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What is $\aleph$  in mathematics?

in Mathematics by Bronze Status (8,080 points) | 145 views

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Aleph is a letter from the Hebrew alphabet. Georg Cantor, a deeply religious man, introduced it to denote cardinalities of sets in his effort to provide a rigorous axiomatization of set theory.

Two of the results of Cantor's work appear paradoxical.

The first paradox has to do with Aleph-0 - the cardinality of all countably infinite sets: Although the sets N, Z and Q are strict subsets of each other, they all have the same cardinality i.e. Aleph-0.

The second paradox has to do with Aleph-1 - the cardinality of R (or just the interval [0, 1]), which is equal to the cardinality of the set of all subsets of Z i.e. 2 to the power Aleph-0.

It appears that the continuum hypothesis (CH - Hilbert 1st Problem)  that speculates on the existence of an infinite between Aleph-0 and Aleph-1, is quite independent of Axioms chosen for Axiomatisation of Set theory (e.g. Zermelo-Frankel + Axiom of choice). CH is thus an example of Godel Incompleteness Theorem: A statement that is neither provable nor disprovable.
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