4 ms·

It feels really uncomfortable that, if you have infinitely many people wearing red or blue hats that they can't see, then they can all guess their own hat color

It feels really uncomfortable that, if you have infinitely many people wearing red or blue hats that they can't see, then they can all guess their own hat color with only finitely many of them being wrong. Not to mention Banach-Tarski and a hundred other strange phenomena. This all militates toward rejecting AC. But then, if we reject AC, we can have infinite sets that are incomparable (i.e., they're not isomorphic and neither is larger than the other). So pick your poison! Thinking about such things for too long makes me feel grateful that I spend most of my time in the finite world (or, let's say, the world of continuous parameters that we only ever measure to finite precision, so that the statements we care about can ultimately be phrased arithmetically). In this world, because of e.g. the Shoenfield absoluteness theorem, anything that can be proved with the help of AC can also be proved without it.

phs 8y ago
As a follow-up, what is your position on intuitionist/constructivist logic?
- ScottAaronson 8y ago
  I don't begrudge others their sincerely held faiths! But I confess that I haven't yet seen the need for nonstandard logics for anything I've personally been interested in.
OscarCunningham 8y ago
Thanks for the answer! For other people reading, here's the blue/red hats puzzle that Scott was referring to (edited from https://en.m.wikipedia.org/wiki/Hat_puzzle#Prisoners_and_hats_puzzle https://en.m.wikipedia.org/wiki/Hat_puzzle#Prisoners_and_hat...): > A countably infinite number of prisoners, each with an unknown and randomly assigned red or blue hat, line up in a single file line. Each prisoner faces away from the beginning of the line, and each prisoner can see all the hats in front of him, and none of the hats behind. Starting from the beginning of the line, each prisoner must correctly identify the color of his hat or he is killed on the spot. The prisoners have a chance to meet beforehand, but once in line no prisoner can hear what the other prisoners say. The question is, is there a way to ensure that only finitely many prisoners are killed? The solution is that it is possible, and you can look on the Wikipedia page to find out how. Even more paradoxical is that the prisoners can still save all but finitely many of themselves even if there are infinitely many possible hat colours. For example you could allow the hat colours to be specified by three real numbers giving an RGB value.
s-shellfish 8y ago
Does it make more sense to use an analogy where we have infinitely many people with a strongest 'thought' (a statement they hold as most true), they can all guess their own thought with only finitely many of them being wrong? My apologies if that sounds ridiculous, I'm honestly not that attentuated to making much sense to myself. > Thinking about such things for too long makes me feel grateful that I spend most of my time in the finite world Hmm.

phs 8y ago

As a follow-up, what is your position on intuitionist/constructivist logic?

ScottAaronson 8y ago
I don't begrudge others their sincerely held faiths! But I confess that I haven't yet seen the need for nonstandard logics for anything I've personally been interested in.

OscarCunningham 8y ago

Thanks for the answer! For other people reading, here's the blue/red hats puzzle that Scott was referring to (edited from https://en.m.wikipedia.org/wiki/Hat_puzzle#Prisoners_and_hats_puzzle https://en.m.wikipedia.org/wiki/Hat_puzzle#Prisoners_and_hat...): > A countably infinite number of prisoners, each with an unknown and randomly assigned red or blue hat, line up in a single file line. Each prisoner faces away from the beginning of the line, and each prisoner can see all the hats in front of him, and none of the hats behind. Starting from the beginning of the line, each prisoner must correctly identify the color of his hat or he is killed on the spot. The prisoners have a chance to meet beforehand, but once in line no prisoner can hear what the other prisoners say. The question is, is there a way to ensure that only finitely many prisoners are killed? The solution is that it is possible, and you can look on the Wikipedia page to find out how. Even more paradoxical is that the prisoners can still save all but finitely many of themselves even if there are infinitely many possible hat colours. For example you could allow the hat colours to be specified by three real numbers giving an RGB value.

s-shellfish 8y ago

Does it make more sense to use an analogy where we have infinitely many people with a strongest 'thought' (a statement they hold as most true), they can all guess their own thought with only finitely many of them being wrong? My apologies if that sounds ridiculous, I'm honestly not that attentuated to making much sense to myself. > Thinking about such things for too long makes me feel grateful that I spend most of my time in the finite world Hmm.