Please clarify the pun in an equation with omega and infinity

Question

I received the following equation:

$\Large \sqrt[\Omega]{\omega} = \infty$

This shall be some kind of mathematical joke but my math (or physics) is not advanced enough to see it. Can someone help?

Answer 1

Thanks to Sebastian Baltes for posting an answer to another question that stuck in my mind so I could answer this one.

The "pun"

This equation appears around here in the section "Infinitesimals and Surreal Numbers" of Rudy Rucker (Sr.)'s "Infinity and the Mind: The Science and Philosophy of the Infinite".

…Conway’s new numbers are called the class of surreal numbers…He even gets a definition of the traditional symbol $\infty$, for potential infinity. $\infty$ is defined as the gap between the finitely large and the infinitely large surreal numbers, and Conway derives the weird equation $\infty=\sqrt[\Omega]{\omega}$, which almost magically ties together the potential infinity $\infty$, the simplest actual infinity $\omega$, and the Absolute Infinite $\Omega$.

The symbol $\omega$ is often used for the first infinite ordinal, or related concepts. (It's also often used for roots of unity, but that's unrelated.) The symbol $\Omega$ was used by Cantor to refer to his idea of the Absolute Infinite. And the symbol $\infty$ is a lemniscate which is used as an infinity symbol in a variety of contexts such as Calculus.

The equation is notable, and somewhat like a pun, because it appears to relate three different conceptions of the vague idea of "infinity".

The meaning

For the material on the surreal numbers (the surreals), Rucker cites, primarily, On Numbers and Games ("ONAG") by John Horton Conway. However, that book doesn't quite have the equation mentioned, though I wouldn't be surprised if some other writing of Conway's did. It does have the similar "$\infty=\omega^{1/\mathbf{On}}$" in the section "Gaps in the Number Line".

I don't know to what extent this equation can be explained to someone who hasn't already studied the surreals, but this might still be useful to those who have.

A surreal, in notation associated with Conway, is a pair of sets of surreal numbers where every number in the first/"left" set is less than every number in the "right" set. A property of the construction is that a surreal lies between the numbers of its left and right sets. A "gap", is an ordered pair of proper classes of surreals, where every surreal is in one of the two classes and every number in the left class is less than every number in the right class.

In ONAG: $\omega$ is often a surreal that corresponds well to the ordinal $\omega$: $(\{0,1,2,\ldots\},\varnothing)$. $\infty$ is "the gap between reals and positive infinite numbers". Equivalently, $\infty$ is the gap where the left class contains every surreal less than some integer and the right class contains those surreals greater than every integer. $\mathbf{On}$ is the gap where the left class is the class of all surreals, "the gap...at the end of the number line". And $\dfrac{1}{\mathbf{On}}$ is "the gap between $0$ and all positive numbers".

He is not entirely explicit around there by what is meant by exponentiation with gaps as in $\infty=\omega^{1/\mathbf{On}}$, but given the definition of $\mathbf{On}$, this is very similar to the idea behind $\infty=\omega^{1/\Omega}$, which you might be tempted to rewrite as $\sqrt[\Omega]{\omega}=\infty$.

The discussion gets confused a bit if we take more writing on the subject into account. In ONAG, $\infty$ is redefined to be a particular game (surreal without the inequality condition) "$\mathbb R\Vert\mathbb R\vert\mathbb R$. And Conway uses $\Omega$ as a generic gap variable. And Conway's notation in ONAG is not entirely consistent with more modern treatments as in Aaron N. Siegel's Combinatorial Game Theory.