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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#211

Post by zeeboo »

hzhou321 wrote: post #194
cvoss wrote: post #186

Earlier quoted context omitted.

I find it helpful to think of countability in terms of the following game: You have a set of items in mind. You propose a (non-terminating) scheme for listing all the items in the set. An adversary attempts to name any item X, hoping your scheme misses it. However, you then show your scheme does, in fact, get to X after a _finite_ amount of time. The set is said to be countable if you prove that your adversary cannot…

Since it is non-terminating, you didn't really prove every item gets counted after finite time, did you? Refer to my other reply, you asserted a requirement of predefined (describable) counting scheme here. Why that requirement has any relevance here (in the context of infinity)?

If I start at 0 and successively add 1, do you agree that I eventually hit any positive integer you could pick after a finite number of steps? Does that not prove to you that I hit every positive integers? Which one do I not hit?

Re: How real are real numbers? (2004)

#212

Post by drdeca »

jawarner wrote: post #199
D_Alex wrote: post #189

Earlier quoted context omitted.

Yes - I was just thinking the same thing. I have not come to a conclusion one way or another on whether it is possible to de-couple the generating program from the generated programs for the purpose of analysing the proof. It seems that there should be a way to do it... unfortunately I cannot spend more time on this now.

Does there have to be a generating program? The set of all finite programs is countable, so it could not possibly describe the uncountable set of real numbers - this would be a surjection from a countable set to an uncountable set. In particular only the subset of computable numbers [1] can be described by the countable set of finite computer programs. Also, any infinite program either passes through a finite number…

What if the function takes input, and for each input, the program halts, but for every instruction, there is an input such that the program when run on that input will execute that instruction? Would that imply a contradiction? It doesn't seem like it to me. Yeah, no, that should be fine.

Example: consider a language where 0 means "if the input register is at most 0, halt with the answer "no", otherwise, decrement the input register and continue to the next instruction" and 1 means the same thing except that it gives the answer "yes" instead.

Then the fractional part of any real, expressed in binary, would be a program that takes an integer as input, and answers whether that digit of the number is 1. This seems like it would count as a program to me.

So, this would allow there to be uncountably many "programs", only, almost all of them would be impossible for us to refer to specifically.

Hey, this way you could define a uniform measure over programs. Hah.

Wait, is that how Solomonoff induction stuff works? (Of course, with a richer language than what I described)

Re: How real are real numbers? (2004)

#213

Post by Confusion »

mietek wrote: post #129
mbid wrote: post #104

Earlier quoted context omitted.

I don't understand this. For a constructivist, "¬A" means "from A, falsity is derivable". You can derive falsity from the statement "there is a countable enumeration of the real numbers". A diagonal function given such an enumeration is constructively definable, and it is also constructively true that it is not equal to any of the enumerated ones (because 0 != 1 is true constructively). Thus, the diagonal is not enum…

> The interpretation of Cantor's result will depend upon one's view of mathematics. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument#I...

A surjection seems silly: doesn't that imply that some reals would be indistinguishable? Or can you indeed not prove that more than the natural number of reals are distinguishable in constructive mathematics?

Re: How real are real numbers? (2004)

#214

Post by Smaug123 »

Smaug123 wrote: post #20
maverick_iceman wrote: post #16

Earlier quoted context omitted.

Our best theories, General Relativity and the Standard Model, say that the world is a continuum.

Then my statement is vacuously true. (This was intentional, but perhaps a bit obscure.)

I'm a bit confused why this got downvoted. I can understand my parent post being downvoted, but the explanation for my parent post? Is it false?

Re: How real are real numbers? (2004)

#215

Post by mbid »

Confusion wrote: post #213
mietek wrote: post #129

Earlier quoted context omitted.

> The interpretation of Cantor's result will depend upon one's view of mathematics. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument#I...

A surjection seems silly: doesn't that imply that some reals would be indistinguishable? Or can you indeed not prove that more than the natural number of reals are distinguishable in constructive mathematics?

Honestly, the result hinted at in the wikipedia article caught me by surprise as well: There are some flavors of constructive mathematics (notably CZF), that are still consistent if you also add the statement "There is a subset A of the natural numbers such that there is a surjection from A onto the real numbers". Note that this does not imply that you can prove this in CZF, it only means that you cannot disprove it. See http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf prop. 8.2 if you're interested.

Wikipedia says that the Cantor argument shows "no more" than that there is no bijection between the natural numbers and the real numbers. As far as I can tell, it also proves that there is no surjection from the natural numbers onto the reals in every system of constructive mathematics I know of, so I think this is slightly misleading.

What do you mean by "distinguishable"? In constructivism, a set A such that x = y or x != y for all x, y in A is sometimes called decidable. And yes, it is indeed not true that the real numbers are decidable. This would imply that there is an algorithm that decides whether two infinite sequences of ones and zeroes will differ at some point or agree indefinitely, which is pretty much equivalent to solving the halting problem. On the other hand, the natural numbers, integers, rational numbers and every finite algebraic extension of the rational numbers (for example Q[i], "rational imaginary numbers") is decidable.

