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# Are two infinities equal? Two mathematicians claim to have proven it

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From Kevin Hartnett at Quanta:

Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories.
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In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers.

Their work also finally puts to rest a problem that mathematicians had hoped would help settle the continuum hypothesis. Still, the overwhelming feeling among experts is that this apparently unresolvable proposition is false: While infinity is strange in many ways, it would be almost too strange if there weren’t many more sizes of it than the ones we’ve already found. More.

It sounds like a bit of sleight-of-math, maybe hype, but what do readers think?

The viability of an infinite past

and

Can the universe be infinite in the past?

LocalMinimum, To build on your post, it also might be useful to distinguish between several different questions relating to "infinity" which one might ask: 1) Could we dispense with some uses of "infinity", i.e., ∞, in areas such as calculus? The answer to this is "yes" (as you point out, I believe). For example, in the statement that the limit of f(x) as x → ∞ equals ∞. 2) Do infinite sets (and infinite cardinal numbers) exist in whatever mathematical system you choose to work in? I suppose this depends on one's choice of system. 3) If infinite cardinals do exist in some particular system, does more than one exist (so that there are different degrees of infinity)? 4) If there are multiple infinite cardinals, is this a useful fact in some sense? 5) We could also ask whether it's possible for infinite collections of things to "actually" exist. daveS
LocalMinimum --If, however, you are taking issue with people making absurd metaphysical arguments I'm guessing your referring to my observation that infinity and eternity have a relationship. Why does that bother you? tribune7
LocalMinimum, I'm totally cool with infinity as a "useful" and "important" tool of mathematical analysis, i.e. a mental exercise.
If, however, you are taking issue with people making absurd metaphysical arguments while ducking behind the knotty, generally ambiguous notion of infinity, or demanding infinities where their existence can’t be justified to explain things in ways that explain nothing, then I’m with you.
Yes. You stated the position better than I did. Andrew asauber
You guys keep sayin stuff like this. I guess you are asking me to take your word for it, because I don’t see any evidence it’s true. And I guess that’s because the realm we’re discussing is all in your heads. I’m all ears and eyes if you have anything further to say.
Integrals are the limits of summations of infinite terms. Derivatives are slopes with run shrunk to practically but not technically zero. It's basic Calculus. Higher powered physics classes are constantly referencing infinity; because, well, you're often building integrals and using them to make areas and volumes out of derivatives, and integrals are the limits of summations of infinite terms and derivates are slopes with run shrunk to practically but not technically zero. I mean, in real analysis, you're not even worried about infinity, you just readily reference and work with it. If it's not actually useful, it's bizarrely pervasive. Just, "Oh, it's infinity again. Well, here's what I can and can't do with it. Oh, that's how this works. Cool." You can say it's not a thing, but, to be consistent, you have to argue that ideas aren't things. You can say it's not useful, but it finds a lot of use. If you can find a way to teach real analysis without infinity, and not some "totally-not-infinity" Ersatz that doesn't reduce to (infinity + some nonsense), that would be very cool. Darboux sums without reference to infinity. Deltas that map to sufficiently rather than arbitrarily small epsilons. I don't know, it's your job if you want it. In the mean time I'll just keep using infinity and take it easy.
I think I’ve gotten over my hump about this. In math, Infinity is a way of thinking about numbers. That’s all it is. People just need to stop the bad habit of confusing a way of thinking about numbers with trying to use an absurdity called Infinity to explain something in real life.
It depends on what you mean by this. Calculus is the workhorse of mathematical physics/engineering, and its fundamental techniques rest upon direct application of the notion of infinity. It's used to explain things in real life left, right, and center. At its most basic it's pretty much adding up infinite sets of infinitesimal quantities. The formulas are derived by pushing things out to infinity/figuring out what happens after you do something an infinite number of times. Infinity is just there, and those very practically applicable descriptions of real things rest on it. If, however, you are taking issue with people making absurd metaphysical arguments while ducking behind the knotty, generally ambiguous notion of infinity, or demanding infinities where their existence can't be justified to explain things in ways that explain nothing, then I'm with you. That's just abuse of a mutual ignorance, and it's vulgar and crooked. LocalMinimum
