cardinals.
43.
This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers.
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For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated ( in its own cardinality ).
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It is also not complete, as illustrated by the in ZFC + " there exists an inaccessible cardinal " theory unresolved continuum hypothesis.
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The continuum hypothesis, first proposed as a conjecture by Cantor, was listed by David Hilbert as one of his 23 problems in 1900.
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In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.
48.
This is especially the case with some of the key notions discussed by Aczel, namely the continuum hypothesis and the axiom of choice.
49.
In the multiverse view it is meaningless to ask whether the continuum hypothesis is true or false before selecting a model of set theory.
50.
A trivial consequence of the continuum hypothesis is that a complete theory with less than continuum many nonisomorphic countable models can have only countably many.
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