

The Banach–Tarski Paradox [video] - selectnull
https://www.youtube.com/watch?v=s86-Z-CbaHA

======
thetwiceler
This video is somewhat misleading. I appreciate the attempt at making Banach-
Tarski accessible to a general audience, but it dwells on the wrong aspects of
what makes Banach-Tarski interesting, making the construction look more like a
magic trick with sleight-of-hand. I wish the video had at least mentioned the
Axiom of Choice somewhere, as that is fundamentally what Banach-Tarski is
about.

The sleight-of-hand comes in around 14:30 into the video, where we are told to
create the sequence for an "uncountably infinite number of starting points."
That's exactly the point where the construction is non-constructive, and the
infamous Axiom of Choice is used. There is no construction - in the sense of
constructive mathematics - that can achieve what is described at this point.

Banach-Tarski is not generally regarded as some deep fact about mathematics, a
point the video mistakenly belabors. Rather, it is a consequence about
particular axiomatizations of set theory which admit the Axiom of Choice.
Banach-Tarski is _only_ valid with the Axiom of Choice, and in fact that is
the main interest in the paradox.

In my personal opinion, the Banach-Tarski paradox isn't much more enlightening
than the simpler construction of the Vitali set (assuming the Axiom of
Choice), which is a non-measurable set of real numbers (with Lebesgue measure,
i.e., length).

Another part of the video I find misleading has to do with the hyper-
dictionary, where he describes the hyper-dictionary by putting some parts of
the dictionary "after" other parts which are infinitely long.

The putative applications of Banach-Tarski to physics are ridiculous.
Uncountable sets are fundamentally unphysical. The Axiom of Choice serves
mainly as a convenience to mathematicians when either a proof avoiding the
Axiom of Choice would be more complicated, or so that mathematicians can state
properties of objects which are set-theoretically larger than anything that
can be relevant to physics anyways.

~~~
tome
> Uncountable sets are fundamentally unphysical.

I think you didn't quite mean to say this, the real numbers being uncountable
yet forming the basis for classical physics.

~~~
jjoonathan
I think he did: you don't need real numbers to formulate classical physics.
Everything measurable has finite precision so you can always get away without
postulating that your limits actually converge to something.

Of course, the reality is that then you would wind up with awkward limit-
taking machinery in your answers. Real numbers encapsulate that complexity so
you might as well use them to simplify both the notation and manipulation of
limits. But you don't need to.

~~~
thetwiceler
Yep, this is what I meant to allude to, and you've worded it much better than
I could have.

Perhaps a nice way to say it is that the mathematical objects necessary for
physics that I can think of are separable (such as the real numbers).
Basically, whenever you have uncountable sets, they come along with some
topological structure which must be handled continuously.

~~~
tome
There's even an argument that separability is sufficient for most of
_mathematics_ let alone physics
([http://arxiv.org/pdf/math/0509245.pdf](http://arxiv.org/pdf/math/0509245.pdf))
but you're going to have to work very hard to persuade me that if we're going
to describe _any_ sets as physical then uncountable ones are less physical
than countable ones.

------
baddox
His initial explanation of uncountable infinity isn't exactly correct, and I'm
afraid it will give people the wrong idea. He says that the real numbers are
uncountable, because even between 0 and 1 on the number line there are an
infinite number of real numbers. But that is also true of the rational
numbers, which _are_ countable! After all, what is the smallest rational
number larger than 0?

~~~
egonschiele
If anyone's curious on how it is possible to list all the rational numbers, it
would go something like this:

0, 1, -1, 2, -2, 1/2, -1/2, 3, -3, 1/3, -1/3, 2/3, -2/3, 3/2, -3/2, 4 ...

at each step you list the numbers where the numerator and denominator are <=
x. For example, if x = 2, we can count 1, -1, 2, -2, 1/2, and -1/2\. Obviously
it is an infinite list, but you _can_ list them.

~~~
baddox
My favorite way to visualize the bijection is
[https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree](https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree).

------
msherry
Q: What's an anagram of "Banach-Tarski"?

A: "Banach-Tarski Banach-Tarski"

------
gus_massa
This video is much better than I expected. If you have 25 minutes you can see
a friendly presentation of the proof.

[The connection with particle physics at the end I a little too much.]

An important detail is that this video is made by Vsauce. He usually has good
videos, but sometimes the connections between the parts are too farfetched.

