Recent content by Warp

Some possible solutions to the paradox come to mind (I have not looked any of this up online, even though I'm sure I'm not the first one to think about this exact problem): Somehow, the set of computable numbers is actually uncountable even though its definition makes it sound like it isn't...

It has to be a terminating algorithm that computes the number up to a given precision (which can be arbitrarily large). The Cantor's diagonal argument "algorithm" is exactly that: It can be used to compute the new number to an arbitrary given precision, and terminates when it has produced that...

Famously, the set of computable numbers is countable. That's pretty much a result of their definition: The decimal expansion of a computable number can be generated to any arbitrary length by a finite algorithm. And since the set of all possible finite algorithms is countable, so is the set of...

Another insight: When using rational numbers to make the Dedekind cut, there are only countably many ways to choose said rational number. When using irrational numbers to make the Dedekind cut, there are uncountably many ways to choose said irrational number. So, in a sense, when you are...

Yeah, even with the additional insight in that post #5, I'm still having a hard time understanding why there are more Dedekind cuts of the rationals using irrationals than there are using rationals, given that the cuts using irrationals are still unique and do not cause duplicates. If you have...

That makes it sound that there would be duplicates. In other words, two different irrational numbers would generate the same split of the set of rationals (and, in fact, for there to be uncountably many cuts, infinitely many irrationals would give the same split of the rationals.) However, as...

Talking a bit more with ChatGPT about this subject, I think I got a sort of understanding breakthrough, after understanding two key facts: Using two Dedekind cuts on the set of rational numbers is, essentially, selecting a subset of the rational numbers (between the two cuts). There are...

I was not talking about the set of real numbers but about the set of irrational numbers, both in the description of the problem and my sentence "this makes it sound like the set of rationals and the set of irrationals are of the same size." If the argument is "since there is no 'smallest...

I have now tried for an hour to make ChatGPT make me understand how this works, without success, so I humbly ask for a clearer explanation. The succinct way to express what I don't understand is this: A "Dedekind cut" simply splits the set of numbers into two, without dropping any values and...

I understand that, but my question was what would happen if you bombard the pure water with free protons. Not HCl, but just protons. And yes, I understand that the protons will not remain free when they hit the water, and will quickly form protonated water clusters or hydronium molecules. But...

Reading a bit about pH and what makes a substance "acidic" or "alkaline", apparently it has something to do with, essentially the ability of the substance to "donate" or "take" extra protons, or something like that. I was more particularly interested in why pure water (at 25°C) is neutral, ie...

Are you saying that if the speed of light is different in different directions, that's completely unobservable? Surely if light moved in one direction at c and in another direction at 1m per year, there would be some observable effect?

If the speed of light were different in different directions, couldn't this be observed in other ways, such as unexpected redshift/blueshift, or unexpected patterns in the cosmic microwave background radiation? (I know this isn't really related to my original question anymore, but anyway...)

Perhaps Rømer's experiment alone doesn't prove that the speed of light is the same in all directions, but if the result is the same as the two-way speed as measured in a different experiment, wouldn't this combination of two experiments prove it? Or would it merely prove that the one-way speed...

That doesn't matter. It doesn't matter where the light originates from, as that has zero effect on the measurement being made. Io could just as well be producing and emitting the light itself (eg. via some volcanic activity), and it would still give the exact same result.