> The claims made in both in your question and the Wikipedia page on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable, and every object is uniquely specified by a property.
Despite arguments about countability, which ignore how difficult it is to pin down "what is definable," it is possible (although not necessary) for all real numbers to be describable/definable in ZFC.
> The vast majority of real numbers can't be described. But it is impossible to give a single example.
If we accept the first sentence and assume the second refers to indescribable numbers, then isn’t it obvious that we have no examples of things we cannot describe?
If the second sentence refers to real numbers in general I can give one example or two.
He just means the phrasing is kind of poor. A general description of an undefinable number could be something like: given a sequence of Turing Machines, T_1, T_2, ..., T_n, where T_1 is the smallest representation of Turing machine over a grammar G, and T_2 is the second smallest representation of a Turing machine over G, and T_3 is the third smallest etc... take the limit of the ratio of said Turing machines that halt to Turing machines that don't halt as n goes to infinite.
I mean that's a description of some number, you could even write it out mathematically or write an algorithm to express that number. Of course neither the algorithm or the formula will ever converge and yet it will also always be bounded between 0 and 1 (hence it doesn't diverge to infinity).
So is that a description of a number? Well sure in one sense I just described it, there is only one single real number that can satisfy that description, and as I said I could in principle write it out formally and rigorously... and yet in another sense it also doesn't describe anything since no matter how hard you try, there will always be at least two real numbers that could potentially satisfy the definition and no way to eliminate one of them.
It’s only the vast majority that can’t be described.
So either it is claimed that it is counter intuitive that you can’t give an example of something you can’t describe. That is not counter intuitive- that is basically the definition of indescribable.
The other way the sentence can be read is that you can’t give an example of a real number. Of course you can. It’s only the vast majority of real numbers that can’t be described. There’s still infinitely many we can describe. 1 is a real number.
This is very interesting but I think it relies heavily on interpretation.
For example there exists models of ZFC where all "real numbers" are definable, but said model does not include all the actual real numbers, it excludes any number that is not definable in ZFC. The issue is that the term "real number" is overloaded. In the formal sense it may refer only to numbers that are members of a model in which undefinable numbers are excluded. In another sense the term "real number" refers to actual real numbers as we humans intend for them to exist but do not have a precise formal definition.
This actual set of real numbers does indeed contain members that are not definable in ZFC or any formal system, the issue is that there is no way to formalize this actual definition.
This is similar to what another poster mentioned about Skolem's paradox:
> In another sense the term "real number" refers to actual real numbers as we humans intend for them to exist but do not have a precise formal definition.
Ah a Platonist in the flesh. Don't see many of you on HN. I don't think real numbers truly, objectively exist and think of them more as artifacts of human thought, but that's a deep deep rabbit hole.
I'm kind of curious then, what do you believe the cardinality of the "real" real numbers is?
I think I'm with you on that. Real numbers don't exist in an objective sense, I mean they exist in the same sense that an Escher painting of a hand drawing a hand exists, but they don't exist in the sense that a hand drawing a hand actually exists.
When I was in high school I remember thinking that computers use the discrete to approximate the continuous and that it is the continuum that is real and the discrete that is an imperfect representation of the continuous. Then a high school teacher blew my mind when he told me to consider the opposite, that in fact it's the continuous that is used to approximate the discrete. The discrete is what's real and we humans invented the continuous to approximate the discrete.
That simple twist in thinking had a profound effect on me that influences me to this day 30 years later.
If anything I may have some extreme opinions that frankly no one takes seriously and I'm okay with that. For example I think the finitists had it right and infinity does not exist. There really is such a thing as a largest finite number, a number so large that it's impossible even in principle to add 1 to it. I can't fathom how large that number is, but there's physical justification to believe in it based on something like the Bekenstein bound:
At any rate, I like thinking about this stuff, I do appreciate it, but I don't take it literally. It's poetic, it can inspire new ways of thinking, but I also remind myself to compartmentalize it to some degree and not take these ideas too literally.
If you're sympathetic to the finitist cause, the idea that all mathematical objects are in fact definable is right up that alley. It's nice that this happens to line up acceptance of infinity, but finitism is basically entirely predicated on definability.
His conclusion (which I agree with) is
> The claims made in both in your question and the Wikipedia page on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable, and every object is uniquely specified by a property.
Despite arguments about countability, which ignore how difficult it is to pin down "what is definable," it is possible (although not necessary) for all real numbers to be describable/definable in ZFC.