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The point is that there are more real numbers than possible finite descriptions, so some real numbers must be indescribable. Related (and more precisely defined) concepts include (non)definable and (non)computable numbers, http://en.wikipedia.org/wiki/Definable_number and http://en.wikipedia.org/wiki/Computable_number respectively.
Of course you can describe them. But describe them in terms of what? I think that's the key. Let's say you have a pencil and you want to describe its length, which will be a unit multiplied by a number. If I were to try to describe the true length in meters it will be: 0.0178(...) * meters. This would be an "indescribable" number the author talks about.
However, I can just describe it in terms of itself and call it one pencil long - we use the pencil itself as the basis of the unit so the number we multiply the unit to is just: 1 * pencil long.
This is good. I really liked it. But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but should there be ? Given the balloon example, he makes the case that as an imaginary god: "...stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more dig…
What you describe has to do with mathematical constructivism[0], and also with the Axiom of Choice, and it's really complicated (to me, I'm not a mathematician, and I only sort-of get it). You may have heard about the Banach-Tarski paradox[1], which tells you that if you assume "Real numbers" are actually reality, and the Axiom of Choice, you can divide a sphere into five pieces (one of which is just a point) and rea…
Yes, but in that construction, not all the sets are Lebesgue measurable. It's a curious fact, already true just in one dimension, i.e., the reals, that we can't have a nice way to assign a length, area, or measure to all the subsets. Instead there have to be some subsets to which we can't assign a length, etc. Here the usual proof on the reals does use the axiom of choice!
With measurable sets, which already can be wildly bizarre, can't do anything like Banach-Tarski.
So, suddenly in this thread, instead of hackers, we are all turning into undergraduate pure math majors!
They are real numbers, for which we have just proven it is impossible to find a description that will match them. We have proven that no description will ever describe them. Of course you can describe them. But describe them in terms of what? I think that's the key. Let's say you have a pencil and you want to describe its length, which will be a unit multiplied by a number. If I were to try to describe the true lengt…
Decimal system :)
Interesting philosophical point though.
The interesting thing here is that it's much harder to put this problem properly into mathematical terms than it is to solve it. The whole insight here is that you a "description" of number is just some finite sequence of symbols from a finite alphabet. Now, if you understand why cardinality of continuum is greater than aleph null, it's totally straightforward to show that there are only countably many descriptions,…
This is good. I really liked it. But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but should there be ? Given the balloon example, he makes the case that as an imaginary god: "...stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more dig…
What you describe has to do with mathematical constructivism[0], and also with the Axiom of Choice, and it's really complicated (to me, I'm not a mathematician, and I only sort-of get it). You may have heard about the Banach-Tarski paradox[1], which tells you that if you assume "Real numbers" are actually reality, and the Axiom of Choice, you can divide a sphere into five pieces (one of which is just a point) and rea…
From there please take a deep look into topos theory and you'll discover again that AC is just one choice of properties that make for a meaningful "foundation of mathematics (of a kind)" and a not particularly remarkable one at that.
They are real numbers, for which we have just proven it is impossible to find a description that will match them. We have proven that no description will ever describe them. Of course you can describe them. But describe them in terms of what? I think that's the key. Let's say you have a pencil and you want to describe its length, which will be a unit multiplied by a number. If I were to try to describe the true lengt…
Describe them (completely) in terms of any finite series of symbols.
Some numbers can be described as "7". Some as "pencil length". Some as "the fourth zero of the third Bessel function of the first kind". These are all describable numbers.
Yet there are vastly more numbers that cannot be described in this way -- vastly more numbers that we miss than that we hit.
Earlier quoted context omitted.
> In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge. Indeed, but by the same argument as the author's, there are more Cauchy sequences than can be described, so it looks like the real numbers are much bigger than necessary to do mathematics :)
No, to "do mathematics", e.g., show that the Riemann integral exists, that e and pi exist, etc., we want completeness. Then we are done: The reals are the only complete Archemedean ordered field! So, we have no choice!
Unlike reals, computable numbers are countable, and are all describable. (That is, there are aleph-null of them, so there are exactly as many computable numbers as natural numbers, and fewer than real numbers). And while almost all real numbers aren't computable (by the argument in the article), essentially every number you'd ever stumble upon in a math class is.
I prefer computable numbers to reals because I have trouble accepting that a thing exists when it by principal cannot be defined.
https://en.wikipedia.org/wiki/Computable_number
http://www.amazon.com/Computable-Calculus-Oliver-Aberth/dp/0...
For me, I like to wonder about the transition point in history where man went from not using numbers to using them. What prompted this? Were numbers used first to indicate order (e.g. my first born son) or quantity? How did someone first teach the idea of numbers to another?
I'm sure there are endless books and articles written on this topic, but I've never had the desire to see what others have to say on the subject. Sometimes it's just nice to let my mind relax and drift to such questions; I sometimes recapture the feeling of wonder and exploration that I had originally...
Sigh...
Also (hehe): quantum theory, everything is quantized. So as far as I understand physics, indescribable numbers aren't god's numbers, they're actually our own invention.