Tuesday, December 27, 2011

Aleph Numbers vs. Beth Numbers

Note: Since the Google Chart API does not seem to support the Hebrew "Beth" character, I will instead use the letter "B" where that character would ordinarily appear.


The difference between Aleph Numbers and Beth Numbers is not one that I ever understood. From what I could tell,
\aleph_{i}=B_{i}
and that was that. Both represent cardinalities of infinite sets. The first such number of each system is the countable infinity, the size of the natural numbers. So what's the difference?

Well, that depends whether you subscribe to belief in the Continuum Hypothesis, and I am sincere in calling it a belief. The Continuum Hypothesis is the somewhat unresolved statement that there can be no set whose cardinality is strictly greater than the cardinality of the natural numbers and strictly less than the cardinality of the real numbers. The work of Kurt Gödel and Paul Cohen in the 1900s proved that the Continuum Hypothesis cannot be decided - that is, cannot be proven or disproven - within the usual formulation of axiomatic set theory used in mathematics, Zermelo-Fraenkel Set Theory (regardless of whether one includes the Axiom of Choice).


Now, it is a fact that the cardinality of the real numbers is equal to the cardinality of the power set of the natural numbers (or any countably infinite set of numbers):
|\mathbb{R}|=|\mathcal{P}(\mathbb{N})|
But is it true that the smallest cardinality following that of the natural numbers is equal to the cardinality of the power set of the natural numbers?

This is the difference between the Aleph Numbers and the Beth Numbers: While the Aleph Numbers range over all successive large cardinalities, the Beth Numbers range over exclusively the cardinalities of successive power sets of the natural numbers. Therefore, the Continuum Hypothesis is exactly equivalent to answering the question of whether
\aleph_{1}\stackrel{?}{=}B_{1}

Indeed, there does even exist the Generalized Continuum Hypothesis which asks whether, for any ordinal i
\aleph_{i}\stackrel{?}{=}B_{i}

That is, for any infinite cardinality, does there exist no smaller cardinality than that of its power set? Of course, since the Continuum Hypothesis itself is undecidable, its generalized form is certainly undecidable as well.

I would also like to point out that, although it has been proven that the Continuum Hypothesis is undecidable in ZFC Set Theory, there are models in which it is decidable. In particular, I recently read briefly about research by noted set theorist William Hugh Woodin which constructs a set theoretic model known as "Ultimate L" in which the Continuum Hypothesis is true. This work is certainly beyond my current level of education, but I do hope to learn enough set theory to eventually be able to understand Dr. Woodin's research.

Final note: For those less familiar with set theory and infinite cardinalities, I'd like to point you towards the much celebrated Cantor's Theorem (proved, of course, by Georg Cantor), which states that the cardinality of the power set of any set, whether the set be finite or otherwise, is strictly greater than the cardinality of the set itself. Pretty awesome stuff.