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- TL;DR Summary
- Possible values of cardinality in different models of ZFC+ ~CH
Ok, so assume we have a model for ZFC where CH does not hold. What values may ##2^{\aleph_0}## assume over said models?
A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if ##\kappa## is a cardinal of uncountable cofinality, then there is a forcing extension in which ##2^{\aleph_0} = \kappa##. However, per König's theorem, it is not consistent to assume ##2^{\aleph _{0}}## is ##\aleph _{\omega }## or ##\aleph _{\omega _{1}+\omega }## or any cardinal with cofinality ##\omega##.
The range of values for ##2^{\aleph_0}## is infinite. This means that there is no upper limit to the possible values that can be obtained by raising 2 to the power of the cardinality of the integers.
No, ##2^{\aleph_0}## cannot be a finite value. This is because the cardinality of the integers, represented by ##\aleph_0##, is an infinite value, and raising 2 to an infinite power results in an infinite value.
The range of values for ##2^{\aleph_0}## is the same as the range of values for ##\aleph_1##, ##\aleph_2##, and any other cardinality that is larger than ##\aleph_0##. This is because all of these values represent infinite sets, and raising 2 to any of these powers will result in an infinite value.
No, the range of values for ##2^{\aleph_0}## cannot be calculated exactly. This is because the cardinality of the integers is a transfinite number, meaning it is larger than any finite number and cannot be represented by a specific numerical value.
The range of values for ##2^{\aleph_0}## has significant implications in set theory and the study of infinite sets. It is also used in various mathematical proofs and has connections to other mathematical concepts such as the continuum hypothesis and the theory of infinite cardinals.