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The Axiom of Choice Is Wrong (2007)

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Re: The Axiom of Choice Is Wrong (2007)

#131

Post by cygx »

naasking wrote: post #129
Jweb_Guru wrote: post #120

Earlier quoted context omitted.

That doesn't prove anything about the continuity (or lack thereof) of spacetime, just indicates that energy is discretized. Also has no bearing on the original statement about infinity. It's also worth pointing out that the original statement is nonsensical because the theory of real closed fields doesn't suffer from incompleteness in the first place, so there's no need to "save" it from paradox the way you have to s…

The Bekenstein bound, as a starting point, addresses the more general point that all of known physics needs only computable abstractions. Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen, which does address your specific question about whether space is truly continuous.

Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen

I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.

Re: The Axiom of Choice Is Wrong (2007)

#132

Post by naasking »

cygx wrote: post #131
naasking wrote: post #129

Earlier quoted context omitted.

The Bekenstein bound, as a starting point, addresses the more general point that all of known physics needs only computable abstractions. Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen, which does address your specific question about whether space is truly continuous.

Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.

We simulate quantum mechanics on classical computers all the time. Superposition requires exponentially more resources in some cases, but it doesn't somehow make QM non-discrete.

Re: The Axiom of Choice Is Wrong (2007)

#133

Post by sn41 »

andybak wrote: post #52
IsaacL wrote: post #36

Earlier quoted context omitted.

Are there any good books on this topic?

Yes. An uncountable number are in the Library of Babel. (actually - I'm guessing there's actually a countable number in the Library of Babel but it didn't read quite so amusingly that way. In any case - all the ones I flicked through were trash.) Edit - The Library of Babel is actually finite isn't it? Fixed alphabet and fixed book length? It's a while since I read it.

The Library of Babel is finite if the books are unique. Interestingly, Borges does mention that one of the books in the library must be an index of the other books. This is similar to the notion of a universal computably enumerable language.

However, I doubt that Borges' claim is accurate. If the set of programs is finite, then I think there cannot be a comprehensive index of all programs. A finite set is a regular language, and there is no universal regular language in the set of regular languages.

Re: The Axiom of Choice Is Wrong (2007)

#134

Post by TheCoelacanth »

waqf wrote: post #88
Tloewald wrote: post #60

Earlier quoted context omitted.

As soon as you start treating the axiom of choice as a superpower for a conscious being you are in conceptual la la land. How do I guess the color of my hat if there's an uncountable number of possible colors? Doesn't that mean that communicating the value of that color involves transmitting an infinite amount of informtion?

Quick, tell me your favourite number between 0 and 1. How did you do that? Weren't there an uncountable number of possible numbers? Ah, you may say, but I obviously wasn't going to choose one with an infinite information content, so all but countably many possible numbers had probability 0. Which is true. But in fact it's true that whenever you have a probability measure on an uncountable space then all but countably…

But the axiom of choice applies to all sets, not just ones that you can easily choose a number from. For instance, what is your favorite non-computable number between 0 and 1?

Re: The Axiom of Choice Is Wrong (2007)

#135

Post by cygx »

naasking wrote: post #132
cygx wrote: post #131

Earlier quoted context omitted.

Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.

We simulate quantum mechanics on classical computers all the time. Superposition requires exponentially more resources in some cases, but it doesn't somehow make QM non-discrete.

The point is that according to ordinary QM, any of the states you can construct via superposition are physical.

Eg for the qubit, the state space is the Bloch sphere, which is continuous.

Re: The Axiom of Choice Is Wrong (2007)

#136

Post by paulddraper »

CJefferson wrote: post #3

As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…

> If the set of all numbers any can express in any sensible way, and the solution to any problem any could ever have is countable

This is trivially true, since there are only countable (finite) expressions and problems.

> why do we need the other uncountables?

You can't do calculus without reals.

http://math.stackexchange.com/questions/1880741/why-cant-cal...

And calculus is pretty important (e.g. for any branch of physics ever).

Re: The Axiom of Choice Is Wrong (2007)

#138

Post by naasking »

cygx wrote: post #135
naasking wrote: post #132

Earlier quoted context omitted.

We simulate quantum mechanics on classical computers all the time. Superposition requires exponentially more resources in some cases, but it doesn't somehow make QM non-discrete.

The point is that according to ordinary QM, any of the states you can construct via superposition are physical. Eg for the qubit, the state space is the Bloch sphere, which is continuous.

Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality.

Furthermore, quantum computing and classical computing are known to have the same computational power, just different computational complexity, ie. any quantum system can be simulated by a classical system with exponential slowdown, at worst.

There are plenty of papers exploring this territory [1,2,3] if you want further details. There are older references but they're not easily available online.

[1] Computability in Quantum Mechanics, 1995, https://philpapers.org/rec/MYRCIQ

[2] Effectively calculable quantum mechanics, 2015, https://arxiv.org/pdf/1508.03879.pdf

[3] Constructive physics, https://arxiv.org/pdf/0805.2859.pdf

Re: The Axiom of Choice Is Wrong (2007)

#139

Post by red75prime »

TheCoelacanth wrote: post #134
waqf wrote: post #88

Earlier quoted context omitted.

Quick, tell me your favourite number between 0 and 1. How did you do that? Weren't there an uncountable number of possible numbers? Ah, you may say, but I obviously wasn't going to choose one with an infinite information content, so all but countably many possible numbers had probability 0. Which is true. But in fact it's true that whenever you have a probability measure on an uncountable space then all but countably…

But the axiom of choice applies to all sets, not just ones that you can easily choose a number from. For instance, what is your favorite non-computable number between 0 and 1?

Inverse of Kolmogorov complexity of thirteenth bit-string with uncomputable Kolmogorov complexity, of course.

Re: The Axiom of Choice Is Wrong (2007)

#140

Post by Tloewald »

waqf wrote: post #88
Tloewald wrote: post #60

Earlier quoted context omitted.

As soon as you start treating the axiom of choice as a superpower for a conscious being you are in conceptual la la land. How do I guess the color of my hat if there's an uncountable number of possible colors? Doesn't that mean that communicating the value of that color involves transmitting an infinite amount of informtion?

Quick, tell me your favourite number between 0 and 1. How did you do that? Weren't there an uncountable number of possible numbers? Ah, you may say, but I obviously wasn't going to choose one with an infinite information content, so all but countably many possible numbers had probability 0. Which is true. But in fact it's true that whenever you have a probability measure on an uncountable space then all but countably…

When we think of numbers between 0 and 1 we aren't thinking of Real numbers.

Name a transcendental number between 0 and 1. Which one springs to mind? Maybe, if you're lucky you can come up with some famous transcendental number and "scale" it to lie within the interval (pi over ten!).

The numbers we live with are a finite dimensional vector field over a bounded subset of algebraics - not even countable.

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