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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#121

Post by gerdesj »

Richard's Paradox seems a bit shaky to me (p4): "Since all possible texts in French can be listed or enumerated"

Unless I have completely missed the point then he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on.

You can easily squint hard enough to generalize to all texts in all languages, now, earlier and possible then allow that grammar, spelling and so can be pretty slack. Now translate that lot into numbers in some way (a bunch of IT bods should be able to manage that!) To be honest French on it's own is probably more than enough.

"How very embarrassing! Here is a real number that is simultaneously nameable yet at the same time it cannot be named using any text in French."

The very act of naming the number (in French) constructs the French text that adds to the set of possible French texts.

I think that the set of possible French texts is exactly as large as the set of reals. So is the set of all language texts and that the "paradox" is merely trying to use the Cantor argument backwards.

Re: How real are real numbers? (2004)

#122

Post by hn_throwaway_99 »

textfile wrote: post #65
option wrote: post #55

Note that the probability of randomly picking a rational number from [0,1] is exactly 0. That is, with P=1 you will end up with an irrational number, not representable in any modern computer.

This is a mind-blowing. So rational numbers are an invention of humans. That is to say, natural numbers simply don't exist unless and until they are explicitly defined. Am I understanding correctly?

Someone else responded to me in a recent Hacker News discussion that really clarified this in my head: A real number essentially has an infinite number of digits after the decimal place - the difference between a rational and an irrational number is that the digits end up in a repeating pattern in a rational number (you can think of rationals that terminate as really having an infinite number of zeros, e.g. 1.5 as 1.5000000...). Thus, to randomly pick a real number, you randomly select digits an infinite number of times after the decimal. Clearly there is no way a random process will produce an infinite number of repeating digits, thus the probability of picking a rational would be 0.

Re: How real are real numbers? (2004)

#123

Post by CuriouslyC »

maverick_iceman wrote: post #16
Smaug123 wrote: post #10

Earlier quoted context omitted.

But in our universe, if it's discrete on a macroscopic level, then it's either inherently a natural number (it can be counted), or it's intelligently created.

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

General relativity and the standard model also produce infinite values, which to me is a pretty sure sign your model is broken. Viewing black holes as infinitely dense hasn't yet caused us problems in terms of predictions (but they're still a bit wild west area), and they fixed the standard model with renormalization (which seems like a bit of a hack). They're good enough to be useful but I'm pretty sure everyone expects them to be radically reformulated at some point.

Re: How real are real numbers? (2004)

#124

Post by surement »

gerdesj wrote: post #121

Richard's Paradox seems a bit shaky to me (p4): "Since all possible texts in French can be listed or enumerated" Unless I have completely missed the point then he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on. You can easily squint hard enough to generalize to all texts in all languages, now, earlier and possible then allow that grammar, sp…

> I think that the set of possible French texts is exactly as large as the set of reals.

How? The set of French texts is countable and the set of reals is uncountable.

Re: How real are real numbers? (2004)

#125

Post by joe_the_user »

gerdesj wrote: post #121

Richard's Paradox seems a bit shaky to me (p4): "Since all possible texts in French can be listed or enumerated" Unless I have completely missed the point then he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on. You can easily squint hard enough to generalize to all texts in all languages, now, earlier and possible then allow that grammar, sp…

It depends how you conceptualize things.

If one takes natural language as a means to definite sets, you quickly get a plethora of paradoxes, for example Russel's paradox("Takes the sets of all sets that don't contain themselves. Does that contain itself?"). So if one takes "natural language" as one's system of defining set, one has to assume it's inconsistent and any statement is provably true and false. Thus "Richard's Paradox" is in the same boat as all statements in our "system of natural language".

That said, I think the proof in the text falls apart in another way - it neglects the distinction made in Skolem's Paradox. The Löwenheim–Skolem [2] theorem show that any system definable with a finite alphabet has a countable model. This is only an apparent paradox because this model is only countable when "viewed" from outside the model, within the model, it is possible to have ostensibly uncountable sets. So ones could certainly haves a countable model of the real numbers while the real numbers themselves remained uncountable as "countable" and "uncountable" were defined in the countable model.

[1] https://en.wikipedia.org/wiki/Skolem%27s_paradox [2] The https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...

Re: How real are real numbers? (2004)

#126

Post by bo1024 »

gerdesj wrote: post #121

Richard's Paradox seems a bit shaky to me (p4): "Since all possible texts in French can be listed or enumerated" Unless I have completely missed the point then he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on. You can easily squint hard enough to generalize to all texts in all languages, now, earlier and possible then allow that grammar, sp…

> he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on.

Nope, because he used the diagonalization technique to make sure the new text was not in the original set, which supposedly contained all texts.

Re: How real are real numbers? (2004)

#127

Post by stcredzero »

whatshisface wrote: post #3

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…

I think Borel has confused names with things.

Pedantic positivist?

Re: How real are real numbers? (2004)

#128

Post by bo1024 »

whatshisface wrote: post #3

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…

You might be reading too much into the phrase "mathematical fantasies". This is not saying Borel claims they literally do not exist. More that they are not "accessible" or "usable".

Re: How real are real numbers? (2004)

#129

Post by mietek »

mbid wrote: post #104
btilly wrote: post #58

Earlier quoted context omitted.

Yeah, I'm a trained mathematician as well. A constructivist would state the result in a variety of ways. But none of them would involve a potentially self-referential construction based on the absolute truth of an infinite number of statements. Which really does rule out Cantor's argument.

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...

Re: How real are real numbers? (2004)

#130

Post by CogitoCogito »

CuriouslyC wrote: post #118
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.

What do you mean by this statement?
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