Aleph, Beth, Continuum

Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are “the same size”. For two finite sets, if there is a one-to-one correspondence — or bijection — between them, they have the same number of elements. Cantor extended this equivalence to infinite sets.

We denote the cardinality of a set {A} by {||A||}. Any two sets {A} and {B} are equinumerate if there is a bijection between them. This term is introduced because such sets may not be “the same size” in the usual sense: Galileo showed, long ago, that there is a bijection between the natural numbers and the even numbers: just map {n} to {2n}. We can write this as {||\mathbb{N}|| = ||2\mathbb{N}||}.

Cantor showed that there is an infinite hierarchy of infinities. Here is how he did it: the power set {P(A)} of any set {A} is the set whose elements are all the subsets of {A}. Cantor showed that there is no bijection between {A} and {P(A)}, so that {||A|| < ||P(A)||}.

We can continue the process of forming power sets indefinitely, to get the sequence

\displaystyle ||A|| < ||P(A)|| < ||P(P(A))|| < \cdots

an unlimited hierarchy of infinite cardinal numbers.

Cantor used the symbol {{\aleph}_0} to denote the cardinality of the natural numbers ({\aleph} is the first letter of the Hebrew alphabet). He also showed that the set of real numbers {\mathbb{R}} and the power set of {\mathbb{N}} are equinumerous:

\displaystyle ||\mathbb{R}|| = || P(\mathbb{N})|| = 2^{\aleph_0}

What is {\aleph_1}?

Is there a smallest cardinal number greater than {\aleph_0}? Cantor showed that, for any fixed universal set {U}, the set of cardinals of the subsets of {U} is well-ordered under the natural ordering. Thus, every set of cardinals has a smallest element. He denoted by {\aleph_1} the smallest uncountable cardinal, that is, the smallest cardinal greater than {\aleph_0}. Continuing, he defined {\aleph_2} to be the smallest cardinal greater than {\aleph_1}, and so on:

\displaystyle \aleph_0 < \aleph_1 < \aleph_2 < \cdots \ \ \ \ \ (1)

Cantor also defined another sequence of cardinals: using {\beth} (the second letter of the Hebrew alphabet), he defined

\displaystyle \beth_0 = ||\mathbb{N}||\,, \quad \beth_1= ||P(\mathbb{N})||\,, \quad \beth_2= ||P(P(\mathbb{N}))|| \,, \quad \cdots

All the {\beth}-numbers are distinct. We see that {\beth_0 = \aleph_0} and {\beth_1 = 2^{\aleph_0} = ||\mathbb{R}||}. There is an unlimited sequence:

\displaystyle \beth_0 < \beth_1 < \beth_2 < \cdots \ \ \ \ \ (2)

How are {\aleph_n} and {\beth_n} related?

It would be beautiful if we could show that {\aleph_n = \beth_n}. Cantor struggled valiantly to prove this, but was unsuccessful. The Continuum Hypothesis (CH) posits that there is no cardinal number between {\beth_0 = ||\mathbb{N}||} and {\beth_1 = ||P(\mathbb{N})||}. CH implies that {\beth_1} is the smallest uncountable cardinal or, symbolically

\displaystyle \beth_1 = \aleph_1

The proof of CH was the first problem on David Hilbert’s famous list of 23 problems presented in 1900. In 1938, Kurt Gödel proved that CH is consistent with the usual axioms of set theory. Twenty-five years later, Paul Cohen proved that the negation of CH is also consistent with these axioms. Thus, CH is independent of the other axioms: we are free to assume that it is true or that it is false.

The Generalized Continuum Hypothisis states that, for every ordinal number {\nu},

\displaystyle \beth_\nu = \aleph_\nu

This implies that the sequences (1) and (2) are identical. We could write it in symbolic form:

\displaystyle [ \aleph = \beth ] \equiv \mathbf{GCH}

The choice to accept or reject CH (or GCH) is reminiscent of the freedom to accept or reject the parallel postulate. This choice leads us to the riches of elliptic and hyperbolic geometry. We can view the freedom to take or leave CH in an equally positive light.

Sources

{\bullet} Bajnok, Béla, 2013: An Invitation to Abstract Mathematics. Springer, 406pp [Chapter 22]. ISBN: 978-1-4899-9560-5.

 


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