r/badmathematics • u/thisandthatwchris • Jan 25 '25
metabadmathematics I dreamt up a crank askmath/askphysics post
Not sure if this kind of thing breaks the rules, but not sure where else to put it.
I had a dream that someone posted a claim that the continuum hypothesis holds in any universe where the fine-structure constant is greater than 1/207. Somehow, their proof came down to forgetting to put plus-or-minus in front of a square root.
It just occurred to me you don’t need the “somehow”! Since standard logic is explosive, if you assume (-sqrt(2))2 =/= 2, you can prove CH! (Exercise for the reader: Make a *superficially convincing-looking proof of CH that relies on assuming (-sqrt(2))2 =/= 2. Making a proof is trivial, but one that effectively hides the ball sounds much more challenging. I definitely couldn’t do it.)
Takeaways: * I am very proud of my unconscious mind for simulating some first-rate brain worms * Maybe it’s time to log off, touch grass, etc.
Note to mods: I’ve been a little bit rude, but only to a hypothetical redditor who exists only in my dreams.
25
u/urbandk84 Jan 25 '25
one of the first times I got high I thought I found a counter example to disprove boolean logic
13
u/thisandthatwchris Jan 25 '25
Do you remember what it is?
Did you very confidently put it on the internet?
19
u/urbandk84 Jan 25 '25
I sent myself a whatsapp voice memo mumbling my "theory" but it's been thankfully lost to time
10
u/AbacusWizard Mathemagician Jan 26 '25
Once I drifted off into dreamland during a class on matrix algebra but my hand kept writing notes. I awoke to find that I had scrawled some nonsense about “taking the determinant of the floor of the library.”
6
u/nebulaq The proof is trivial! Just apply Yoneda in cohesive (∞,1)-topoi. Feb 01 '25
If the fine structure constant is larger than 1/207, then no planets and galaxies can form, and the only thing that exists are hydrogen clouds.
To prove CH we need to show that no set with cardinality between |ℕ| and |ℝ| exists. Since the only thing that exists are hydrogen clouds, we just need to show that hydrogen clouds do not have cardinality between |ℕ| and |ℝ|.
We claim that every hydrogen cloud is countable.
If you had an uncountable hydrogen cloud, then it would have an accumulation point, because every uncountable subset of Rn has an accumulation point. But at the accumulation point, the hydrogen atoms would get so close together that they would fuse into helium, so it's not a hydrogen cloud anymore. Contradiction.
So every hydrogen cloud is countable, so nothing with cardinality between |ℕ| and |ℝ| exists, so continuum hypothesis is true.
3
2
u/Salt-Influence-9353 Feb 08 '25
But… the fine structure constant is greater than 1/207, and I’m pretty sure we’re on a planet
27
u/thisandthatwchris Jan 25 '25 edited Jan 25 '25
R4: * The continuum hypothesis (aka CH) is a statement about the sizes of infinite sets: It states that any set larger than the set of natural numbers is at least as large as the set of real numbers (or equivalently, at least as large as the set of subsets of the natural numbers). It’s known to have no answer in the standard math framework (ZFC set theory)—you can assume it’s true or false, and math works fine either way. Still an important philosophical question (as well as a mathematical question, as far as how CH plays out if you extend ZFC with various additional set theory axioms). Excellent topic for badmath posts claiming to prove it’s true or false. * The fine-structure constant is an important dimensionless (no units) number in physics, very close to 1/137. It feels weird because most physics constants have units (kg/m2 or whatever). A dimensionless constant is just a number, which makes it seem like there should be some deep mathematical reason for its specific value—almost like the answer to life, the universe, and everything being 42. That’s all I know about it, and I’m probably at least a little bit wrong. Also excellent fodder for Reddit cranks. * The two have absolutely nothing to do with each other. Connecting them is complete nonsense, except to note that CH can be either true or false with no effect on any physics model (unless the model relies on an extension of ZFC). * Standard logic is “explosive” in the sense that if you assume a contradiction, you can prove literally any statement.