How the continuum hypothesis could have been a fundamental axiom

[FillMaths]

The pdf file is available at: https://arxiv.org/pdf/2407.02463

[drpossum]

I argue this should have been the link posted rather than to some blog that says very little other than to host the link

[Maxatar]

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:

https://risingentropy.com/the-continuum-hypothesis-is-false/

[jerf]

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.

[codeflo]

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?

[dang]

We've changed the title now. Submitted title was "A mathematical thought experiment for accepting the continuum hypothesis".

Submitters: "Please use the original title, unless it is misleading or linkbait; don't editorialize." - https://news.ycombinator.com/newsguidelines.html

[vitus]

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.

[ajkjk]

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.)

[yantrams]

Thanks for sharing this blog. Tons of interesting stuff.

[pnin]

Please read it again, carefully. Hamkin is not trying to convince anyone of accepting CH.

[drewcoo]

HN title is misleading.

[FillMaths]

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?)

[xeonmc]

This hypothesis could've been an email.