Doesn't the fact that the Banach-Tarski theorem can be proven to be true within our current mathematical framework mean...

Doesn't the fact that the Banach-Tarski theorem can be proven to be true within our current mathematical framework mean that we need to rethink some of our geometry and/or set theory axioms?

Attached: banach-tarski-paradox.jpg (1321x1676, 161K)

Other urls found in this thread:

en.wikipedia.org/wiki/Initial_singularity
hawking.org.uk/the-beginning-of-time.html
lmgtfy.com/?q=electron
vixra.org/abs/1703.0073
twitter.com/SFWRedditImages

No. Fucking stupid shithead. Fuck you, you garbage turd algebraist crap.

How do you think the big bang happened?
You best start believing in Banach-Tarski paradoxes.
You're in one.

Retard

Prove it wrong, faggot.

Everyone already knows that the axiom of choice was a mistake.

How is the big bang related at all?

Banach-Tarski theorem perfectly valid -- under the assumption that zero-magnitude "points" exist.
The assumption was Euclid's.

Just doesn't happen to match the real world.
Doesn't mean anything is wrong.
Math is all about coming to logical conclusions, given the initial assumptions.

Why the universal homophobia?

Allah is the creator of our universe.

What was the volume of the universe at the earliest moment of the big bang? What is the volume of the universe today?

Retard

Not an argument. Don't be jealous you didn't think of this first.

Big Bang has nothing to do with B-T.
The Bang MUST have had some finite dimension unless you simply dismiss QM as rot. (And it you do, please cite something better.)

But doesn't this completely invalidate the concept of volume?

en.wikipedia.org/wiki/Initial_singularity
>The initial singularity was a singularity of infinite density thought to have contained all of the mass and space-time of the Universe[1] before quantum fluctuations caused it to rapidly expand in the Big Bang and subsequent inflation, creating the present-day Universe
>infinite density

And if it doesn't invalidate the concept of volume, it's a contradiction, plain and simple.

How is it a contradiction exactly?

I agree. Axioms should serve the purpose of making what we consider intuitively obvious precise so that we are able to derive exact results. If our axioms contradict lead to results that contradict our intuition then they aren't serving their purpose.

Maths need not to always follow physical reality. Example: quotient space.

Banach-Tarski merely proved that subsets of R^n exist which cannot be attributed a measure that satisfies certain reasonable conditions.

That's incredibly retarded considering intuitively obvious beliefs are proven wrong all the time.

And in fact, you could make a good argument the shit nature of our intuitions is a major reason why we even have so much investment in mathematics and science in the first place.

>when you add two numbers, they always make a bigger number!
>hey, look! we thought of negative numbers. Now, sometimes you add two numbers and get a smaller number.
>Veeky Forums: FUCKING MATH IS BROKEN FUCK ADDITION I HATE THIS SO MUCH ITS A PARADOX AAAAAAAGGHHHH
it's just due to the strangeness of set measures, you fucking tools.

Doesn't matter what you parrot.
A black hole is supposed to have a point of "infinite density" at the center. Maybe.
But both Relativity and Quantum Mechanics say BHs have a finite size. The two theories cannot be completely reconciled as yet, but they definitely agree on that.

Maths is BASED on empirical reality, but goes far beyond it as a pure intuition. As Kant would say in his Prolegomena:
>The problem of the present section is therefore solved. Pure mathematics,
as synthetic cognition a priori, is possible only because it refers to no other
objects than mere objects of the senses, the empirical intuition of which
is based on a pure and indeed a priori intuition (of space and time), and [4:284]
can be so based because this pure intuition is nothing but the mere form
of sensibility, which precedes the actual appearance of objects, since it in
fact first makes this appearance possible. This faculty of intuiting a priori
does not, however, concern the matter of appearance – i.e., that which
is sensation in the appearance, for that constitutes the empirical – but
only the form of appearance, space and time.

/thread

because 1V=/=2V

What is the finite density you're claiming here specifically? And why does it disagree with Stephen Hawking's claim of an initial singularity of infinite density?
hawking.org.uk/the-beginning-of-time.html
I wouldn't call that parroting so much as using actual sources, which is something I'd like you to try.

What are 1V and 2V when V is 0?

I'm not saying that axioms should make true everything that seems obvious, I'm saying that we for example have a very concrete intuition for what volume is. Our axioms and definitions should make this vague intuition concrete so that we can do actual math, however if we realise that using our definitions and axioms we can prove things that contradict our basic intuition we should reexamine whether it is sensible to define and assume the things we did. For example the reason that the axioms of basic geometry were reexamined during the early 20th century was that they lead to conclusions that didn't make sense to us, even though they were properly derived.

