Grains of sand

[marcmagus]

It seems to me I've heard reference to something being like "the infinite grains of sand on the beach". This is, of course, wrong. So then it caused me to think of the numerous grains of sand and the numerous stars in the sky and wonder which is more numerous.

Consider: the grains of sand on all the beaches on earth are both countable and finite. Last I checked, we were uncertain whether the stars are finite. (Question for someone with a bit more math at hand: if they're infinite, how infinite are they?)

Comments

[marcmagus]

I seem to recall, although it's really been a while when I looked at it, that the set of all coordinates in two dimensions (x,y) where x and y are integers was of the order of the Reals, not the Integers. I could be misremembering.

I'm trying to think now if there can be a coordinate system which can identify any location in 3-space using only the Integers. I know you can get from the Naturals to the Integers by doubling every Integer then using the 1s digit (base 2) as a sign. It seems in my head as though you can get to 2-space by defining a spiral to be walked from your origin, so a single Integer gives you a precise coordinate. I suppose there's probably something similar you can do in 3-space to define a walk which uniquely visits every integer location, which would mean that, yes, it's countable.

(I found your argument that it must be countable because it's discrete unconvincing. This is because there's more than one direction on the outside in which to increment arbitrarily, and I was unsure this didn't bring us above Aleph-null.)

Hmmm...do massive singularities (see black hole) count as stars? If you can have a star with no volume, then we're looking at something of the size of the Reals at a minimum, because you can have arbitrarily many singularities between two singularities; at least for the moment before their mutual gravitational attraction sucks them into a single, more massive, singularity. Infinitesimals are at least as weird as infinities, especially when dealing with physical objects...

[soulchanger.livejournal.com]

Yeah, I'm pretty sure that Z^n is countable. This stuff sort of gives me a headache.

But about the singularity idea. I would think that in order for us to say that there were uncountably many singularities, it wouldn't be enough that they could be there - we'd have to prove that they are there, and that uncountably many are there. If you look at the proof that R is uncountable, you'll see what I mean.

[marcmagus]

Hmmm. I agree that I wouldn't be comfortable asserting that there are uncountably many singularities simply from what I've said. However, I'm comfortable with it as a refutation to your claim that there must be countably many stars. If singularities count, then the packing limitations no longer apply and we're potentially in R.