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Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question

Part 1:

Are there real numbers that can't be defined. What I mean by defined is that there exists some sequence of a finite set ( i.e. $0$-$9$,$\cdot$, $-$, $\sqrt{}$, $\pi$, etc.) that describes the value. This sequence need not be finite itself. Only that its representation exists and is unique.

Part 2:

Given all the numbers that meet part 1, are there any for which there does not exist some algorithm to arrive at it's value.

Asaf Karagila
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8bitwide
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    MOST real numbers are what you say. – Edward Jiang Oct 06 '14 at 23:56
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    See this answer at MathOverflow. – Vladimir Reshetnikov Oct 07 '14 at 00:05
  • From your definition, there are a countable number of definable numbers. – Taladris Oct 07 '14 at 00:07
  • @Vladimir Reshetnikov thanks, while very very interesting, a lot of that was somewhat over my head. But it appears that it is leaning toward the answer I was expecting. That while the answer to my question may be "none", it hasn't been proven one way or the other. Or am I just confusing myself?? – 8bitwide Oct 07 '14 at 00:13
  • All fields and aspects of mathematics come down to and are either derived or expanded from this expression: 1+1 = 2... where the addition of a single entity upon itself gives replication in the form of a linear transformation and in this specific case because the applied operation is addition which is linear happens to be a 1 dimensional linear transformation specifically being horizontal translation. From this expression and operation we have already introduced without even knowing it the following: multiplication, powers, integration, polynomials, trigonometric, geometry and their inverses. – Francis Cugler Mar 03 '17 at 09:40

1 Answers1

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Usually definablility refers to a fixed language (often first-order, but not necessarily), in a fixed interpretation for the language (also called a structure). The meaning is that there is a formula (which is by definition a finite string of characters) which is only satisfied by a unique element of the structure.

Just writing digits, multiplication, "etc." doesn't quite tell us what the language is. Moreover, by allowing an infinite definition you easily can write each real number as an integer along with an infinite decimal expansion. You don't even have to appeal to $\sqrt{}$ or $\pi$ or the "etc." part. Just note that a decimal expansion fully decides the real number's value. And although some real numbers have several decimal expansions (e.g. $1=1.0=1.000=0.\overline{999}$) remember that $2=1+1=3-1=4-2=5-3$ and so on. So having several definitions is not a big deal.

Finally, algorithm is a finite sequence from a finite possible list of operations. So there are only countably many algorithms to begin with, therefore the majority of real numbers are such that there is no algorithm to compute them (or their decimal expansions, if you prefer to think about it that way). Of course that you can talk about infinitely many possible operations, or infinitely long computations. The answer may change, depending on the exact definitions and set theoretical intricacies.

Asaf Karagila
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  • I am not convinced that there are countably many algorithms. Example pi + 1, pi + 2, pi + .1, pi + .2; pi being a value for which every school child learns a finite algorithm for. Does not the fact that an algorithm could include one( or more ) of the natural numbers + (Other stuff) show it is not countable? – 8bitwide Oct 07 '14 at 00:47
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    (1) There are only countably many natural numbers. (2) An algorithm, traditionally, is a finite sequence from a finite list of operations. Since there is a bijection between the natural numbers and set of all finite sequences of any finite alphabet with at least two distinct characters, there can only be countably many algorithms. (3) The question on how many algorithms (or how many C++ programs are there, if you prefer) has been asked several times before. (4) Your comment shows that you might want to review the definitions of "countability" and "algorithm" again. – Asaf Karagila Oct 07 '14 at 00:49