Ordinal Number Theory

The title is to be read where word concatenation is right-associative; i.e. Ordinal (Number Theory). We’re studying ordinals in my set theory class (and have been for a while because, well, that’s what set theory is), and I noticed that you could talk about ordinal divisibility. So I started to wonder what the infinite primes looked like. Let’s make a definition.

For ordinals α and β, α |L β if there exists an ordinal γ such that β = α×γ. This is to be read “α left-divides β”. Similarly, α |R β if there is a γ such that β = γ×α, and is read “α right-divides β”

So what properties does ω have? Recall that 2×ω = ω, but ω×2 > ω. This is true for all finite values of 2 :-). Therefore, we can see that ω is a right-prime (has no right divisors other than 1 and itself), but every finite ordinal is a left-factor of ω (it is about as far from being a left-prime as possible).

ω+1 is both a right-prime and a left-prime. 2×(ω+1) = 2×ω + 2×1 = ω+2, so 2 |L ω+2 and ω+1 |R ω+2, so ω+2 is not a left-prime or a right-prime. In fact, for every finite ordinal n, n |L ω+n and ω+1 |R ω+n, so ω+1 is the largest prime of the form ω+n.

This is just a teaser. I encourage people to discover more about this theory and comment here :-). I will be, supposing school doesn’t get in my way.


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