Dumb rockets

So today I learned that the entire time I've been playing Space Alert, I've been getting a rule wrong. It's something that doesn't affect all that much, I think, and we're likely going to just keep playing the same way? I dunno.

It's just funny because like -- I'm like the guy who knows all the rules and interactions in Space Alert (well, in the base game; the expansion is way too complicated for that, also I don't even have it), but, uh, this one rule I had wrong because there was never any reason to look it up!

And that is: Rockets. When Oren taught me the game, he said, the rockets are smart rockets, so if there's nothing within range for them to hit they don't explode, they just circle back and return to the bomb bay. Only if there's something to target are they used up.

Turns out that's not what the rules say! The rules say that when you shoot a rocket you do in fact use up that rocket, no matter what; so you can only shoot rockets 3 times per game, that's it.

...hm, I guess that does maybe affect that a bit, because it means you can't to the same extent spam rockets as a vacuum action. But then, 3 is still quite a bit -- many games rockets don't get used at all -- so uh actually you still likely can? Are you really going to fire more than 3 rockets all that often?

So yeah I dunno. Probably not going to change it, everyone I know uses the "smart rockets" rule -- usually because I taught them. :P I dunno where Oren got the rule from! But yeah.

Two unexpected formulae

Let's start with the less surprising one: A formula for primes (thanks to Jeff Lagarias for bringing this to my attention).

Everybody know there's no "formula for the prime numbers", right? Most mathematicians would I think view the question of a "formula for primes" as a bit silly -- the obsession with "closed-form formulas" is for high-schoolers, and what constitutes a "closed-form formula" isn't well-defined anyway, outside of very specific contexts. Most mathematical functions don't have formulas; high-schoolers always ask about a formula for this, a formula for that; you have to just tell them, look, stop worrying about a "formula" for it, it is what it's defined to be and that's it, OK? Math isn't about formulas.

Well, uh, it turns out that one of the premises above is wrong. And in fact this information isn't new -- the article I linked cites a result of Mazzanti from 2002 (!) that any Kalmár elementary function can be expressed in terms of addition, multiplication, exponentiation, monus (minus, but saturating if you try to go below 0), and floored division.

That's a huge class of functions -- implying that, contrary to the point of view expressed above, most number theoretic functions like divisor count, sum of divisors, etc, do have "formulas"! And yes, that includes the n'th prime function. The paper I linked isn't proving the existence of such a formula, which follows from the Mazzanti result; rather it's just trying to optimize it a bit.

Now all this assumes you're OK with monus and floored division. But:
1. I think very little of the power here is coming from monus; I think pretty much every instance of monus used to construct p_n (and lots of these other number-theoretic functions) could just be minus instead, I don't think they ever really perform a subtraction without knowing that the difference is nonnegative.
2. If you're not OK with monus, but you are OK with max or min, well, then monus is just max{n-m,0}
3. If you're not OK with that, well, we're allowing exponentiation, so you can write |n-m| as √((n-m)²) and then say it's (|n-m|+(n-m))/2 (this is a little questionable in my book but hey it would satisfy a high schooler :P )
4. As for floored division, unlike monus it really is quite essential to making this work -- evidently floored division is way more powerful than I ever realized! But like, would a high schooler be OK with floors in their formula? I'm pretty sure the answer is yes. :P

So, wow, that sure overturned some things I thought! But wait, here's something crazier:
"Any function I can actually write down is measurable, right?"

Like, every mathematician knows, that if you can explicitly write down a function, you don't really have to worry about whether it's measurable right? Not only do all the things one might normally do to make a function preserve measurability, but it's actually impossible to constructively prove the existence of a non-measurable set (and therefore function). So if you can write down a function explicitly, you can't prove it's non-measurable.

Somehow it never occurred to me before there's a loophole there: No, it can't be provably non-measurable... but maybe it's not provably measurable. And that's exactly what this paper claims to do -- it writes down an explicit function whose measurability is independent of ZFC. I haven't yet taken a good look at this one to understand how it does it, but I should really sit down sometime and do that.

But, even having looked inside a bit, I was still confused -- how could it be the case that it wouldn't be provably measurable, given that the definition wasn't anything too crazy, and all the common ways of making functions preserve measurability? I think the answer here is that while the infimum or supremum of a countable collection of measurable functions is again measurable, there isn't any principle that says that if f:X×Y→R is measurable, then so is x↦inf_y∈Y f(x,y). Huh! I would never have guessed that that operation need not (provably) preserve measurability. Crazy!

Sometimes, the right formula can do a lot!