Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Banach-Tarski and the Paradox of Infinite Cloning

quantamagazine.org

Post reply on HN
Print view View original

121–130 of 148 posts

Re: Banach-Tarski and the Paradox of Infinite Cloning

#121

Post by wizzwizz4 »

danparsonson wrote: post #117
layer8 wrote: post #112

Earlier quoted context omitted.

Integrals over a finite interval can have (and often do have) a finite size even though the interval contains an infinite number of points, with an infinite number of different values at those point.

Right, because the integral of a function is not a straight sum of values of that function evaluated for every number in the interval; the integral of y=x dx for 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons have a fixed energy, so cramming an infinite number of them into a finite space necessarily requires infinite energy.

I'd argue you can't cram any electrons into any finite space without infinite energy. But we're not really talking quantum physics, here.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#122

Post by layer8 »

danparsonson wrote: post #117
layer8 wrote: post #112

Earlier quoted context omitted.

Integrals over a finite interval can have (and often do have) a finite size even though the interval contains an infinite number of points, with an infinite number of different values at those point.

Right, because the integral of a function is not a straight sum of values of that function evaluated for every number in the interval; the integral of y=x dx for 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons have a fixed energy, so cramming an infinite number of them into a finite space necessarily requires infinite energy.

Well, I was referring to the electron field, not to electrons. According to QFT, particles are excitations of an underlying quantum field. It’s the field that is fundamental, not the particle. See e.g. [0]. And those fields are continuous, not discrete, i.e. can only be described by an infinite number of points and values.

[0] https://www.quantamagazine.org/what-is-a-particle-20201112/

Re: Banach-Tarski and the Paradox of Infinite Cloning

#123

Post by DerekL »

perl4ever wrote: post #119

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. In the paragraph on nonstandard analysis in the Wikipedia page on infinity, it says: "Th…

> 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.

They really aren’t connected. The first statement (the positive integers can be partitioned into two sets, each of which has the same size as the original set) follows from the usual axioms of set theory (ZF), while the Banach–Tarski paradox cannot be proven to work without the Axiom of Choice or a similar axiom.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#124

Post by robertlagrant »

hprotagonist wrote: post #51
danparsonson wrote: post #34

Earlier quoted context omitted.

OP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.

it's OK. electrons don't "exist" discretely, either. At best, when you "hand me 7 electrons", you're directing me towards the fat part of 7 probability distributions, so we're back to math again...

But is that not the locations that are inexact, not the quantity of electrons?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#126

Post by danparsonson »

layer8 wrote: post #122
danparsonson wrote: post #117

Earlier quoted context omitted.

Right, because the integral of a function is not a straight sum of values of that function evaluated for every number in the interval; the integral of y=x dx for 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons have a fixed energy, so cramming an infinite number of them into a finite space necessarily requires infinite energy.

Well, I was referring to the electron field , not to electrons. According to QFT, particles are excitations of an underlying quantum field. It’s the field that is fundamental, not the particle. See e.g. [0]. And those fields are continuous, not discrete, i.e. can only be described by an infinite number of points and values. [0] https://www.quantamagazine.org/what-is-a-particle-20201112/

Great read, thanks. I think I see the point you're making now - any continuity necessarily requires an infinity by virtue of being continuous.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#127

Post by danparsonson »

wizzwizz4 wrote: post #121
danparsonson wrote: post #117

Earlier quoted context omitted.

Right, because the integral of a function is not a straight sum of values of that function evaluated for every number in the interval; the integral of y=x dx for 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons have a fixed energy, so cramming an infinite number of them into a finite space necessarily requires infinite energy.

I'd argue you can't cram any electrons into any finite space without infinite energy. But we're not really talking quantum physics, here.

I'm not sure what you mean - any battery you have would seem to contradict you - would you elaborate?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#128

Post by Smaug123 »

bigbillheck wrote: post #120
ncallaway wrote: post #64

Earlier quoted context omitted.

> The Banach-Tarski theorem is a consequence of things we want Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want. You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axio…

> Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” OK, which axiom do you want to replace?

It's not about the axioms, which are perfectly fine as set theory. It's about the encoding of geometry in set theory, for which there are many options, some with nicer properties than others. Cf the original link in the thread, https://twitter.com/andrejbauer/status/1428471658088738818, which argues that you should model locales rather than points (and I guess if your foundation is set theory then you should use set theory to model those locales, though it sounds rather painful to me, a layman).

Re: Banach-Tarski and the Paradox of Infinite Cloning

#129

Post by ithkuil »

tsimionescu wrote: post #101
ithkuil wrote: post #90

Earlier quoted context omitted.

I'm not talking about proving that there exist infinite things. I'm talking about using the abstract concepts of infinity as a useful mathematical tool to produce predictions. Notable example: calculus

Sure, but calculus makes infinity sufficient, but not necessary for describing the physical world. Integers, rationals, and apparently complex numbers (presumably those with rational components) are actually necessary for describing the physical world, given our current understanding. Irrational numbers and infinities are extremely useful, but not strictly necessary.

Sorry, I'm not getting it. A large use case for complex numbers is describing things that rotate, literally or not, like oscillations, waves etc. Trigonometry lies deeply in that math and the irrational number pi pops out left and right. An approximation of pi wouldn't cut it, would it?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#130

Post by tsimionescu »

ithkuil wrote: post #129
tsimionescu wrote: post #101

Earlier quoted context omitted.

Sure, but calculus makes infinity sufficient, but not necessary for describing the physical world. Integers, rationals, and apparently complex numbers (presumably those with rational components) are actually necessary for describing the physical world, given our current understanding. Irrational numbers and infinities are extremely useful, but not strictly necessary.

Sorry, I'm not getting it. A large use case for complex numbers is describing things that rotate, literally or not, like oscillations, waves etc. Trigonometry lies deeply in that math and the irrational number pi pops out left and right. An approximation of pi wouldn't cut it, would it?

Actually, for any practical use case or possible observation, there is an approximation of Pi that is good enough. The ancient Egyptians apparently did quite well in their architecture approximating Pi as 22/7 (3.(1428571)).

You only need the exact number Pi if you want to measure something like the ratio between the length of a perfect circle and its radius with infinite precision. But you can't be sure your measurement has infinite precision with a finite number of measurements, and so you can't observe the difference between a perfect circle and a many, many sided X-agon, even if perfect circles do exist in the geometry of the universe.

Just as a fun aside, even if perfectly circular shapes do exist, it's unlikely that perfect circles would exist in physical objects - at best, you would have ellipses, and there is no (known?) way to compute the ratio between the length of an ellipse and the properties of its foci.

Post reply on HN