------
ColinWright
A couple of months ago I posted this[0]:

    
    
        The Point of the Banach-Tarski Theorem
            – not just a curiosity
    

That spawned a lot of discussion. Indeed, there are many many submissions[1],
and some spawn considerable discussion, others are still-born.

The Banach-Tarski theorem is a lovely result, and I look forward to seeing
what people say about this new presentation of it.

========

[0]
[https://news.ycombinator.com/item?id=9674286](https://news.ycombinator.com/item?id=9674286)

[1]
[https://hn.algolia.com/?query=banach%20tarski&sort=byDate&pr...](https://hn.algolia.com/?query=banach%20tarski&sort=byDate&prefix=false&page=0&dateRange=all&type=story)

------
madez
I consider uncountability an artefact of a flawed approach to mathematics. I'd
recommend looking into constructive mathematics based on intuitionistic logic.
All fruitful insights based on other mathematics can be proven by it, from all
of it's results we can easily extract methods and it is much less mystic. I
think it is much more fun, too.

~~~
baddox
I'm aware of constructivism, but what is "intuitionistic logic?" I am under
the impression that the real numbers (and their uncountability) are generally
accepted by constructivists.

~~~
madez
Intuitionistic logic is nearly classical logic. That is, we start by assigning
statements one of the two values ‘True‘ and ’False’. While truth in the
classical sense is abstract, it is concretized in intuitionistic logic with
the meaning of ’we can prove it’. We know that there are statements that can
neither be proved nor disproved, so we cannot make use of the law of excluded
middle "a statement is true or it is not". Add that we are consistent, that is
"not (a and not a)" for all a, and then negation must not be the inverse of
itself, because we'd be able to proof the law excluded middle otherwise. You
see, it is basically classical logic with some minor adaptions to take into
account what we’ve learned.

I understand under "constructive reals" the computable reals, and there are
only countably many of them.

~~~
thetwiceler
It seems misleading to say that intuitionistic logic assigns statements to one
of the two values, True or False. There's no symmetry between the notions of
truth and falsity as there is in boolean logic.

You need to be very, very careful when talking about the size of the
constructive reals. If you are working within constructive mathematics, if you
describe the real numbers as setoids of Cauchy sequences with Cauchy-
equivalence as the equivalence relation, then there are uncountably many real
numbers. From a meta-theoretic perspective, it is obvious (as we are working
in constructive mathematics) that every real number is in some sense
"computable".

Your notion that there are countably many constructive reals probably comes
from definitions of the constructive reals from within classical set theory,
wherein you must internalize some notion of what it means for a real number to
be constructible, and so you are, in a sense, working meta-theoretically. Then
it is no surprise that the computable real numbers are countable. After all,
Skolem's paradox says that, meta-theoretically, we could have countable models
of the classical real numbers as well.

Additionally, meta-theoretically, we see that our definition of real numbers
in constructive mathematics will also have a countable model when seen from
the outside.

~~~
madez

       if you describe the real numbers as setoids of Cauchy sequences
       with Cauchy-equivalence as the equivalence relation, then there
       are uncountably many real numbers.
    

I start with computable sequences (or — if want — with turing machines),
define natural numbers based on them, go to the rational numbers, define real
numbers as a setoid of cauchy-sequences (note: all these sequences will be
computable by the way we constructed them) of rational numbers, and end up
with countably many. So, it depends on what you start your constructive world
with. I see no way to start with something uncountable.

You talk about meta-theory and models. I never saw the reason to complicate
things with that. Care to elaborate?

~~~
thetwiceler
ihm's response to you I think did a great job explaining what I meant to say,
but let me elaborate further.

Just like you might define computable real numbers, you may similarly describe
the "definable" real numbers, that is, those numbers which are uniquely
specified by logical statements. You will inevitably find that the definable
real numbers, like the computable reals, appear to be countable.

So the fact that something is computable, per se, isn't exactly what makes the
real numbers countable rather than uncountable.

I use the vocabulary of meta-theory and models, because the practice of
defining either the countable reals or the definable reals within a formal
system looks a whole lot like defining a formal system within itself (i.e.,
metatheory). So the fact that the computable reals and the definable reals
appear countable, is, at least to me, much like the statement that there are
countable models of your favorite formal system.

~~~
madez
HN doesn't allow comments with multiple parents, so I needed to choose one to
reply to. ihm, please consider this comment also a reply to yours.

The more I talk about constructivism the less I feel it's adequate to label
what I'm talking about, even though my approach to mathematics is completely
constructive. Please don't treat every word I use as totally definite.