Re: How real are real numbers? (2004)

#216

Post by drdeca »

hzhou321 wrote: post #197
drdeca wrote: post #195

Earlier quoted context omitted.

> Since you never can finish counting Not what countable means. The meaning meant is the one about having a map from the set to the integers, where no two things of the set map to the same integer. The correspondence is meant as either a full thing, or at least a well defined specification which could be applied to any of the things, if you want to get philosophical about ontology or something. Also, the specificatio…

Why should you get to choose your real number -- specifying more and more digits -- implies an un-exhaustive process, while I am not allowed to do the same? An unfair game will have unfair winners, how would it mean anything? If we both are allowed an un-exhaustive process of specifying what we have, this goes back to the counting game. As we never can finish, how does it make countable (or not)?

The reason that the real number being defined can be defined in terms of the function is because, that is the situation being considered in the statement.

For any way of doing the mapping, there is a real number that the mapping misses. Alternative statement: "there is no such mapping that doesn't miss any of the reals".

This is shown because, given a mapping, I can find a real that the mapping misses.

When one says "for all x, there exists a y such that P(x,y)", the y is allowed to depend on the x.

That's what this is.

Why wouldn't your objection apply to the proof that the halting problem is uncomputable ? The program that the halting checker can't check is defined in terms of the halting checker. Why is that allowed? Because that is what the statement is saying. For any purported halting checker, there exists a program it doesn't decide the halting of.

Similarly here, for any purported bijection between the integers and the reals, there is a real that the purported bijection misses.

I don't know if you are using the word "countable" in the standard way, so I don't know what you mean by that last sentence.

Re: How real are real numbers? (2004)

#217

Post by esoterica »

hzhou321 wrote: post #181
mbid wrote: post #45

Earlier quoted context omitted.

Mathematician here. If you mean to say that you cannot constructively prove "the real numbers are not countable", then you're wrong. As a rule of thumb, you can usually prove negative statements constructively as you would prove them classically. A constructivist would probably state the result more positive (and stronger, constructively): To every countable set M of real numbers, there is a real number not contained…

I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since…

> Since you never can finish counting

You seem to have serious trouble wrapping your head around the concept of words having different meanings in a layman context and a technical context. Just because you can't "count" (in the counting-out-loud-on-your-fingers sense) the naturals doesn't mean they aren't countable in the rigorous, set-theoretic sense. Your entire argument basically boils down to "this word means something different from what I'm used to it meaning in daily conversation".

> I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity.

How narcissistic do you have to be to assume that an entire academic discipline must be wrong just because you can't personally grok it?

> I don't have to describe it, I can hand-wave it just as the way mathematicians hand-waved the infinity.

It isn't hand-waved at all. Just because you don't understand it doesn't mean it's not rigorous. Go read a set theory textbook or take a MOOC or something.

Re: How real are real numbers? (2004)

#218

Post by Animats »

CogitoCogito wrote: post #40

The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers.…

The continuity of real numbers provides a clean theoretical basis for continuity of functions.

Exactly. Real numbers are an invention to make mathematics simpler and cleaner. If you don't have continuous reals, it takes extensive case analysis to prove relatively simple things for, say, a binary floating point representation. It's possible to do that; Boyer and Moore did work like that to formalize floating point and check its correctness. (That work was funded by AMD, after the famous Pentium divide bug.)

Newtonian mechanics assumed that the physical universe is described by real numbers. But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.

Maybe reals should be viewed merely as a useful convenience for analysis, not some fundamental part of mathematics.

"God created the integers. All the rest is the work of Man." - Kronecker

Re: How real are real numbers? (2004)

#219

Post by deepsun »

E6300 wrote: post #101
deepsun wrote: post #67

Earlier quoted context omitted.

What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter? What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure? What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore? I just m…

> What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter? Would you mind producing such a phrase? > What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure? > What about phrases that specify one and the same number for a specific person, but once that person reached age of…

> Would you mind producing such a phrase? Sure: "Current quarter of the Moon". Or "How Giants scored last season". Or "One, if P=NP, zero otherwise".

Well, I see your point, if we limit our phrases to only formal language, then you're right.

Re: How real are real numbers? (2004)

#220

Post by [deleted] »

E6300 wrote: post #101
deepsun wrote: post #67

Earlier quoted context omitted.

What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter? What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure? What about phrases that specify one and the same number for a specific person, but once that person reached age of 40, then he/she is not sure anymore? I just m…

> What about phrases that specify some numbers on Thursdays, and another numbers when Moon is in second quarter? Would you mind producing such a phrase? > What about phrases that some people agree specifies a number, while other people think it's another number, and yet another people just aren't sure? > What about phrases that specify one and the same number for a specific person, but once that person reached age of…

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