Some text about math that I like

While experts might consider this too obvious to spell out, Gödel’s and Cohen’s analyses of CH aren’t so much about infinity, as they are about our ability to reason about about infinity using finite sequences of symbols. The game is about building self-contained mathematical universes to order—universes where all the accepted axioms about infinite sets hold true, and yet that, in some cases, seem to mock what those axioms were supposed to mean, by containing vastly fewer objects than the mathematical universe was “meant” to have.

— Scott Aaronson, The Complete Idiot’s Guide to the Independence of the Continuum Hypothesis: Part 1 of <=Aleph_0.

⦗🔗 quoted article⦘  Scott Aaronson has been a math-and-computers prodigy since age 11.  In this article, he calls himself a “doofus computer scientist” who is not an actual expert on the Continuum Hypothesis.  I added the hyperlink to the quote above, and also its associated tool-tip and ❓ cursor (when using a mouse, not a touch-screen).  The repeated “about about” is his.

It is quite rare that I come across an opinion about infinity that I can actually believe in.  Yes, infinity is a game you play in imaginary universes.  ∞ is not real, despite the fact that there is a representation for it in the IEEE floating-point data-type that in FORTRAN is called “real” — this bit-pattern is not actually a number, but a representation for an error-state.

I don't know why he included the text “<=Aleph_0” in the title of his blog post, when he clearly is capable of making his computer say “≤ℵ_₀”.  Perhaps, if I read some more of his blog, there might be some old story of how he once got burned when he put fancy 𝕌𝕟𝕚𝕔𝕠𝕕𝕖 in a blog post title.  That sort of thing used to happen a lot, but blogging software is better now.  I so want to fix those low-res characters in his post title — but no.  This is the plain-ASCII artistic form that he chose.  Also, Dr. Aaronson uses single spaces after his sentential periods, but he is only 39 years old so that's probably appropriate typography for him and I shouldn't “fix” it when I quote him.

I don't think I actually have an opinion on whether there is an infinity whose size is between number-of-integers and number-of-reals.  It doesn't bother me that there seems to be a great big void between ℵ₀ and 2^ℵ₀ with no named numbers in between — this does not cause me to feel like the universe is “mocking” me, perhaps because I don't quite believe that real numbers actually exist as described.  The rational numbers can't actually have a density property because the universe is quantized; sometimes I think that means there can't be any numbers that are truly irrational, even though it makes math more internally consistent if you choose to believe in them.  But, as Professor Aaronson says, it's all a great big game and you can play if you want to.

I *do* seem to have an opinion about ZFC, which the Continuum Hypothesis is independent of (this is the subject of Dr. Aaronson's post).  Whether CH is true is independent of whether ZFC is true, so you can believe in ZFC or not — and I don't.  Even within ZFC, most of mathematics is independent of whether you believe in the C part — which I don't.  The axiom of choice is ridiculous and no one should believe in it.  How can it be that you can take a sphere, chop it up into an infinite number of pieces, and then re-assemble them into a new sphere with the same mass but twice the volume?  Maybe the problem is that you can't actually chop something up into an infinite number of pieces?  What happens when the pieces are smaller than ℓₚ?

In other news, the pandemic continues, so let's all take refuge in abstract mathematics.

Updates to recent posts

Predicting US elections: This essay began with "I remember", which was supposed to indicate that it is a reminiscence rather than a fact-piece.  Still, things seem to be going my way on Wall St.: we are less than 1% away from the target pre-election low that I gave.

Continuum Hypothesis: The inestimable Dr. J reminds me of the Banach–Tarski Paradox, which seems to say that you *don't* need to chop up a sphere into an infinite number of pieces in order to somehow reassemble it into two spheres of equal size.  However, as stated at Wikipedia, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.  It's a finite number of pieces but each piece is an infinite collection of disconnected points.  How long would it take to select the points for each piece?  Can this be done in finite time?  Does it matter that the sphere isn't actually made of an infinite number of points but rather a finite number of atoms that cannot be split without destroying them?