Mathematics is the science of formal reasoning. I call the rules of reasoning,
that is the rules of manipulation of statements including their lexicon,
syntax and semantic, a logic. I see intuitionistic logic as the most
intuitive, universal and fruitful logic, in short: _the_ logic.

To reason, we need something to reason about. Said somethings should be
respresentable, so we can communicate about it. Let me call the things we
reason about objects. What do we use to represent objects? Symbols, pictures,
sounds, smells and many more things we have senses for. We transmit objects as
a literal, or as an algorithm ("a turing machine") of how to recreate the
object. All turing machines can be encoded with finite bit arrays.

It turns out we are able to make digital computers recreate symbols and
pictures and sounds to a fidelity such that we can't notice any difference
compared to the original. I see no fundamental difficulty besides physical
limitations to recreate other literals. All literals in a computer can be
encoded with finite bit arrays.

Finally, I conclude that finite bit arrays are the most general objects we can
reason about; they have rich semantics by the possibility to interpret them as
turing machines. What we end up with looks a lot like common programming
languages.

Based on intuitionistic logic, finite bit arrays, and the ability to interpret
them as turing machines, we can follow the canonical construction of the
reals. In this world, every object is a finite bit array, plus some annotated
interpretation ("a type"), which is just notation for a bigger finite array.
There is simply no place for non-countability.

I long for a software that lets me mix logic and usual programming seamlessly,
without unnecessary like meta-theories, meta-logic. Coq is near, but the
programming is too unnatural.

One can reason about the logic of a Rust programm, discuss it with others, and
even the compiler understands some of the meaning, but richer intuitionistic
logic statements about the output or behaviour of the code are sadly beyond
the compiler.

I'm about to start applying for my graduate studies of mathematics. Do you
have any recommendations which university to apply to to research in the area
I talked about?

------
roflmyeggo
Vsauce makes a good point that we just aren't made to intuitively understand
this type of stuff. Recognizing this fact, in my opinion, is a key driver in
helping to wrap our minds around these concepts. It's important to understand
that these concepts are valid in both our visible world and the hidden quantum
world, the main difference is scale.

For example, I could never wrap my head around the fact that electrons can
have multiple paths/histories simultaneously when travelling. The same is true
of a baseball thrown in the air, the only difference is that on the visible
scale that we are used to the chance of that baseball taking a different
path/history is so small that it will never happen.

~~~
ihm
I actually think it's very possible to have an "intuitive understanding" of
Banach Tarski (I would say I have one, but perhaps we disagree on what is mean
by such an understanding).

My "intuitive understanding" of this comes via an intuitive understanding of a
paradoxical decomposition of the free group and its Cayley graph, which is
flashed briefly in the video here[0] but sadly not discussed at length.

[0]:
[https://www.youtube.com/watch?v=s86-Z-CbaHA&feature=youtu.be...](https://www.youtube.com/watch?v=s86-Z-CbaHA&feature=youtu.be&t=116)

------
amelius
Infinity is a concept, not a number. Confusing the two is what gets you into
trouble. And, unfortunately, it is easy to confuse them because in
mathematical notation, infinity is often used in place of a number.

~~~
grumpy-buffalo
There are plenty of senses in which infinity IS a number -- or rather, many
numbers. See e.g. the Wikipedia articles on cardinal numbers, ordinal numbers,
hyperreal numbers, and surreal numbers.

~~~
amelius
> There are plenty of senses in which [...]

Yes, perhaps. But still it IS not a number.

Calling infinity a number is a "hack" done by mathematicians.

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acconrad
This is crazy as I was just thinking about uncountable infinity when I was on
the bus this week, having no idea this was part of such a paradox.

The things you consider when you're not distracted by a cell phone...

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anotheradhoc54
from layperson's intuition - isnt this just a trick performed by extracting
the extra elements from an infinity ? sort of the opposite to losing
information through common scaling by zero ? 2 = 1 because (2)0 = (1)0 a
matter of defining rules and staying consistent to them. the paradox arises
from expecting a contrived model to manifest in physical reality

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prezjordan
It's so refreshing to see a video like this have over a million views. Really
love the stuff Vsauce puts out - this might be his best yet.

~~~
roflmyeggo
It's his passion for the material that always keeps me watching. Listening to
someone explain a topic that they find deeply intriguing is a pleasure.
Reminds me of Feynman.

