I went into this hoping for a mathematical thought experiment, but rather this is merely a historical thought experiment in the sense of "Wouldn't it be nice of mathematicians accepted CH early on?". It seems the big selling point of accepting CH is that mathematicians would be less hesitant to use nonstandard analysis.
For an actual thought experiment that rejects the continuum hypothesis, I rather enjoy the explanation found at:
That sort of argument makes me a nervous. One of my favorite mathematical quotes is a sort of related one about the Axiom of Choice, referenced and explained at https://math.stackexchange.com/a/787648: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" That sounds like the "obviously false" branch of a similar debate about the continuum hypothesis.
I don't think this is a slam dunk. For this argument to work, the dart probability must be 100% for any function. This is supposed to be clear "intuitively", and then, by constructing a counterexample using the CH, it's concluded that the CH is false.
But the space of functions from R to countable subsets of R is so vast (and so far removed from the physical world) that I don't think it's possible to have any "intuition" of what's possible in that space. And indeed, we see that there's a construction of a function f that doesn't conform to the "intuition". If there's an "intuitive" line of reasoning and a formal one, and they disagree, shouldn't we just conclude that our intuition is flawed?
> shouldn't we just conclude that our intuition is flawed?
Alternatively, we might conclude that our intuition is right and instead our definition of real numbers isn't exactly what we want for some cases/questions.
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.
Also, this entire argument seems to be founded on 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)."
> 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.
Mostly I just find these arguments to be evidence that 'measure theory is not very interesting', that is, it's concerned with proving things about mathematical objects that you won't find in reality and therefore I don't care about.
I wonder sometimes if there is a concrete version of the statement: 'there is an infinite number of interesting theorems', which would suggest that perhaps doing 'all the math' is not a good idea and we should only do the math which we find important.
(of course, others would disagree that measure theory is unimportant, anyway. Shrug.)
You need measure theory for probability, economics, QFT and Physics, etc. And who is doing "all the math"? The vast majority of resarchers who "do math" are largely in PDEs and other fields that simply use the technology of math for "things that you find in reality" like engineering problems or machine learning and so forth. And most mathmaticians would agree that it is some of the most uninteresting and ugly kind of math.
Whereas the relative minority of people who study really abstract things like say k-theory or large cardinals in set theory are largely doing it out of interest in it's intrinsic beauty. And this is especially true for idk, some esoteric subfield of tropical geometry or modal logic or something, who's relevance to "things you find in reality" are completely orthogonal as to the motivations of those people who chose to spend their lives uncovering the truths within them.
Math research isn't about blindly marching from proof to proof by mechanical deduction with no conception of the larger picture like a uniform bubble spreading outwards, it is done by small communities of scholars who hack away at a specific nexus of interesting problems and structures for their own sake.
Sometimes, like with spin bundles or lie algebras or non-abelian geometry, yeah you can apply it to "real" problems, but that's not how the theory was developed, and as a theoretical physicist I will tell you that you will find no greater blindness to the underlying structure or ugliness in the use of the technology than those people that exclusively wield the technology against "real" problems, instead of appreciating it for its own sake.
The main problem with a statement like that is that "interesting" is extremely subjective. Personally, I often find math and CS to be more interesting when it's further from reality. To each his own, I suppose.
Sorry, I tried my best. I wanted to mention the thought experiment part, since that is the most interesting bit. (But I'm not sure why it was misleading?)
It’s a thought experiment for how mathematicians could have assumed the continuum hypothesis, and how dangerously close they came to making that mistake. It’s not an argument in favor of CH.
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For an actual thought experiment that rejects the continuum hypothesis, I rather enjoy the explanation found at:
https://risingentropy.com/the-continuum-hypothesis-is-false/
But the space of functions from R to countable subsets of R is so vast (and so far removed from the physical world) that I don't think it's possible to have any "intuition" of what's possible in that space. And indeed, we see that there's a construction of a function f that doesn't conform to the "intuition". If there's an "intuitive" line of reasoning and a formal one, and they disagree, shouldn't we just conclude that our intuition is flawed?
Alternatively, we might conclude that our intuition is right and instead our definition of real numbers isn't exactly what we want for some cases/questions.
Submitters: "Please use the original title, unless it is misleading or linkbait; don't editorialize." - https://news.ycombinator.com/newsguidelines.html
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.
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.
I wonder sometimes if there is a concrete version of the statement: 'there is an infinite number of interesting theorems', which would suggest that perhaps doing 'all the math' is not a good idea and we should only do the math which we find important.
(of course, others would disagree that measure theory is unimportant, anyway. Shrug.)
Whereas the relative minority of people who study really abstract things like say k-theory or large cardinals in set theory are largely doing it out of interest in it's intrinsic beauty. And this is especially true for idk, some esoteric subfield of tropical geometry or modal logic or something, who's relevance to "things you find in reality" are completely orthogonal as to the motivations of those people who chose to spend their lives uncovering the truths within them.
Math research isn't about blindly marching from proof to proof by mechanical deduction with no conception of the larger picture like a uniform bubble spreading outwards, it is done by small communities of scholars who hack away at a specific nexus of interesting problems and structures for their own sake.
Sometimes, like with spin bundles or lie algebras or non-abelian geometry, yeah you can apply it to "real" problems, but that's not how the theory was developed, and as a theoretical physicist I will tell you that you will find no greater blindness to the underlying structure or ugliness in the use of the technology than those people that exclusively wield the technology against "real" problems, instead of appreciating it for its own sake.
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