The first letter of the Hebraic alphabet, Aleph, is no exception. Aleph is a silent letter, and is composed of three other Hebraic letters, two Yuds -one of them being upside-down- and a diagonal Vav.

Aleph is made of Yud, Vav, Yud

Aleph

The top Yud symbolizes God (יהוה), the bottom Yud symbolizes the Jew (יהודי), and the middle Vav is a pipe, or a ladder, representing faith connecting them.

In other words, the very letter Aleph embodies the gateway between the material world (“the Earth”) and the spiritual world (“the Sky”). In the Jewish tradition, not only Aleph possesses divine characteristics[1], but it conveys by itself the notion of Infinity.

This symbolism hasn’t been unacknowledged in mathematics.

Natural Numbers

Giuseppe Peano was an Italian mathematician. He contributed defining the set of natural numbers in the end of the 19th Century.

How did he work that out? Let’s 0 be an element, called zero. And let S be a unary function, called successor. S takes a natural number as an argument, and returns the next successive natural number as a result.

Then let us define numbers according to the following axioms [2].

  1. Zero is a number.
  2. If a is a number, the successor of a is also a number.
  3. Zero is not the successor of a number.
  4. Two numbers of which the successors are equal are themselves equal.
  5. If a set S of numbers contains Zero and also the successor of every number in S, then every number is in S (induction axiom).

With these axioms, natural numbers are built as a series of nested successor functions.

\(1 = S(0); 2 = S(S(0)); 3 = S(S(S(0))) …​\)

We’ll name \(\mathbb{N}\) the list resulting from 0 and all its successors. \(\mathbb{N}\) is actually a set, coined set of Natural numbers. \(\mathbb{N}\) has an infinite number of elements, hence is deemed infinite set.

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no elements is the empty set, noted ∅; a set with a single element is called a singleton.

If we define a binary operator +, named addition, along the following rules:

  • \(x + 0 = x\)
  • \(x + S(y) = S(x+y)\)

it can be proved that \(\mathbb{N}\) endowed with the internal law of addition constitutes a group.

Integers, Rationals, Algebraic, Reals

In the same way, one can define more sets of numbers.

\(\mathbb{Z}\) as the set of Integers 
defined as the set of natural numbers and additive inverses.
  \( (\mathbb{Z},+,\times) \) is a ring.

\(\mathbb{Q}\) as the set of Rational numbers
defined as the set of ratios of two integers. 
\( (\mathbb{Q},+,\times) \) is a field.

\(\mathbb{A}\) as the set of real Algebraic numbers 
defined as the set of roots of polynomials with integer coefficients. 
\( (\mathbb{A},+,\times) \) is a field.

\(\mathbb{R}\) as the set of Real Numbers
 defined as the completion of \(\mathbb{Q}\) with respect to the \(|x-y|\) metric.
\( (\mathbb{R},+,\times) \) is an ordered field.

Sets of Numbers

Sets of Numbers

Numbers go up: theory of cardinals

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. The cardinal of a set S is noted |S|. In the case of a finite set, its cardinal number, or cardinality, is therefore a natural number.

\( |∅| = 0 \)

\( |\{0\}| = 1 \)

\( |\{0,1\}| = 2 \)

\( |\{0,1,2\}| = 3 \)

\( ... \)

\( |\{0,1,...,n\}| = n+1 \)

\( ... \)

Finite cardinals can be quite large. See for instance the following video on Graham’s number.


Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements.

Sets of equal cardinality

Two sets X and Y with equal cardinality

In the case of infinite sets, the behavior is more complex. The study of infinite sets has been pioneered by the German-Russian mathematician Georg Cantor.

By Cantor’s convention, the Hebraic letter Aleph is used for cardinality of infinite sets [3].

Aleph Null is defined as the cardinal of Natural numbers.

\( \Huge{|\mathbb{N}| = \aleph_0} \)

A fundamental theorem due to Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.

Indeed, it can be shown by constructing suitable bijections that:

\( |\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| = |\mathbb{A}_R| \)

Moreover, using a technique called Cantor’s diagonal argument, one can prove that:

\( |\mathbb{N}| < |\mathbb{R}| \)

The cardinal of Real numbers is named Aleph One [4].

\( {|\mathbb{R}| = \aleph_1} \)

Is there any cardinal between Aleph Null and Aleph One? This question has been coined Continuum Hypothesis and has no definitive answer. It leads in fact to a fork in mathematics, one branch where the hypothesis is true, and another where it is false.

Bulls on Parade

After Aleph Null and Aleph One, other cardinals appear, corresponding to greater and greater infinite sets.

Now, do you remember that modern Aleph is made of two Yuds and a Vav?

It took some evolution to arrive here. It has been discovered that the letter was originally shaped as an Ox head in its proto-Canaanite form.

Everything is a number, according to Pythagoras. And we saw the symbiotic relationship between letters and numbers. Could Aleph (Ox) prefigure hexadecimal notation (0x), this is something no one will ever know.

Further readings

Borges, Jorge Luis, The Aleph

Hofstadter, Douglas, Gödel Escher Bach: an eternal golden braid

Penrose, Roger, The Road to Reality

Sources

Raskin, Rabbi Aaron L., The first letter of the Hebrew alphabet

Weisstein, Eric W. "Peano's Axioms" From MathWorld -- A Wolfram Web Resource.

Wikipedia contributors. Cardinal number (Internet). Wikipedia, The Free Encyclopedia

Notes

[1] In Hebrew, the gematria of a word is the number resulting from summing the numerical values of its letters.
Aleph is made of two Yuds and a Vav. Yud is the 10th letter in the Hebrew alphabet, and Vav is the 6th letter. So the shape-value of Aleph is 2*10+6 = 26.
The gematria of God (יהוה) is 10 + 5 + 6 + 5 = 26.
Same in plain English: G = 7, O = 15, D = 4 hence GOD = 7 + 15 + 4 = 26.

[2] Here are the same five Peano axioms written using mathematical formalism:``\({0 \in \mathbb{N}}\\\forall x \in \mathbb{N}, S(x) \in \mathbb{N}\\\nexists x: S(x) = 0\\x = y \Leftrightarrow S(x) = S(y)\\\forall K \subseteq \mathbb{N}, 0 \in K \wedge \Rightarrow K = \mathbb{N}\)

[3] The greek small letter omega (ω) is used for the smallest infinite ordinal.

[4] Aleph One is the pseudonym of the author of the great Phrack article Smashing the stack for fun and profit (1996).