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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#111

Post by enugu »

upquark wrote: post #96
enugu wrote: post #56

Earlier quoted context omitted.

> In my view, every real number is well-defined... How so? The set of definable numbers in any formal langauage might not be clear concept. But you are making a stronger statement. For any given language, like for instance ZFC, we can say that definable numbers are a countable subset. Hence measure zero.

Then we mean different things by define. I am saying the set R (with all its elements) is an uncontroversial, well-defined construction within ZFC. I am leaving out any linguistic or Turing-computability aspects out of this, and people try to bring it back in, mixing computability with definability. For instance, Chaitin's constant is a perfectly well-defined number, albeit uncomputable by construction: https://en.wi…

Defining the set of real numbers is very different from defining all real numbers. Yes, Chaitin's constant is defined(with a computable system as a parameter).

But that's the point - we cant produce such a definition for almost all reals.

Re: How real are real numbers? (2004)

#113

Post by Smaug123 »

gjm11 wrote: post #106
Smaug123 wrote: post #18

Earlier quoted context omitted.

Nitpick: the infinite countable union of countable well-ordered sets is countable. This statement is immune to the failure of Choice.

The formulation I prefer: a countable union of counted sets is countable. (Any countable set has a well-ordering, because a bijection with the natural number gives you one in an obvious way. The trouble is that you need well-orderings for all of them together. The unusual term "counted" emphasizes that we need the actual "countings" to do the job, whereas for me "well-ordered" is sufficiently commonplace that it does…

In my mind, "well-ordered" is fully distinct from "well-orderable" :) but "counted" is unambiguous, you're right.

Re: How real are real numbers? (2004)

#115

Post by tromp »

GregBuchholz wrote: post #61
mcherm wrote: post #51

Earlier quoted context omitted.

Consider the set of all real numbers which you can actually specify IN ANY FASHION AT ALL, so that the person writing a paper about it and the person reading the paper are talking about the same number. This includes easy ones like "3" or "Square root of Pi". It includes oddballs like Chaitin's constant -- the probability that a randomly constructed program will not get stuck in an infinite loop (for some particular…

>Chaitin's constant -- -- which is so ornery a number that we cannot (even in THEORY) figure out a single digit of it (other than its being between 0 and 1). You might be interested in: https://www.cs.auckland.ac.nz/~cristian/Calude361_370.pdf "A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a com…

Unfortunately, that is not a Chaitin Omega, since the notion of program length in that paper is flawed. In particular, it counts a 0 or a 1 provided as binary data in the program as adding 7 bits of length to the program. Later papers by Calude fix this problem, while reducing the number of computed bits to 43.

Re: How real are real numbers? (2004)

#116

Post by klodolph »

AnimalMuppet wrote: post #85
klodolph wrote: post #52

Earlier quoted context omitted.

The Standard Model has discrete energy levels but continuous spacetime.

ELI5: Where does the Standard Model say anything about spacetime being continuous? I thought it was only about what particles exist.

It is definitely far, far more than just what particles exist. The Standard Model describes how those particles interact with via the three forces other than gravity. The particles themselves are modeled using quantum fields and the properties of particles arise from operators on those fields. Those operators don't work unless the field is continuous. The operators will have a spectrum which describes how the corresponding observable is quantized.

That's not really ELI5, but remember learning derivatives in calculus? Just like you can't take the derivative of a function which isn't continuous, you can't use the Standard Model if spacetime isn't continuous.

Re: How real are real numbers? (2004)

#117

Post by tossaway322 »

AnimalMuppet wrote: post #42
dbcurtis wrote: post #26

1. Given any two real numbers on the real number line, you can find another real number between those two points. 2. The Planck length is the smallest unit of distance with any meaning. 3. The universe has finite diameter. Discuss. 4. For extra credit: Given 2 and 3, above, it follows that both the diameter and circumference of the universe can be expressed in Planck lengths as integers with a finite number of digits…

1-3: You can define a (mathematical) real number that cannot be interpreted as a position in the universe that can be physically realized, taking the Planck Length into account. 4: At the precision of the Planck Length, you have two integers that are the closest physically-meaningful values whose ratio approximates pi. In both cases, you're confusing mathematical abstractions with what is physically realizable in a d…

The article unfortunately does not address the question of whether current mathematical analysis is an appropriate framework for a description of space/time. We are, in the end, dealing with elements "smaller than" (if that's the right phrase!) the Planck length. Of course, you can choose to ignore that and use what's already been provided, but to do so is "whistling past the graveyard". This has ramifications for all of string theory, quantum gravity et al.

Re: How real are real numbers? (2004)

#118

Post by CuriouslyC »

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.…

Unfortunately, physicists seem to believe that infinity and the infinitesimal are real things.

Re: How real are real numbers? (2004)

#119

Post by upquark »

enugu wrote: post #111
upquark wrote: post #96

Earlier quoted context omitted.

Then we mean different things by define. I am saying the set R (with all its elements) is an uncontroversial, well-defined construction within ZFC. I am leaving out any linguistic or Turing-computability aspects out of this, and people try to bring it back in, mixing computability with definability. For instance, Chaitin's constant is a perfectly well-defined number, albeit uncomputable by construction: https://en.wi…

Defining the set of real numbers is very different from defining all real numbers. Yes, Chaitin's constant is defined(with a computable system as a parameter). But that's the point - we cant produce such a definition for almost all reals.

> Defining the set of real numbers is very different from defining all real numbers.

I'm saying that ^ sentence makes no sense to me, I don't know how to parse it formally. If you start talking about the set of "definable" numbers (not computable, but specifically "definable"), I believe you're gonna run into paradoxes as it's an ill-defined concept, similar (in spirit) to "all integers described under 100 words". In fact, the linked article actually talks about it in 2.3.

> For any given language, like for instance ZFC, we can say that definable numbers are a countable subset. Hence measure zero.

If I can describe a set of objects, then we're all set as far as I'm concerned (mathematically speaking). Being able to efficiently construct individual elements of this set using Turing machines or other computational devices is an orthogonal problem.

Also, I don't think having only countable number of utterances in ZFC precludes you from having well-defined uncountable sets described in that system (quite obviously, for any set S take 2^S which is very well-defined).

Re: How real are real numbers? (2004)

#120

Post by CuriouslyC »

gertef wrote: post #80
7373737373 wrote: post #59

Earlier quoted context omitted.

Really? Interesting. I would have expected it to be something like epsilon.

The measure of epsilon is 0. Proof: what else could it be? If it's not 0, there's a smaller number, contradicting your (intuitive) definition of epsilon.

Wouldn't it be more accurate to describe epsilon as an infinitesimal?
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