1

I can explain to you every natural number (in theory) in the sense, that I could describe it and you would know exactly which number I'm talking about, e.g. by writing it down, this can be done in a finite amount of time.

Same holds for all numbers ins $\mathbb{Z}$ and $\mathbb{Q}$. Furthermore all real number I know have this property. I can describe $e,\pi,\sqrt{2}$ in a finite amount of time. Essentially it holds for all algebraic numbers and certainly for all computable numbers (as we can first list all programms, which are finite and then order their outputs).

There are however some non-computable numbers that are "known", e.g. Chaitins constant.

Are the non-computable numbers, that we can "describe" countable?

And more generally, are the numbers we can describe "countable"?

If so, why do we really need all those other numbers, we have no possibility to ever describe i.e. use them? Of course, that the real numbers have no "holes" is nice, but we can never actually hit a hole, as by hitting we would have to describe it.

Jürg W. Spaak
  • 1,328
  • 1
    Every system of description that we have available to us uses finitely many symbols, and uses the symbols finitely many times. We cannot parse an infinite list of symbols, unless we can simplify it down to a finite description. Similarly, we can't use infinitely many symbols, unless we have a finite way of generating them. So yes, under any reasonable, human-readable system of description, it will only be able to describe countably many things. – Theo Bendit Sep 14 '18 at 13:07
  • @MauroALLEGRANZA Not necessarily. Like in pointwise definable models (see reference in Carl's answer), it could be that we have all the individual descriptions but don't have a/the sequence of all of those descriptions which would witness countability. – Stefan Mesken Sep 14 '18 at 13:15

1 Answers1

8

If we pick a particular description system, it seems naively that the set of real numbers that will be describable is countable. But, at the same time, it will seem that by referring to that description system itself we can describe even more real numbers that were not describable in the original system. Indeed, given any sequence of real numbers, the proof of the Baire category theorem allows us to concretely describe a real number that was not in the original sequence.

The topic of describable or definable real numbers is full of technical difficulties that can appear paradoxical. For example, in some models of ZFC every real number is definable, and at the same time that model (like all models of ZFC) believes its set of reals is uncountable. See https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb for one more thorough explanation.

As a particular example of a subtle technical difficulty, the question reads

Of course, that the real numbers have no "holes" is nice, but we can never actually hit a hole, as by hitting we would have to describe it.".

This argument reverses the roles of the construction and the description system. If we begin with a fixed description system, there is no reason to think that every real number "hole" we can concretely construct as a limit of describable numbers will be describable by that particular description system. The real number "hole" may only be describable by some other description system. That leads us to look at unending sequences of description systems, which changes the setting significantly, because we are no longer expecting every number to be describable by the same system.

Carl Mummert
  • 84,178