Words and Numbers

※ 원 출처는 thousandmaths (인듯 하다. Tumblr에서 글을 리블로그하는 구조를 잘 몰라서 원 출처가 맞는지 확실하지는 않다.)

I wasn’t particularly convinced by the evidence, but I eventually realized a proof that $n$ and $n+1$ share at least one letter for all $n$.

First, large numbers:

• $n \geq 1,000,000$ is clear because all such numbers have at least one -ILLION.
• (If you use the long scale, there’s no problem at the transitions between the -ILLIONs and the -ILLIARDs; they share an L, for instance.)
• Noting that this shares an N with THOUSAND, we get $n = 999,999$ which allows us to extend all the way down to $n \geq 1,000$.
• Then THOUSAND shares a D with HUNDRED so we get $n = 999$ and hence down to $n \geq 100$.

Second, negative numbers: The statement is true for all $n \leq -2$ because all negative numbers will have MINUS or NEGATIVE or ANTI. These all have Ns, so even if you switch conventions arbitrarily it still works. We check the shared letters between $-1 \leq n \leq 12$ by hand, and this immediately gives it up to $n = 18$ (because of -TEEN).

$n =-1$ — O

$n = 0$ — O

$n = 1$ — O

$n = 2$ — T

$n = 3$ — R

$n = 4$ — F

$n = 5$ — I

$n = 6$ — S

$n = 7$ — E

$n = 8$ — E

$n = 9$ — E

$n = 10$ — E

$n = 11$ — E

$n = 12$ — E

For $n = 19$, we have a shared N, and then all numbers between $20$ and $99$ have a -TY. So the only number left to check is $n = 99$, for which there is a shared N.

By the way, I don’t see any reason to stop at infinity.

In particular, this rule also extends a couple steps into the countable ordinals, even if we make the rule for limit ordinals as strong as* “every sequence converging to $\beta$ has all but perhaps finitely many elements sharing a letter with $\beta$":

• OMEGA contains an O and an A, one of which* is shared by every number greater than $1,000,000$. So that gets you $n = \omega$, and hence up to $n < \epsilon_0$.
• OMEGA and EPSILON share an O so that gets you $n = \epsilon_0$, and hence up to… somewhere. This is about where my understanding of ordinals gets fuzzy, but I think that everything is a sum of epsilon numbers up to $n < \Gamma_0$?
• Assuming that statement is right, there are two reasonable transliterations of $\Gamma_0$, but either of them work: GAMMA-NAUGHT and FEFERMAN-SCHÜTTE both have an N, common with EPSILON (and also common with VEBLEN, which is convenient)

* If you use the short scale, you can improve the rule to “every sequence converging to $\beta$ has all but perhaps finitely elements having a common letter shared with $\beta$”, since every sufficiently large finite number has an O.

But I’m not sure if you can guarantee this if you use the long scale, since perhaps by some clever alternations of -ILLIONs and -ILLIARDs you could avoid having any of the letters O, M, E, G, or A, in infinitely many of the numbers. It seems unlikely because of the increasing complexity of the prefixes, but starting at $10^{6000}$, the rules are made up and the names don’t matter. (…well actually, the names do matter, which is the problem XD).

Similarly, going substantially** beyond $\Gamma_0$ becomes difficult in that there doesn’t seem to be a standardized naming convention for these moderately-sized ordinals.

** Of course, it’s not hard to go (a long way) past $\Gamma_0$, but it’s not clear to me when we start running into possible troubles again: if it is the Ackermann ordinal or something smaller. And yes, I’m getting all of my knowledge of ordinals at this scale from Wikipedia, so that probably isn’t helping things.