--I can try to understand them,-- I think the problem is that they worship science which is like worshipping a screwdriver. The screwdriver is useful but if you put it on altar, bow before it and say it's the answer to everything, not only will it not be used but the worshipping will prevent other problems from being solved. tribune7
tribune7, Eternity will be the state in which we ultimately reside. We are not there now. God is the only guy who could make such a state possible. We are way beyond the realms of science and practical application with this talk. It's amazing (but not surprising) to me that people continue to try and square this circle. I cannot relate to the reasons why. I can try to understand them, but all I get are claims of 'useful' and 'important' and nothing after. Andrew asauber
Andrew, actually I think it applies more to real life than most math. It's basically the same as eternity. If you believe you have a soul that will exist forever/infinite/eternity you will make real life decisions far different than if you think you only are cognizant for a few minuscule moments, and they are all that matters. tribune7
I think I've gotten over my hump about this. In math, Infinity is a way of thinking about numbers. That's all it is. People just need to stop the bad habit of confusing a way of thinking about numbers with trying to use an absurdity called Infinity to explain something in real life. Andrew asauber
But it’s been and remains a very useful tool
You guys keep sayin stuff like this. I guess you are asking me to take your word for it, because I don't see any evidence it's true. And I guess that's because the realm we're discussing is all in your heads. I'm all ears and eyes if you have anything further to say. Andrew asauber
Whatever. Your confidence is misguided. First you have to find someone who cares enough to even look for a use. No one has yet. Finding a use for number theory was easy enough. But that is something else as the infinite cardinality thing is but a minor aspect of set theory. Number theory is screaming "use me, that is what I am here for". ET
ET @ 11:
Cantor’s work is over 100 years old and no one has found a use for it yet. I wouldn’t hold your breath.
You say that as if 100 years is a long time. Historically speaking, it's nothing. A good 1500+ years were spent trying to derive Euclid's Fourth Postulate from the first three. They didn't stop when a minority of great minds explored non-Euclidean geometries, which they did in secret because it was "whacko" and unacceptable to established philosophies. It wasn't until the last century that we found a practical use for such things (General Relativity). Calculus is itself a generalization/systematization of methodologies that are known as far back as around 2500 years ago. Pi was originally calculated by a method of integration. And number theory itself, G.H. Hardy's "Queen of Mathematics", whose hands remained unsullied by common toil until recently, is probably as old as history. Properly picture yourself as an ant, and those 100 years as the year of your life. The big picture is simply beyond our view. But we can contribute handy tools and cool toys for others beyond our own short lives to discover uses for. LocalMinimum
asauber @ 19: Infinity may be an illusion. Given a Planck time step and a Planck length you can argue the universe is discrete at some resolution. But infinity is used all over the place in math as basic and necessary to modern society as Calculus. Maybe it's just a clever way of figuring things that makes no sense outside of the process of figuring. Maybe there's some way to do it that "makes more sense". But it's been and remains a very useful tool, underlying the math that underlies all physics and engineering; and in that it's real enough. LocalMinimum
so understanding it is important to make sure the rest of mathematics makes sense
I don't think mathematics needs infinity to function correctly. Andrew asauber
Umm, there isn't any "getting to infinity" ET
Ah, I see what you mean - getting to infinity is a journey. That's not always how it's used, though. Bob O'H
Bob O'H- There isn't an end. Like the proverbial energizer bunny it keeps going and going and going. Oleg Tchernyshyov is a better person to talk to about this topic (infinity being a journey) than I. But I don't know where he hangs out these days. Try Johns Hopkins ET
asauber - infinity is certainly abstract (just like information!). But that doesn't mean it isn't useful or important. It crops up in mathematics all over the place (I even occasionally use it in statistics), so understanding it is important to make sure the rest of mathematics makes sense. ET - I'm afraid I don't even know what that means. In what sense is infinity a journey? Bob O'H
Bob O'H:
If we want to understand numbers, then the issues of the sizes of infinity are important.
Boloney. If you want to understand infinity you first must understand that infinity is a journey. ET
then the issues of the sizes of infinity are important
Bob O'H, But Bob, the things you are defending are entirely imaginary. The importance is imaginary. I hope your brain is not too far out in infinity to understand this. Andrew asauber
asauber - it's not mumbo jumbo. If we want to understand numbers, then the issues of the sizes of infinity are important. And, as LocalMinimum has pointed out, there can be important uses that we only find later. Bob O'H
