So it is in fact possible and consistent with ZFC that all reals are definable.
See: https://mathoverflow.net/questions/44102/is-the-analysis-as-...
And, I imagine that the set of real numbers of the meta theory could be (in the meta theory) the same set as the set of real numbers in the model?
(for an extreme example of this, by the Löwenheim–Skolem theorem there are countable models of ZFC)
So you can do what you are suggesting, and you will of course get a countable set of reals (or what are reals according to the inner model), but they might not be countable according to the inner model. They might not even be a set according to the inner model, and there are even inner models that think you've got all of the reals!
(see https://mathoverflow.net/questions/351659/set-of-definable-r... pretty heavy reading)
So the statement "the set of definable reals is countable" is nonsense - you're talking about things that live in different universes of meaning.
And I think by virtue of this, the statement should have meaning.
As like, a statement in the meta-language that models of ZFC which have as their sets of reals, the (according to the meta-theory) set of reals, that the set of reals definable within ZFC, is a countable set of the meta-theory.
Also, did someone downvote your comment?? I don’t know why if so. It seems a productive comment to me.
By definition an inner model consists of some domain (a set of sets) and some choice of mappings from all the function/relation symbols of ZFC to functions/relations on this domain, satisfying the axioms of ZFC.
You are suggesting to enumerate every formula of ZFC, evaluate them against this inner model, and take the set of all reals that are uniquely picked out by some formula (according to the model).
The trouble is that even though you can make the set of reals the same, your chosen interpretation of all the functions/relations will not match the meta-theory, and in fact cannot match it (i.e. the meta-theory cannot provably construct an inner model like this, by Tarski's undefinability of truth theorem).
So you will get a set of reals, and they will be reals according to the meta-theory too, but the meta-theory cannot relate this set to the definable reals of the meta-theory.
As far as I can see this is the strongest statement you can actually prove: "the set of reals in any inner model of ZFC uniquely definable by a formula (according to the interpretation of the inner model) is countable (according to the interpretation of the meta-theory)".
> Also, did someone downvote your comment??
Someone did, yeah, but I don't mind =) I probably sound like a crackpot to the uninitiated.
It's certainly a brain-bender that even in the unit interval if we imagine filling in all the the rationals and then adding in the describable-irrationals like PI/4, sqrt(2)/2 and so on.. that this still does not even come close to covering the unit interval - or any interval - of Real numbers! My imagination sees a line with a heck of a lot of dots on it, but still knowing that there clearly still uncountably-more values that are not covered/described! Amazing! The continuum (Real numbers) is such a fascinating concept!
Sure you can assign them to an arbitrary set, but you don't have access to the value.
It is a hay in the haystack problem, where you really only have access to the needles, not the hay.
Take the (uncountable) set of Real numbers. Remove the normal numbers, which is almost all of them in the sense that the probability that "a uniformly randomly chosen real number is normal (and therefore also undescribable)" is 1. The remaining set of numbers, which has measure 0 in the Real numbers, is still uncountable, meaning that the proability of randomly choosing a describable number in that set is again 0.
I'm not sure how deep this chain can go. Google AI says "only 1 steps" but it's not admiting the case described in this comment.
Plenty of normal numbers have a finite representation from which digits can be efficiently extracted. E.g., Champernowne's constant (in any base) is normal, and you can find its digits with a relatively simple algorithm.
All computable reals can similarly have their digits extracted by some algorithm or another, even though it may take a long time. I wouldn't call that "not having access to the value". Of course, uncomputable numbers are a different story, but they have nothing to do with normality in any base.
And of course, radix representations are not the only way to evaluate real numbers. E.g., you could represent them with simple continued fractions (which would still allow addition, multiplication, comparison, etc.), and then you could write out any quadratic irrational with a periodic expansion.
Meaning it holds for all elements of a set except for a subset that has measure zero.
Yes some normal numbers are in the constructable reals, but it is a measure zero subset.
You are putting your hand in the haystack and only finding needles, finding the hay in the haystack is the problem here.
> Yes some normal numbers are in the constructable reals, but it is a measure zero subset.
Almost all irrational computable reals (in the sense of natural density) will be normal, for any sane enumeration. Just because a real number is computable doesn't mean it's less likely to be normal.
> All computable reals can similarly have their digits extracted by some algorithm or another, even though it may take a long time.
It was an intentional abstraction to avoid a self referencing claim, not to say that they are equivalent to anything.
The nice thing about constructible reals is after the construction you can typically forget how you constructed them, be that through Axioms, Cauchy sequences, Dedekind cuts etc...
The computable reals by definition can be computed to within any desired precision by a finite, terminating algorithm. That is why I said it is begging the question.
> Almost all irrational computable reals (in the sense of natural density) will be normal, for any sane enumeration. Just because a real number is computable doesn't mean it's less likely to be normal.
While some have Conjectured claims close to this, looking into why there have been no proofs for even a single number that was not explicitly created to be normal in any base may be a good lens in to the hay-in-the-haystack problems I was refrencing.
From: "Distribution Modulo One and Diophantine Approximation (Cambridge Tracts in Mathematics, Series Number 193)" Page 81, section 4.1 "Equivalent definitions of normality"
> Lemma 4.3: Let b and r be integers greater than or equal to 2. If a real number is simply normal to base b^r, then it is simply normal to base b.
That there may exclude many of what appear to be simply normal numbers from actual ones. As a graduate level text that book may be a bit expensive for what it is, but a good reference in my experience.
The 'Normal' property that is the full measure set of the reals is far more constrained than natural density. Which is why it is so surprising that is is the property of almost all of them.
That is why it was offered as a lens, specifically one that was worked on before computability as a subject, as an intentional way to gain distance from the almost intractable polysemy problems there.
The set of all problems that can be described by a finite description is countable. Why would we care about the rest of them?