Transfinite Moduli

Question

I'm looking at arithmetic $\mod ω$ (where $ω := |**N**| = 1+1+1+... =$ Lim($n$) : n ∈ N).

Specifically, I'm trying to show that

$... + \frac 18 + \frac 14 + \frac 12 + 1 + 2 + 4 + 8 + ... ≡ 0 \mod ω$,

and hence that $1 + 2 + 4 + 8 + ... ≡ -1 \mod ω$.

I can't find any literature on this topic but it certainly feels like something worth studying; does anyone know how I could go about investigating this? I've been thinking about this for weeks now and still only have 2 ideas:

1. The fact that N is not the power set of any set seems to suggest that ω is not a power of 2, but c (:= |R| = 2^ω) is, and (I think) c is divisible by ω.

2. The "infinite binary tree" fractal over the interval (0,1) illustrates

   • the fact that |R| = 2^ω (real numbers x : 0 < x < 1, expressed in binary)
   • the fact that $\frac 12 + \frac 14 + \frac 18 + ... = 1$ (its width)
   • and it has $1 + 2 + 4 + 8 + ... $ nodes.

This seems like a really important object for my purposes, but I can't find a way to link the number of nodes with the width.

Any help is very much appreciated, this has been driving me crazy.

EDIT: assume all multiplication is right-multiplication. I'm writing it like this for ease of reading; my apologies.

Answer 1

Arithmetic mod $\omega$ is tricky business. If $1 + 1 + 1 + \cdots = \omega$, then shouldn't $-1 + (1 + 1 + 1 + \cdots)$ be $-1$ mod $\omega$? In which case, we have $-1 \equiv -1 + 1 + 1 + 1 + \cdots = (-1 + 1) + 1 + 1 + \cdots = 0 + 1 + 1 + \cdots = 1 + 1 + 1 + \cdots \equiv 0$, so $-1 \equiv 0$ modulo $\omega$. The usual rules of modular arithmetic, therefore, do not apply mod $\omega$ - either we have to accept that $-1 \equiv 0$ mod $\omega$ or one of the above steps does not apply mod $\omega$.

The major issue here is that addition and multiplication do not behave well when applied to ordinals and cardinals. Basically every rule is broken - for example, while $2^{\omega} = \mathfrak{c}$, $3^{\omega} = \mathfrak{c}$ as well. No finite power of two can also be a power of three, except $1$ itself. And yes, $\mathfrak{c}$ is a multiple of $\omega$ - but only because $\mathfrak{c} = \mathfrak{c} \cdot \omega$, which is a situation that can't happen with finite numbers.

This means that if you want to ask this sort of question, you need to work with some definition of $\omega$ and so on that permits the operations you want to ask about. The setting I recommend is called the surreal numbers; this gives a definition of $\omega$ that allows expressions like $\omega - 1$ to make sense. Unfortunately, limits are a bit questionable in the surreal numbers; it's arguable whether $\lim_{n\to\infty}n$ would really be $\omega$.

The surreal numbers were an invention of Conway's; Googling the term "surreal numbers" should give you plenty of useful resources.