Camels With Hammers
Philosophy, Ethics, Atheism, Nietzsche
The concept of infinity is fascinating.
That is so cool.
I already knew the content, but the video is pretty well put together.
Incidentally, a day or two ago the video had an incorrect statement of what the Continuum Hypothesis was, which has since been fixed. You can still see evidence of this older version if you hover over the 6:16 mark in the seek bar.
Very well done video.
Yes. This is why I enjoy being a mathematician.
After subjecting it to this video on an early Friday morning, my brain doesn’t like me no more. X_X
It’s not a coincidence that the father of set theory, Georg Cantor, spent a considerable portion of his time in hospital for mental illness…
(TBH, it wasn’t just the craziness one would expect necessary to first come up with the infinity concepts; there was also a considerable amount of depression, personal tragedy and professional difficulties).
Gottlob Frege, another late-19th-early 20th c. pioneer in the (modern) foundations of mathematics and logic, is said to have gone into a permanent depressive state following Bertrand Russell’s telling him about the antinomy now called by Russell’s name. Frege’s wife died at about the same time, so his illness may really have been brought on by a double disaster.
I had a physics professor who said the practical definition of infinity was “big enough”. As an example, he gave a camera company (may have been Kodak, but I don’t recall) that used a stop sign several hundred feet down the road to calibrate the “infinite” focus setting on their lenses. So for them, infinity was only a few hundred feet.
The good prof was apparently talking about the infinity out at the right-hand end of the real line, rather than the set-theoretic infinite cardinals or ordinals. Given the resolving power of the optics of the cameras and the inhomogeneous atmosphere, a few hundred feet was probably good enough. For mathematics, not so good.
That’s nice work, and a good introduction to how our finite minds can comprehend the infinite, or at least some kinds of infinities.
Georg Cantor named the infinite cardinalities (set sizes) aleph-0, aleph-1, aleph-2, … I’ve seen beth-0, beth-1, beth-2, … for the sequence of power sets (sets of all subsets), starting with beth-0 = aleph-0.
Beth-0 / aleph-0 is the cardinality not only of integers, but also finite-length lists of integers, rational numbers, algebraic numbers (solutions of polynomial equations with integer coefficients), computable numbers (numbers that can be approximated to arbitrary precision with a finite number of steps), and definable numbers (numbers that satisfy certain sorts of finite-length conditions).
Beth-1 is the cardinality of the real numbers, finite-length lists of real numbers, and continuous functions of real numbers onto real numbers.
Beth-2 is the cardinality of *all* functions of real numbers onto real numbers.
I’ve yet to see any comparable interpretaiton of beth-3 or other higher beth numbers.
Beth-0 / aleph-0 is the cardinality not only of integers, but also finite-length lists of integers
Sorry, what exactly do you mean by this? Because if you are using “finite-length list” in the way I’m thinking of it, aleph_0 is definitely not the cardinality of a finite-length list of integers.
I think he means the cardinality of the set of finite-length lists of integers, not the cardinality of any given element of that set.
Aha! This is why I asked. I can see that now.
That’s what I meant – sets of all finite-length lists of integers.
However, a countably-infinity-long list of integers has cardinality beth-1.
Aleph-0, beth-0 — countable infinity.
Beth-1 — continuum or C.
Continuum hypothesis: aleph-1 = beth-1
Aleph-1, incidentally, is the cardinality of the set of all countable ordinals; there are provably no infinite cardinals between aleph-0 and this number, even without the axiom of choice; with the axiom of choice you can make the stronger statement that aleph-1 is the smallest cardinal greater than aleph-0.
Beth-3 would be (assuming the axiom of choice) the cardinality of the set of all operators O : F -> F where F is the set of functions over the reals. (In general the set of functions from X to Y is Y^X; (beth-2)^(beth-2) = (2^(beth-1))^(beth-2) = 2^((beth-1)*(beth-2)) = 2^(beth-2) = beth-3). I can’t think of any more “natural” sets of this size.
One might be tempted to wonder how big the collection of all aleph-* numbers (or all ordinals) is: in fact the answer is that it’s so big that it cannot be a set, only a proper class, so there is no way to define its size as a cardinal number. (The “n” in aleph-n or beth-n isn’t restricted to natural numbers, it can be any ordinal, even uncountable ones.)
Thanks people for these explanations. Thanks for so rudely rubbing my nose into my helpless ignorance. This will not stand, ya know. I will do something about it.
I like infini
The end of the video should have specified that the unanswerability of a question in mathematics is relative to the assumed axiom set, like ZFC. We can always choose to go to stronger systems that actually can answer many of the unanswerable questions in weaker systems, though no matter how strong we make a system it will always have the weakness of incompleteness (either that or it will be inconcistent, via Godel).
