A deliberately silly example of the λ-calculus:

\[ \begin{align*} \text{Let } \Omega & = \omega \ \omega \\ \text{where } \omega & = \lambda x.x \ x \end{align*} \]

Set Theory

The cardinality of any set's power set (set of all subsets) is \(2^c\) where \(c\) is the cardinality of the given set. Every set's cardinality is strictly less than that of its power set.

\[ \begin{align*} \forall s & \left( |\mathcal{P}(s)| = 2^{|s|} \right) \\ \forall s & \left( |s| < |\mathcal{P}(s)| \right) \\ \text{where } & \mathcal{P}(s) = \text{ the power set of } s \end{align*} \]

The set of natural numbers, \(\mathbb{N}\) a.k.a. \(\omega\):

\[ \begin{align*} \mathbb{N} & = \omega = \{ x | ( x = 0 ) \lor \exists y ( ( y \in \mathbb{N}) \land ( x = y + 1 ) ) \} \\ |\mathbb{N}| & = |\omega| = \aleph_0 \end{align*} \]

A countably infinite set is one for which there is a bijective (i.e. one-to-one) mapping between its members and \(\mathbb{N}\). By definition, the cardinality of all countably infinite sets is \(\aleph_0\), i.e. equal to \( | \mathbb{N} | \), since any two sets have the same cardinality if there is a bijection between them.

Cantor's approach hinged on identifying sets that were infinite and naturally ordered but not countable. To show that the set of real numbers, \(\mathbb{R}\), is such a set, start by assuming that you can define a countable set of real numbers between any two real numbers in the continuum (points on the number line). It does not matter what rule you use to construct this list, since the crux of Cantor's argument is to show that any such rule cannot represent all the points of the real number line between the chosen end points. As for any countable set, we can look at it as a mapping from natural numbers to elements of our set. Real numbers are represented as potentially infinite sequences of digits, as in:

\[ \begin{align*} <0, \ &0.427 \dots> \\ <1, \ &0.613 \dots> \\ <2, \ &0.590 \dots> \\ \vdots & \end{align*} \]

Cantor's argument for showing that any such list cannot include all the real numbers within the selected range proceeds by constructing a new sequence of digits representing a real number that cannot already be in our list. He does so by applying a simple for adding each successive digit to the new real number based on the sequences of digits representing the real numbers that are already included. Specifically, the rule is that the \(n \text{th}\) digit in the new sequence must differ from the \(n \text{th}\) digit in the \(n \text{th}\) entry in the original list.

\[ \begin{align*} <0, \ &0.\mathbf{4}27 \dots> \\ <1, \ &0.6\mathbf{1}3 \dots> \\ <2, \ &0.59\mathbf{0} \dots> \\ \vdots & \\ \hline <r, \ &0.\mathbf{502 \dots}> \end{align*} \]

In the preceding example, the first digit in our new sequence, \(r\), is 5. The only signfigance to 5 here is that it is different from 4, the first digit in the first entry in the original list. The second digit is 0, which differs from the second digit, 1, in the second entry. The third digit is 2, which differs from the third digit, 0, in the third row. And so on. The new sequence of digits is guaranteed to differ in at least one decimal place from every sequence in the original list, so it cannot already be included. If we add \(r\) to our list, we can repeat the procedure to construct yet another new sequence, ad infinitum. This means that there are infinitely more real numbers between any two given ones than can be bijectively mapped to the natural numbers. \(|\mathbb{R}|\) is infinitely larger than \(|\mathbb{N}|\).

Cantor's demonstration that the set of real numbers, \(\mathbb{R}\) is not countable means that \( |\mathbb{R}| > |\mathbb{N}| \). In particular, \( |\mathbb{R}| = 2^{\aleph_0} \), which also happens to be the cardinality of \(\mathcal{P}(\mathbb{N})\). The Continuum Hypothesis is that \( \aleph_1 = 2^{\aleph_0} \). If true, that makes the the cardinalties of the sets of natural and real numbers the two smallest infinite quantities. But since all such hypotheses involving "completed infinities" (as mathematicians used to say) involve definition and reasoning by analogy from finite sets, the Continuum Hypothesis will remain a widely held conjecture but cannot actually be proven formally for the same reason that one must accept the axiom of choice for infinite sets — or not. (Once upon a time, the axiom of choice was considered controversial exactly because it extends a common-sense feature of finite sets to ones with infinite cardinalities. One accepts the axiom if one accepts the legitimacy of saying that what is true of finite sets must be true of infinite ones, but there was a time when the mainstream view of infinite sets among mathematicians was similar to the proverbial reaction of the farmer seeing a rhinoceros for the first time: "there ain't no such animal!" Modern mathematicians almost universally accept the axiom of choice because it is both intuitive and they have become accustomed to reasoning about infinite sets in this way without any qualms about the reality of "completed infinities.")

Generic Math

Rationale for division rules in IEEE 754:

  1. Given that

    \[ \begin{align*} & { \lim_{x \to \infty}\frac{1}{x} } = 0 \\ \therefore \ & { \lim_{x \to 0}\frac{1}{x} } = \infty \end{align*} \]

  2. and

    \[ \begin{align*} & { \frac x x } = 1 \\ \text{where } & x \neq 0 \end{align*} \]

  3. but since \(\infty \neq 0\):

    \[ \frac x {\pm 0} = \begin{cases} \mathtt{NaN} & \text{where } x = \pm 0 \\ \pm \mathtt{INFINITY} & \text{otherwise} \end{cases} \]


    • NaN is the IEEE 754 constant meaning "Not a Number"

    • INFINITY is the IEEE 754 constant meaning \(\infty\)

    • Both 0 and INFINITY are signed in IEEE 754

    • Both 0 and INFINITY follow the usual rules of sign-agreement for division in IEEE 754