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The Point of the Banach-Tarski Theorem

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Re: The Point of the Banach-Tarski Theorem

#101

Post by moloch-hai »

quchen wrote: post #95
moloch-hai wrote: post #29

Earlier quoted context omitted.

I am not a Platonist, or any sort of theist. e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world. I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no bet…

> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. It’s like that for all numbers, not just the fancier ones! I have never experienced a 1.

Even spiders and newborn babies count; it is a natural animal behavior. Axiomatic systems, not so much.

Re: The Point of the Banach-Tarski Theorem

#102

Post by moloch-hai »

wnkrshm wrote: post #82
moloch-hai wrote: post #14

Earlier quoted context omitted.

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.

It's still astounding to me that the 'fractal trick' starts working in R3 and not already in R2.

Agreed, each added dimension adds wacky surprises.

Packing spheres into a cube leaves you a pretty comprehensible space in the middle. Pack hyperspheres into a hypercube, and all hell breaks loose.

Re: The Point of the Banach-Tarski Theorem

#103

Post by Smaug123 »

thaumasiotes wrote: post #77
Smaug123 wrote: post #69

Earlier quoted context omitted.

You're using "being named" in a very unusual sense. Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. Most reals "have infinitely long names" under your definition, and there is no finite time at which you've distinguished between reals with the same initial segment of "name", so really what's the use of the naming scheme at all? (For…

> 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?

Honestly, I really am trying, but as far as I can see you are using the word "named" in a sense that allows most numbers to be called "0". I really can't see how your words make sense given any other definition. Is that correct?

Re: The Point of the Banach-Tarski Theorem

#104

Post by Smaug123 »

riversflow wrote: post #86
jiggawatts wrote: post #78

Earlier quoted context omitted.

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…

What does naming have to do with AC though? This seems completely tangential to the axiom. It’s about the nature of sets, not the semantics of how you make the selection and reference it. 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…

This is all in response to a comment which basically said "why do people think AC is obvious - if you can't even name all elements of the set, why would you expect to be able to choose them". (Which is handwavy, of course, and leads to the wrong intuition about uncountable-but-well-ordered sets, and depending on how you look at it it may even lead you to conclude that you can't ever choose even a single element from an uncountable set.)

Re: The Point of the Banach-Tarski Theorem

#105

Post by [deleted] »

thaumasiotes wrote: post #56
jiggawatts wrote: post #49

Earlier quoted context omitted.

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. Counter-intuitively, this is not true. The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number…

> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. So what? That doesn't stop them from being named.

[deleted]

Re: The Point of the Banach-Tarski Theorem

#106

Post by [deleted] »

thaumasiotes wrote: post #56
jiggawatts wrote: post #49

Earlier quoted context omitted.

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. Counter-intuitively, this is not true. The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number…

> The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. So what? That doesn't stop them from being named.

[deleted]

Re: The Point of the Banach-Tarski Theorem

#107

Post by ttctciyf »

moloch-hai wrote: post #29
ttctciyf wrote: post #24

Earlier quoted context omitted.

> Wherever it comes down to actual numbers Wait, are you saying e and pi aren't "actual numbers"?

I am not a Platonist, or any sort of theist. e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world. I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no bet…

I don't think you have to be a Platonist to allow that numbers that do not exactly correspond to humanly measurable quantities can yet be part of a calculation.

Another example would be Binet's formula for Fibonacci n[1], which though it involves Phi, sqrt(5), etc., always evaluates to an integer, given integer n.

The existence of these irrationals in Platonic heaven doesn't follow from the formula, which afaik depends only on the usual rules of algebra and logic, though I admit I'm not sure what constructivists make of this type of calculation.

1: see https://r-knott.surrey.ac.uk/Fibonacci/fibFormula.html or https://mathworld.wolfram.com/BinetsFibonacciNumberFormula.h... for example

Re: The Point of the Banach-Tarski Theorem

#108

Post by thaumasiotes »

WastingMyTime89 wrote: post #93
thaumasiotes wrote: post #45

Earlier quoted context omitted.

> Actually, an ever better example is: "The set of reals that are not the solution to any equation that can be written with a finite number of symbols." -- an infinite set that has no nameable members! That's not a good example; the problem you're creating is due to sloppy use of language, not any cleverness in the definition. All real numbers, and all numbers of any other variety, can be written with a finite number…

> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name. This is untrue and not particularly hard to prove by contradiction. Suppose every elements in R can be named by a finite number of symbols. You can build a bijection between R and the names of its elements (by definition a name points to a unique element and if eleme…

> Or, you can also easily build a bijection between a finite number of symbols and N. That's just an encoding like ASCII or UTF-8. Therefore, the set of names is countably infinite.

You've confused two different senses of "a finite number of symbols". The requirement is that it takes a finite number of symbols to write down any given number, not that it takes a finite number of symbols to write down every number at once. (Though in fact that also takes a finite number of symbols; it is the precise meaning of ℝ.)

Symbols are, in their simplest interpretation, bounded two-dimensional curves; there is no danger of running out of them before we run out of numbers.

Re: The Point of the Banach-Tarski Theorem

#109

Post by thaumasiotes »

Smaug123 wrote: post #103
thaumasiotes wrote: post #77

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?

Honestly, I really am trying, but as far as I can see you are using the word "named" in a sense that allows most numbers to be called "0". I really can't see how your words make sense given any other definition. Is that correct?

That is essentially correct, and we actually do call many different numbers "0" in different contexts, but it's not necessary for any two numbers to share a name this way. You can give them all unique finite names with an infinitely large alphabet if you prefer.

You might consider that in the existing Chinese writing system, 口 and 囗 are distinct symbols.

Re: The Point of the Banach-Tarski Theorem

#110

Post by yellowapple »

quchen wrote: post #75
yellowapple wrote: post #34

"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…

Had a laugh about that too. It’s sometimes hard to tell what’s »non-technical« when you’ve spent a lot of time on something, and even then what it means to be »technical« depends a lot on the audience. Non-technical here means knows basic math but is not familiar with measure theory; it does not mean _anyone_. I like that they added my favorite math joke in their (linked at the very top) limited audience jokes sectio…

> Non-technical here means knows basic math but is not familiar with measure theory; it does not mean _anyone_.

I guess "knows basic math" is relative; I tend to draw the "basic" line at "it's required for a high school diploma or GED", but I suppose I can blame my country's education system for that bar being much lower than what mathematicians seem to assume :)

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