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I am not remotely convinced by this argument. The first flaw I see is that the author is imprecise by commingling probabilities (0%, 100%) with absolutes (poss
by vitus 2y ago
I am not remotely convinced by this argument.
The first flaw I see is that the author is imprecise by commingling probabilities (0%, 100%) with absolutes (possible, impossible, none, never, etc).
> After all, probability-zero events do happen. Not a problem! Just pick two new real numbers! And if this fails, pick again!
Probability-zero events happen all the time. The probability of getting any specific value selected uniformly at random from the unit interval (say, 0.232829) is zero.
Probability-zero events should not be conflated with properties that exist nowhere.
> We can now state that for any such mapping, none of the three reals is in the countable set assigned to the others. And this entails that we can prove that |(ω)| > |ω2|! In other words, we can prove that there are at least TWO cardinalities in between the reals and the naturals!
That's... not how cardinalities work. Just because you have two sets with different elements does not mean they have different cardinalities. For instance, consider the set of integers {..., -1, 0, 1, 2, ...} vs the set of half-integers {..., -1/2, 1/2, 3/2, 5/2, ...}. These clearly have different elements, but you can easily construct a bijection between the two (just add 1/2 to each element in your set of half-integers), so you can demonstrate that they have the same cardinality.
> We define f(x) to be {y | y ≤ x}
Um, no. This demonstrates the existence of one such mapping. It does not demonstrate that the set of such mappings covers any substantial portion of the entire space of possible mappings.
- vitus 2y agoAlso, this entire argument seems to be founded on https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry. It is not clear that Freiling himself accepts this axiom -- "Freiling's argument is not widely accepted because of the following two problems with it (which Freiling was well aware of and discussed in his paper)."
- cubefox 2y agoIt is clear that Freiling accepts it: > Freiling claims that probabilistic intuition strongly supports this proposition while others disagree.
- baduser 2y ago> Probability-zero events happen all the time. The probability of getting any specific value selected uniformly at random from the unit interval (say, 0.232829) is zero. I would strongly challenge that claim. First, you did not choose that number uniformly at random, you chose it from at best a countably infinite subset, or more realistically, from a finite subset. And secondly, I do not think you can describe a situation where a number is actually chosen uniformly and randomly from the unity interval.
- rbanffy 2y agoBut, if the set is infinite, won’t the odds of any individual number to be selected be effectively zero?
- 998244353 2y agoIt is, and if you choose the number uniformly at random, it's not just "effectively" zero, it is precisely zero. GP's point, as I understand it, is that it is not actually possible to choose a number from [0, 1] uniformly at random in "real life". I think you could argue that, e.g. in the dartboard thought experiment, the probability of choosing individual points doesn't really matter: only probabilities of measurable subsets with positive measure matter.
- baduser 2y agoI guess, but, the set is not even countably infinite. "Selecting at random" is something that happens in the real, non-infinite world, not in the mathematically rigorous would where infinities can exist. So, no, probability-zero events do not happen in either.
- rbanffy 2y agoNot necessarily - you might just come up with a number you know is in the set, say pi/4. I know it's in the set because it satisfies the conditions that define it. Still, the odds of that particular number being picked up are zero.