TEDSF Interview Skills Q&A Platform
1 like 0 dislike
38 views

What is $\aleph$  in mathematics?

in College Questions by Platinum (54.4k points) | 38 views

1 Answer

1 like 0 dislike
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.
by (60 points)

Related questions

0 like 0 dislike
0 answers
asked Mar 22 in GMAT Questions by TEDSF Platinum (91.9k points) | 7 views
Welcome to TEDSF Skills Q&A, a platform, where you can ask skills questions and receive answers from other members of the community. On TEDSF the youth, students, teachers, policy makers and enthusiasts can ask and answer any questions. Get help and answers to any interview skills problem including mathematics, computer science, data science, web development, physics, chemistry, African development and more. Help is always 100% free!

3.8k questions

1.3k answers

63 comments

27k users

3,814 questions
1,313 answers
63 comments
26,964 users