81–90 of 116 posts
"So it really seems that when we switch to constructive theory, it actually does something even better, it eliminates this pathology altogether, making balls behave more sensibly?"
https://math.stackexchange.com/questions/175675/intuitionist...
I like theories without crazy theorems. And I like theories with crazy theorems.
Edit: the link below from soVeryTired to Andrej Bauers post is better
Earlier quoted context omitted.
> It relies on uncountably-infinite division of an object But the theorem claims finite division and not infinite? In R³, given a solid ball B of radius R it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R
Finitely many pieces, but on infinitely variable boundaries. It was clever to make the proof allow a finite number in that place. Without, it would have attracted no attention.
"So for those of you who don't know the result, here it is in simple, non-technical terms:" proceeds to immediately use a character that can't even be copy/pasted due to needing MathJax to render it, refuses to elaborate further Thankfully, it ain't hard to find the Banach-Tarski theorem in actual simple, non-technical terms, so for those wondering what in tarnation that character is: it just means three-dimensional…
I don't have direct evidence for my speculations, but I presume the reason [fraktur](https://en.wikipedia.org/wiki/Fraktur) was more common in mathematics back in the days is largely down to articles having to be type-set using movable type. If you insisted on using ℝ over 𝕽, you were likely to make life considerably harder for your printer (which in turn meant higher printing costs), since they would be considerably more likely to have to cast new types. As printing was modernized and movable type was replaced by more flexible printing-technologies, this pragmatic reason for preferring one glyph over the other went away. Another explanation/contributing factor is that the switch seems to have occurred in tandem with the barycenter of mathematics switching away from continental Europe and towards the US in the post-WWII period (at least if we disregard Soviet mathematics which also flourished in this period, but which was largely published in Russian). The average American would probably be less familiar with 𝕽 and other fraktur glyphs than the average German.
Give every point in the Sphere B a room in the infinity hotel. If the room number is odd, it belongs to sub-sphere B1, otherwise sub-sphere B2. Odd things happen when you divide by infinity.
This has always been my struggle with all of this "infinity" maths. Infinity/Infinity is anything you like and many things you don't and these things always seem to boil down to dividing by infinity. Visit each room in the hotel in turn, each one for half the total time spent going to and visiting the last. Without explanation of why we can divide infinity by infinity when we want to and not get total garbage. Banach…
This is super basic, because innumerate shepherds can use it to count sheep. As your sheep go out in the morning, for each sheep, take a small rock from a pile and put it in a bag. When the sheep come back in, take a small rock from your bag and put it back on the pile. Any rocks left in your bag at the end of this? If yes, you've lost a sheep and have to go find it. If a sheep comes back and you don't have any rocks left, oops, you've got someone else's sheep, so you should probably see if any other local shepherds are missing any.
Or, how can you tell if you've got as many students in your auditorium as there are seats? You look for students without seats, or seats without students. If there are no spare students or spare seats, you have the same number of both.
In both cases, you don't actually need to know how many sheep, rocks, students or seats there are. You just know that the sets of "sheep that went out" and "rocks in a bag" and "sheep that came back" are the same size, and "students" and "seats" are the same size, because you can establish a 1-to-1 mapping between every element of each set.
This is the important result - if you can establish a 1-to-1 mapping between every element of two sets, those sets are the same size
Therefore, if you can show a 1-to-1 mapping between every element of "all integers" and every element of "all even integers", then those sets are the same size. And you can. For each x in the set of integers, you map it to 2x in the set of even integers. Every member of either set has a single equivalent member in the other.
And that's how you can make an infinite amount of space in your infinite-but-full hotel, by moving every guest from room x to room 2x.
Yes, infinity is weird.
As the author points out that Banach-Tarski theorem is an example of hard-to-accept result that comes out of the easy-to-accept axiom of choice. There is a popular quote that related to this: > The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? From https://en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice, the well-ordering principle and Zorn'…
The Axiom of choice has never felt completely self-evident to me. E.g.: what if you have sets where the elements are non-computable? How do you "choose" objects that cannot even be named ? Something like: "the set of all programs that cannot be proven to halt" and the like can be used to create pathological sets where the set itself obviously exists, but you cannot name any of the members. Actually, an ever better ex…
That's easy: pick the TM which is minimal according to some lexicographic ordering on the specification of TM's. That's not computable (obviously) but it's perfectly well defined.
> The set of reals that are not the solution to any equation that can be written with a finite number of symbols
Yeah, that one is harder :-)
(I would simply say "The set of numbers that cannot be described by any finite number of symbols" in order to short-circuit arguments about what constitutes an "equation" and whether or not Chaitin's constant is the solution to some equation.)
Earlier quoted context omitted.
> You're using "being named" in a very unusual sense. No, I'm not. > Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?
Naming a number in mathematics has different rules to assigning names to objects in the physical world. You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2". This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would requ…
AC says: if there is a set of non-empty sets y with length x, I can construct a new set z of length x by taking a member from each set in y.
How does being able to name the constituents z or any member of y matter to the logic of the axiom?
Earlier quoted context omitted.
> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. Thank you! I was beginning to think I was the only one who believes this. So many people seem to mistake the map for the terrain.
I probably confused matters with my undefined expression "actual numbers". e and pi are as actual as i or 0. I think Turing introduced "computable numbers" which are a lot like the reals, but countable. You can write a program that produces each. So it includes integers, rationals, polynomic irrationals, and lots of transcendentals. But the set any particular person (or computer) operates on over the course of their…
Despite it being a non-starter from a pragmatic standpoint, we could for instance easily imagine a novel numeral type that encodes the set S = {a + b·π where a and b are integers} (we can encode integers quite easily and all we need to reposesent such a number in silico is to encode a and b). Using such a numeral type, we are able to do exact arithmetic if our operations are restricted to addition and subtraction (and if we are content with fractional representation of numbers as being considered "exact", we can also do division and multiplication although we would have to work within the larger set S' = { (a + b·π) / (c + d·π) where, a, b, c, and d are integers and c·d ≠ 0} rather than within S).
You make an infinite list of numbers between 0 and 1 chosen at random. Apparently you can make a new number that was never seen in the list before if you pick a digit from each number in the list and add one to it.
Say the list has numbers 0.36285728.. 0.95825597.. 0.47264112.. .. I can make a new number by taking the 3 from the first number the 5 from the second and the 2 from the third num and so on. 0.463.. I never understood
I never understood this example used to explain it, vsauce made a video on the banach tarski theorem. You make an infinite list of numbers between 0 and 1 chosen at random. Apparently you can make a new number that was never seen in the list before if you pick a digit from each number in the list and add one to it. Say the list has numbers 0.36285728.. 0.95825597.. 0.47264112.. .. I can make a new number by taking th…