I highly recommend <a href="https://twitter.com/andrejbauer/status/1428471658088738818" rel="nofollow">https://twitter.com/andrejbauer/status/1428471658088738818</a> and follow ups.<p>Yes, this stuff is fishy, and yes we can blame ZFC which is a bad formalization in comparison to what we've developed since. But the real scandal is why does our definition of geometry "leak" the underlying set theory it's built atop so much? Surely it's bad to have such a leaky abstraction in pure math!<p>The series goes on to show that by abandoning "points" — which pull all the funny set theory stuff into geometry/topology/whatever is the topic at hand, one can still have a classical foundation — e.g. with the axiom of choice and law of excluded middle — that makes mathematicians feel at ease, but also purge this Banach–Tarski gobbledygook.
To me this is proof that infinity is something only present in our math and not in the universe.<p>Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite.<p>Happy to hear disagreements tho.
Can someone correct me if im wrong?<p>What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume.
This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large.<p>The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).<p>This seems incorrect.
> How can you double the volume of an object just by decomposing and rearranging it?<p>That part is easy - for each point on a unitary sphere move it to a point at position 2x ( ie. to a corresponding location on the sphere of the 2 units radius) - you've just doubled the volume, i.e. you've just built a 2 units radius sphere out of the points belonging to 1 unit radius sphere. Banach-Tarski of course more fun and illustrates much more than just volume.
Until recently I never questioned the idea that, say, the positive integers and the odd positive integers are equivalent because they can be paired, but this cloning thing seems like something that falls out of that. And it seems like that view of infinity isn't actually necessary if Cantor style cardinality is not the last word.<p>In the paragraph on nonstandard analysis in the Wikipedia page on infinity, it says:<p>"The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers"<p><a href="https://en.wikipedia.org/wiki/Infinity" rel="nofollow">https://en.wikipedia.org/wiki/Infinity</a><p>I can't say anything precise or mathematical, but after I read the above, I have an "obvious in hindsight" feeling. If H=inf is different from H + 1, how much different is it? 1/inf or an infinitesimal amount! And an infinitesimal is not nothing.<p>The quanta article says "You can add or subtract any finite number to infinity and the result is still the same infinity you started with" but this seems like just a dogma for non mathematicians?
If I remember rightly there'a a Feynman anecdote where he points out that as the real universe is quantised this is a purely mathematical notion.<p>I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
Spoiler: the cut line between the two apple-halves is fractal in shape, with infinite surface area and taking forever to cut at any finite cutting speed.<p>It's sort of hilarious to see a <i>physics</i> site mention the Banach-Tarski paradox. It is, after all, the most obvious hole poked in the most basic working assumption used by physicists: that space and time are measured with real numbers.<p>I've seen physicists go to pretty absurd extremes to avoid thinking about the problems this creates. Fixing it properly is not easy: simply dropping the axiom of choice leaves you unable to do useful physics. Getting back to a useful state, making all sets Lebesgue, can only be done with large cardinals:<p><a href="https://www.jstor.org/stable/1970696" rel="nofollow">https://www.jstor.org/stable/1970696</a><p>Large cardinals are pretty exotic even by the standards of mathematicians. In many departments they are in fact the domain of <i>logicians</i>. In fact, the existence of certain classes of Woodin cardinals is equivalent to the Axiom of Determinacy (AD), which is the "mathematically respectable" way of investigating logics with infinitary conjunction/disjunction. In fact, AD is precisely the Law of Excluded Middle (A or not-A) for logics with infinitely-long conjunctions.<p>Quite odd that something so ethereal would be connected to a tangible act like cutting an apple in half.
Can someone explain the flaw in my reasoning here?<p>Assume I have a sphere made of pure iron. I divide the sphere into individual iron atoms. I divide this group of atoms into two groups of atoms. I take each of those groups of atoms and form them into 2 spheres. How is it that these two new spheres are not either less dense or smaller that the original sphere?
Vsauce made a great video about this <a href="https://youtu.be/s86-Z-CbaHA" rel="nofollow">https://youtu.be/s86-Z-CbaHA</a>
I was initially appalled by Banach-Tarski.<p>Looking into it more closely, it turned out to be both trivial and not notably meaningful, like most surprising results involving uncountable infinity. Nothing that affects us involves actual infinities, so infinities are just a convenient approximation that often produces correct-enough answers. Anything infinities imply that seems crazy trivially is.
All of this is still less crazy than quantum mechanics itself. Some people may cringe, but the proof is valid anyway (not sure what in principle might be wrong with ZFC?). I wouldn't be surprised if it revealed some hidden aspects of reality we still aren't aware of.
Great video on this: <a href="https://www.youtube.com/watch?v=s86-Z-CbaHA" rel="nofollow">https://www.youtube.com/watch?v=s86-Z-CbaHA</a>
After reading the approach it seems like too much work! Anyone willing to critique my suggested simpler proof (which didn’t occur to me until after reading the article).<p>TLDR; It is basically the same as proofs that all countable sets have the same cardinality. (TLDR of that: map set of positive integers x to the even numbers by doubling, and the odd numbers by doubling and subtracting 1. Take the union of even and odd and you end up with the set you started with, the positive integers x).<p>For a circle:<p>We can identify all the points on a circle as the points p associated with the [x,y] coordinates of the complex numbers p = e^(2.c.i.pi), where 0 <= c < 1. (And . is multiply.)<p>If we take each of those points p and rotate it by doubling its c, we now have the same points represented by the expression p = e^(2.c.i.pi), where x <= 0 < 2.<p>So the same number of points, but two passes around the circle, 0 <= c < 1 and 1 <= c < 2. We can move the second set of points in the x positive direction by 2 or more to avoid the overlap.<p><i>We have now rearranged points of one circle into two.</i><p>For the surface of a sphere:<p>We simply divide a sphere up into points defined by a stack of circles at real-valued vertical z positions, z <= -1 <= 1. And their real [x,y] points are the real and imaginary parts of each circle e^(2.c.i.pi).circumference(z), where 0 <= c < 1, and circumference(z) is the cos(z).<p>Again, rotate the points by doubling c, so that they are now located at c, where 0 <= c < 2. There are now two overlapping sphere surfaces. We can move the second in any direction by 2 to avoid the overlap.<p>Similar generalizations work for including the volume.<p>Anyone understand why this simpler proof is wrong, or why the more complex proof in the article does something better?
Infinity is the axiom of paradox. Does the inclusion Infinity complete an otherwise incomplete set of axioms? It solves the halting problem for a finite Turing Machine.<p>I don't buy the diagonalization proof as anything more than the Pythagoreom Theorom. You have infinite rows, and infinite columns. Infinity is Schrodinger's Cat. Once you check in on the state (nth row by mth column) the only thing you can say about the diagonal number is that is hasn't occurred in the rows up to that point, not beyond, nor in the columns (if n > m).<p>Ergo, Infinity is a paradox, and only mathematical in the absurd.