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I have been trying to extend the countable set of algebraic numbers, by adding a countable amount of transcendental numbers (so that the resulting set is also countable).

Now, of course I could simply add some transcendental number or numbers while keeping the resulting set countable, for example:

  • $\pi$
  • All integer multiples of $\pi$
  • All rational multiples of $\pi$
  • All algebraic multiples of $\pi$

But I am looking for a more elaborate way to do it (let's leave the non-mathematical definition of "elaborate" aside for now, because it's not the main point in this context).

In any case, I have finally managed to come up with what I believe to be a countable extension, but it left me with a few (practical as well as philosophical) questions.

Consider the set of all the numbers that can be calculated using a formula which contains a finite amount of:

  • Natural numbers
  • The basic arithmetic operations ($+,-,\times,\div$)
  • The infinite-repetition operator (e.g., $\sum\limits_{n=1}^{\infty}$ or $\prod\limits_{n=1}^{\infty}$ or continued-fraction)

In fact, we only need $\left[1,+,-,\sum\limits_{n=1}^{\infty}\right]$ but I would like to keep this definition easier to understand.

In any case, this set contains all the algebraic numbers, as well an infinite amount of transcendental numbers (including $\pi$, $e$, etc).

I'm pretty sure that this amount is countable, since we are using a finite amount of symbols in order to represent every element in the set, but I'm not sure how to prove it.

From a philosophical point of view, it seems like the uncountable "majority" of real numbers are in fact values that we can never "lay our hands on" - some sort of "unreachable numbers in another dimension" (like particles that go undetected in every sensor that we could possibly build, if I may use metaphors here).

My questions are:

  • Is the set above indeed countable as I speculate?
  • If no, then where did I go wrong claiming that it was?
  • If yes, then:
    • How can we prove it?
    • What other research has been conducted on this?
    • What is "the mathematical point" in declaring that there are uncountably many real numbers, when the uncountable part of them contains values that we can never describe in any conceivable manner?

I have previously posted some of the insights above in response to this question (though I had them in mind before I bumped into it), which follows this question.

The latter essentially asks the same thing that I am asking in the last section above, but without the context that I am providing here in order to "justify" it.

Thank you very much.

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barak manos
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    It seems to me that what you are trying to reinvent, is the concept of Computable number https://en.wikipedia.org/wiki/Computable_number If I understand you ideas, all numbers you describe are computable, and there are only countably many of the latter – Peter Franek Dec 29 '14 at 11:20
  • @PeterFranek: I can't see how any irrational number could be accurately computed and represented on a numerical base. – barak manos Dec 29 '14 at 12:00
  • a computable number is such that, roughly speaking, there exists a (possibly never-terminating) algorithm that will compute its decimal expansion (in infinite time). Most numbers you are familiar with (such as $\pi, e..$) are computable. But, of course, there are only countably many such numbers as there are only countably many algorithms. – Peter Franek Dec 29 '14 at 12:04
  • @PeterFranek: How can the decimal expansion of $\pi$ be computed in finite time if it is infinite and non-periodic??? – barak manos Dec 29 '14 at 12:06
  • Barak Manos: not in finite time. This is a standard concept in theoretical computer science. Look at the wiki link. – Peter Franek Dec 29 '14 at 12:07
  • @PeterFranek: Oops, sorry, I read "finite time". Thanks. – barak manos Dec 29 '14 at 12:08
  • @PeterFranek: In your first comment, you say that there are indeed a countable amount of such numbers (as I speculate), which comes in contradiction with Asaf Karagila's answer below. Can you please refer to that? Thank you. – barak manos Dec 29 '14 at 12:10
  • No, in Asafs answer, he just made the trivial observation that any real number can be expressed as a countable sum of rationals. But it doesn't mean that there is an algorithm (finite code) that would generate those "partial sums". I really think that "computable number" is what you talk about: all rationals, algebraics, $\pi$, $e$, "closed formulas" etc are computable. – Peter Franek Dec 29 '14 at 12:13
  • @PeterFranek: Yes, I just read your link. I believe it accurately describes what I was trying to describe in my question. I also added it in my answer to this question, as I figured it might be relevant to it. Thank you for the enlightening information :) – barak manos Dec 29 '14 at 12:31

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Once you allow infinite sums, you already have uncountably many different sums to work with, and it will generate you all the real numbers. Even if you only limit your infinite sums to sums of rational numbers, every real number is the infinite sum of rational numbers.

Disallowing infinite sums will keep you at the countable level, of course, but it will simply be the field generated by adding those transcendental numbers. It's not particularly special or interesting. Just a field.

Asaf Karagila
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  • If what is inside the sum is a closed formula in $n$ (in order to be able to write in "in practice"), this gets us back to countably many different sums. It is probably what the OP has in mind when adding this operator. – Denis Dec 29 '14 at 11:34
  • Yes, if you want only sequences which are computable, or definable somehow, then it's a different story. – Asaf Karagila Dec 29 '14 at 11:37
  • Thank you, but can you please explain where have I gone wrong claiming that we are using a finite amount of symbols in order to represent every element in the set? The symbol "$\infty$" is just one symbol. I did not say that we should actually expend the infinite sum or product. – barak manos Dec 29 '14 at 12:03
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    @barak: While $\sum_{n=1}^\infty a_n$ has indeed only finitely many symbols (seven according to my count), the $a_n$ is sort of a placeholder, and what happens is that you take a countable subset and sum its elements in a particular order. But the countable set constitutes, in fact, as infinitely many parameters, from which you can define each real number. Of course, you can circumvent this in several ways, but you need to be very accurate in what you're saying. And for this, I believe, you need to first improve your skills in logic and foundations. – Asaf Karagila Dec 29 '14 at 14:36