Do there exist "random" real numbers?

Question

According to a common perception of the real numbers, it contains any totally random sequence of digits that a monkey could type on a keyboard in infinite time, like $$0.298403840284023840238402480234802348023480240480328402348230\ldots $$ But does it even make sense to say this? Does ZFC guarantee that the result of this randomness is a number.

I hope the question made sense. Creative interpretations of my question are welcome.

Your random number does contain none of the digits $1$, $5$, and $7$ at the first $60$ places after the decimal point. The probability that such a thing happens is about $5.1\cdot 10^{-10}$. — Christian Blatter, Jul 13 '14 at 20:15
Numbers aren't random; procedures for producing numbers can be random. (See, for example, http://xkcd.com/221/.) — Qiaochu Yuan, Jul 13 '14 at 20:28
The question makes perfect sense. The type of number you describe is called a 'choice sequence'. They were invented by Brouwer and are the basis of intuitionistic analysis. — , Jul 13 '14 at 20:36
"Arbitrary" would be a better way of describing it than "random". — smithkm, Sep 14 '15 at 20:53
Related: https://math.stackexchange.com/questions/997173/can-you-pick-a-random-natural-number-and-a-random-real-number — Hans Lundmark, Jun 16 '21 at 11:01

score 5 · Accepted Answer · answered Jul 13 '14 at 20:05

Yes, every such infinite sequence of digits (including random ones) is a number, according to how we build them. The set of reals is build using ZFC, by taking all Cauchy sequences of rationals, and quotient them by the relation: two sequences are equivalent if their interleaving is also Cauchy (this is one way to do it). So in particular, your random number is obtained in this construction from the Cauchy sequence $0.2$, $0.29$, $0.298$, $0.2984$, and so on...

The problem of actually describing entirely this number is another one, and indeed not all real numbers can be described in a finite way.

An other remarks is that not only there are random reals, but almost all reals are random, in the sense that the set of non-random real numbers has measure $0$.

But then the question becomes: Does the number I wrote actually exist? This number corresponds to a map $\Bbb N\to{0,1,2,\ldots,9}$ mapping $n$ to the $n$th digit. Such a map is again a set. But does ZFC guarantee that this very set exists in the world of sets? — Gaussler, Jul 13 '14 at 20:10
It depends what you mean by "exists", in some models of ZFC there won't be, see for examples http://www.math.dartmouth.edu/~doyle/docs/random/random.pdf — Denis, Jul 13 '14 at 20:28

Do there exist "random" real numbers?

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