2

Would a theoretic true-random generator of digits for a number always (or almost always) construct an undefinable number in the infinite?

(This question is similar to Random and non-computable numbers , but extended for non-finitely describable numbers)

For context: I am pondering an intuitive approach to the undefinable numbers, for which the internet seems hard pressed to give some example (see my reference below). In one of the references zero-sharp 0# was mentioned as a non-finitely describable real number, but I can't say it helps with intuitive understanding.

Computable real numbers contain the algebraic numbers, which contain the Constructible numbers, which contain the rationals, which contain the integers. A Definable real number also called finitely describable number is not necesarily a computable numbers. But all computable numbers are definable (to compute anything on a Turing machine, a finite state table is required, which can count as description of the generator of digits).

As far as I can tell, the set of all definable numbers is not well defined and thus also not researched (as opposed to the set of all computable numbers). I don't understand ZFC enough to tell whether or not it can yield a useful definition, though it seems closest for that purpose. Chaitin's construction is (as far as I can tell) an example of a definable but not computable number (which is not what I want). I could not yet find any formal definition of the full set of definable numbers or a statement about it's cardinality, but I think it's likely countably infinite (e.g. if the formal languages are at most countably infinite and each language can only define countably infinite numbers).

The digits of a Chaitin's constant seem to have alrogrithmic random properties, which gave me the idea of using random to informally describe some non-definable numbers.

So I am wondering if it's also viable to define a generator of non-definable numbers based on infinite true random. So such a generator generates infinite digits after 0 (wlog) using some source of true random. This would in the infinite always lead to a non-definable number x (not the same specific number, each run would yield a different one).

I would easily allow that such a generator is not mathematically well-defined since there is no mathematic definition for true random, but since I look for an intuitive access to the non-definables, using a non-well definable argument may be a benefit rather than an obstacle. Still I don't know enough about mathematical grasp of random to tell if this has other flaws. Or maybe there is an algorithmic generator of numbers that is well-defined and can be used here instead for the same effect? So it might not be provable mathematically that such generators would produce only non-definable numbers (also because the set of all definable numbers also seems unclear), but I feel like it could be reasoned like:

If the generator could produce a finitely describable number, that description would allow to predict all infinite digits of the number and establish a relationship between the digits, violating the randomness of the digits in the infinite.

Another approach might be to prove that the numbers generated by such a generator are uncountably inifinite, and that the definable numbers are countably infinite, so that at least we know that most numbers generated with the random generator would not be definable.

And I think this is linked to continuous random variable being a member of any countably finite subset of its domain is zero, while the probability of it being in the difference between it's domain and any countably infinite subset is 1.

Despite the issues, is this nevertheless a useful approach to understand non-definable numbers intuitively?

References on the network:

tkruse
  • 121
  • 3
    If you can't define random, how do you expect to get an answer to the question you are posing here? How would you even recognize a correct answer, if one were posted? But let me recommend Knuth's essay on randomness in the Seminumerical Algorithms volume of The Art of Computer Programming. – Gerry Myerson Aug 13 '24 at 09:41
  • Note that integers (and even rational and algebraic numbers) are always definable , but if you produce truely random digits (say after $0.$) , the real number you eventually get is undefinable with probabiity $1$. – Peter Aug 13 '24 at 09:57
  • And in fact , all deterministic random generators are of course pseudo-random because in principle all digits can be predicted. – Peter Aug 13 '24 at 09:59
  • @GerryMyerson: My personal ignorance on random does not imply that everyone else is as ignorant. There might be definitions in the field of random that could allow for a reasonable and verifiable correct answer (that can be accepted), even if it might not be complete. – tkruse Aug 13 '24 at 10:46
  • @Peter: I look for some reference or proof about " if you produce truely random digits (say after 0.) , the real number you eventually get is undefinable with probabiity 1" – tkruse Aug 13 '24 at 10:47
  • Where did you read that $0^\sharp$ is not finitely describable? It is very much (assuming it exists in the first place). – Jonathan Schilhan Aug 13 '24 at 17:19
  • @JonathanSchilhan: the claim was made in an answer here, I think: https://philosophy.stackexchange.com/questions/29780 – tkruse Aug 13 '24 at 20:08
  • 1
    @tkruse Interesting, that's an incorrect remark then. It even says it on the wikipedia page that it is a $\Delta^1_3$ set of natural numbers, which means that it consists of those $n$ so that $\exists x \forall y \exists z P(n,x,y,z)$ for some simple property $P$, and $x,y,z$ range over $\mathbb{R}$. – Jonathan Schilhan Aug 13 '24 at 20:37

0 Answers0