LocalMinimum- Cantor's work is over 100 years old and no one has found a use for it yet. I wouldn't hold your breath. ET
ET @ 8: Funny thing about "useless" math; it often finds uses down the road. G.H. Hardy reveled in the uselessness of number theory. Now it has all sorts of vital uses in software engineering and cryptography. A great many very useful tools in our mathematical toolbox started out as curiosities. The thing about mathematical discoveries is, that you're actually discovering some symmetry, some logical necessity. It's a tool that has a use, perhaps unknowable in the present; even if that use is to show what is useless (e.g. trying to figure out a fractional form for Pi). If nothing else, it is a sublime poetry, wrought with logic and rigor, representing toil and diligence. It can only be a stunt if the proof is wrong and the authors knew/should've known it. That happens, too, though. Like that guy who was trying to sell his quotient to divisions by zero, and even teaching it to his public school class. Yuck. LocalMinimum
technically it’s called “cardinality”, and is defined well enough that mathematicians know what they are talking about
Bob O'H, I'm curious as to why you feel the need to defend this mumbo-jumbo. If you are so inclined, you can explain yourself. Andrew asauber
I would say that the proof is in the use of it. What would go al haywire if infinities were not equal, for example if prime numbers had a lower cardinality than real numbers? If the answer is "nothing" then this "proof" is useless and just a vanity stunt. ET
EvilSnack - if you're interested, give A Mathematician's Apology a read. There are only 2 proofs in it: one that there is an infinite number of prime numbers, and one that that's not enough numbers (i.e. a proof that sqrt(2) is irrational). A part of me doesn't trust Cantor's Diagonisation Theorem either: it's a little too elegant. asauber - Harnett explains what is meant be "size": technically it's called "cardinality", and is defined well enough that mathematicians know what they are talking about. I try not to enter these discussion about infinity because it's a difficult technical subject, and I don't know enough to be able to explain the issues. Bob O'H
Since first hearing the theorems claiming that the countability of the reals is of a different order than the countability of the integers, I was skeptical. The extrapolation from finite sets to infinite sets was too hand-wavy for my comfort. But math isn't my specialty so I didn't take the time to learn more and work on it. EvilSnack
I always thought that 1/infinity = infinity/1 It was one of those whoa dude things. tribune7
two different infinities are equal in size
FWIW, An infinity wouldn't have a size. (I predict my statement will be lost on Infini-Heads.) Andrew asauber
It sounds like a bit of sleight-of-math, maybe hype, but what do readers think?
Higher math in general sounds like sleight-of-hand. The term I've held on to from my schooling was "mathemagic". I've seen enough brought to life through software that I don't sweat my shallow initial perceptions. You can't really know without actually engaging the proof itself. I realize I have uncommon faith in the fidelity of the mathematics community, as I believe them when they make a pronouncement. My experience with mathematicians is that their primary functions are properly, universally doubting and nit-picking; and it's exceedingly difficult to match them at it. Dissent is actually sought after and appreciated, demanded even, in this field. LocalMinimum
It sounds like a bit of sleight-of-math, maybe hype, but what do readers think?
My initial reaction: If Shelah says it's true, it's true. I have no hope of understanding this proof fully, but I would like to at least get the gist of it. daveS
The article is actually pretty good for the subject, though it gets itself contorted in a few weird ways. It ends strangely, saying:
While infinity is strange in many ways, it would be almost too strange if there weren’t many more sizes of it than the ones we’ve already found.
I think this misunderstands the whole thing. There are a number of sizes of infinity, and we've known that since Cantor. The question isn't the number of sizes of infinity (there are infinite sizes of infinity), but whether or not there exist intermediate sizes of infinity. I am not familiar with this work in particular, but Cantor thought that there was not an intermediate size (the author said he didn't know - he couldn't prove it, but he thought it was the case). I tend to agree with Cantor - the nature of infinity actually makes an intermediate size hard, because it can expand indefinitely through any intermediary you want to make. To put it simply, the set of integers is the same size as a number of other infinite sets, such as the number of rational numbers (numbers that can be expressed as a ratio), the number of positive integers, etc. However, the set of integers is smaller than the set of all real numbers. The question is whether there is an intermediate set size. Cantor answered "no" but couldn't prove it. These mathematicians proved that two sizes of infinities were the same that were previously unknown. johnnyb