Readers may be interested in a couple of questions I asked on MSE, namely how we know of the existence of an Aleph_1 (the very first cardinal greater than Aleph_0, the latter of which is the size of the natural numbers 1,2,3,…), and if we can put a bound on the number of infinities that may exist between Aleph_0 and the cardinality of the continuum (i.e. all the real numbers).
Thank you for pointing that out. I don’t know this stuff very well, but I felt a bit uneasy about that part the video.
Judging by what the articles on Wikipedia and Wolfram math world say there seems to be disagreement over whether or not there is anything meaningful left to discover about the Continuum hypothesis.
One thing about the initial post, my current understanding of infinity is that, strictly speaking, it’s not a number. It basically just means an unbounded set. You can say that a real number line goes to infinity, but there’s no point anywhere on that line that infinity lies. It’s not a number itself.
Like, I cheat on the game Battletoads with a Game Genie code to get “infinite lives”. There’s not enough RAM in the NES to actually store an infinite number if it existed. All it means is lives are not limited any more. They won’t decrease on my death. Lives are now unbounded.
The whole concept of cardinality blew my mind. Not just in how fundamental it was, but in the fact that it’s literally one of the easiest things to teach someone about “higher mathematics” I’ve ever read about. I usually just scratch my head at mathematical proofs and wonder “when did they start using triangles in equations?”, but I don’t even need to know basic arithmetic to get this proof of non-matching infinities. The only concepts needed are how to make a grid, how to make a list, and how to count. I showed kids this and they got it instantly. It’s that simple and that powerful.
So, why isn’t this being taught in school? I for one was never taught set theory in my math classes. My mathematics stopped at algebra, but that’s a failing of having gone to a private school that wasn’t exactly very challenging.
A more expansive and historically aware perspective is that what exactly constitutes ‘numbers’ has shifted and been reinterpreted over time. The integers just happen to be in the intersection of a wide array of all sorts of aspects that numbers may have – information that tracks size, order, length, arithmetic, topology – and as we explore outwards we find many things that do not fit all of these descriptions simultaneously. Rationals, algebraics, number fields, reals, complex numbers (quaternions, etc.), extended reals or the Riemann sphere, surreals, p-adics and adeles, ordinals and cardinals, etc. In most cases you will lose at least something, be it commutativity, subtraction, order, divisibility, etc. There is a zoo full of exotic numbery structures. So we may very well say infinity is a number (or infinities, more properly).
Set theory isn’t very standard subject material because (a) the concepts and notation of elementary set theory are usually squeezed into many other courses already (any serious analysis, modern algebra, or topology class at least), and (b) modern set theory has little practical value outside of special theoretical interest to researching mathematicians and students. At least with many other forms of pure mathematics we can cite, say, theoretical physics or cryptography as applications, but not so much with set theory. We may cite maybe fuzzy set theory for its statistical value, pure logic related theory for its computer science value, and set theory at large for value in being a foundation for mathematics (like an assembly language) and for sparking a lot of philosophy of mathematics intrigue, but things like infinite cardinals and the continuum hypothesis are largely irrelevant to anything of the ‘real world.’
The big problem with “infinity” is the fact that the word has so many legitimate meanings inside math: there are the infinite cardinals and ordinals, the “+/- infinities” that you adjoin to the real numbers to do measure theory and probability, the point at infinity that gives you the Riemann sphere, the line at infinity for projective planes, etc.–and then there are the uses of the word in theological woo. To put the kind of elementary fun stuff that this post deals with into K-12, you’d have to take stuff out that college-bound kids need–and many of them already need remediation as freshpeople. Now imagine what the fundies would do with this material–they already don’t like biology, and wait until they find out that there are controversies in the foundations of math–no, my head hurts already.
Well I learned something new here. Thanks for the info.
My mathematical skills are rather lacking. The thing is, learning just a little basic set theory helped my critical thinking skills quite a bit, and for that reason alone seems valuable to teach at the lower levels.
Also, there are some lulls in many class rooms a few very basic lessons could be put into. I don’t think it would require so much time that anything of great importance would need be cut. I mean, how about gym? Do we really need that?
… learning just a little basic set theory helped my critical thinking skills quite a bit, and for that reason alone seems valuable to teach at the lower levels.
I agree, and so did the “New Math” movement of the 1950s-60s. For various reasons–IMHO the most crucial being that the K-12 teachers (or the professors of education) thought that it was not important for them to *understand* the new material–that movement ended in a fiasco. Many professional mathematicians have become traditionalists about K-12 math ed because so many tries at improvement have not worked out well–and don’t get them started on “math experts” in the ed-school establishment.
I mean, how about gym? Do we really need that?
I managed to avoid it, no small feat (heh heh).
Wow, two blogs about set theory in one day! Apparently certain religious fundamentalists have a problem with set theory and unanswerable problems.
It’s a good video.
One minor point. Toward the end, he says that all research mathematicians accept these ideas. That’s not quite right. The constructivists and intuitionists don’t accept them.
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