This

if V=0, then the ball does not exist.

>if V=0, then the ball does not exist.
What is the volume of an electron?

what is an electron?

lmgtfy.com/?q=electron

you can't be serious

you mean photon? electrons have mass

Not an argument.

>mass is the same thing as volume

Attached: 29.png (403x448, 53K)

>not getting what i meant
how do you define your "electron" for the purpose of whatever you're trying to illustrate in mathematical terms?
ball? point? something else?

It has nothing to do with volume.

I'm not "illustrating" anything, I'm just asking you to tell me what the volume of an electron is since you seem convinced an object having 0 volume means said object doesn't exist.

you're not too bright, are you

Calm down and try again little buddy.
Argument. Make one.

forget volume. The theorem, if accepted as valid, implies that one sphere is equal to two spheres, and by extension any whole number of spheres. THe paradox is that we have other, more fundamental theorems, which show that

[eqn]1\neq N[/eqn].

The paradox is the you can use the BT theorem to show that

[eqn]1=N[/eqn].


I wrote a little about the BT theorem here:
>On The Riemann Zeta Function
>vixra.org/abs/1703.0073

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so we're not talking about classical euclidean geometry then?

Nice to see retard physicists fighting

so we're taking those balls as infinite sets of points rather than regular 3d objects with a volume?

Hawking is neglecting quantum effects here. (Yes, he mentions them, but not in this context.)
Below a certain dimension (likely the Planck length or something similar) "size" doesn't mean anything.

But it still has nothing to do with B-T, which is simply a mathematical artifact arising out of the definition of "point".
A line 1 inch long has an infinite number of points. So does a line 2 inches long. And they are the same "infinity" since they can be mapped one-to-one. Trying to add an infinite number of zeros to get a finite number is what got Zeno into trouble.

Infinity doesn't occur in the real universe (as opposed to the matheverse). It's just a warning that the logic has broken down.

Hawking knows this very well --- and has said so. The speech you're quoting is the "simplified for the public" version.

>implies that one sphere is equal to two spheres
No, it says that one sphere decomposed into pieces with indeterminate volume can be re-arranged into two identical spheres. This does not say or imply that one sphere is equal to two spheres. The entire crux of the argument is on decomposition into scatterings of points with no volume, meaning it had nothing to say about volume or real life.

Exactly right.

An electron is an excitation in the electromagnetic field it is not localized, it is not an object, it does not have a volume, it has energy, and it has a mass. It's not a tiny fucking sphere you idiot.

but in that case it's trivial to say you can always take any piece of that "ball" and make another one with equal "volume"
no need to get any more complicated than that

>Below a certain dimension (likely the Planck length or something similar) "size" doesn't mean anything.
So you can get weird counter-intuitive results like the Banach-Tarski paradox in a singularity then. Or something like the big bang could happen.
>Infinity doesn't occur in the real universe (as opposed to the matheverse). It's just a warning that the logic has broken down.
I don't see how you could ever prove that claim.

I never called it a "sphere" and I never said it wasn't an "excitation." I didn't call it anything except an electron, and that's all I need to call it because the only thing I'm disputing is your claim that "0 volume" is the same as saying "non-existent."
>it does not have a volume
Exactly.

divide a sphere in half. It makes two hemispheres. Divide in half again, and again and take the limit of infinite divisions. Those two hemispheres turn into an infinite number of points. It's not an assumption. The sphere IS composed of an infinite number of points.

The sphere --- not a ball --- in the BT theorem is 2D not 3D. In any case, the volume enclosed by the 2D surface is V=4/3 pi R^3. Why do you say "rather than?"

More specifically and to go back to your original claim here:
>it's a contradiction, plain and simple
>because 1V=/=2V
You've proven this wrong by admitting 0 as a valid volume. So 1V equally 2V isn't a "contradiction."

I don't see what that comment had to do with it being Euclidean or not, but no, the axioms used are non-Euclidean.

why are you saying "1V" instead of "one sphere?"
you're making it more complicated than it needs to be

NO NO and NO
the banach-tarski does NOT say that
>the axiom of choice is wrong
>the math is wrong
>the math doesn't describe reality well
>you can actually dissasemble a solid ball and arrange it into two balls
the ONLY thing that banach tarski implies is
>you cannot assign a volume in a meaningful way to ALL subsets of R^3 without causing strange things like the banach-tarski. but given how extremely weird can arbitrary subsets of R^3 be, this is completely okay.