That's not a good example; the problem you're creating is due to sloppy use of language, not any cleverness in the definition.
All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name.
Counter-intuitively, this is not true.
The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number of reals in between them. If you say, okay, sure, but some of those might be named, then pick the two closest and then there is still an infinite number of other reals in between those two!
Another way to look at it is that the amount of information (measured in bits) between any two real numbers is literally infinite. If the reals were represented with binary digits, then a sequential subset of them would have a common finite prefix, and then all possible infinite bit strings would be the suffixes!
So what? That doesn't stop them from being named.
Edit : actually does not even work, courtesy of the Chaitin's numbers
(For a more compelling argument, the equality of real numbers is uncomputable, so it must be impossible to name them. If we could name them usefully, then we could determine equality by asking which ones had different names.)
No, I'm not.
> Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name.
What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?
You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2".
This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would require an infinite number of digits, but it can be defined ('named') with a finite number of bits.
It turns out that this is a rare property for real numbers. The vast majority do not have a shorthand name like this, only the infinitely long description exists.
AC says: if there is a set of non-empty sets y with length x, I can construct a new set z of length x by taking a member from each set in y.
How does being able to name the constituents z or any member of y matter to the logic of the axiom?
You might consider that in the existing Chinese writing system, 口 and 囗 are distinct symbols.
This is untrue and not particularly hard to prove by contradiction.
Suppose every elements in R can be named by a finite number of symbols.
You can build a bijection between R and the names of its elements (by definition a name points to a unique element and if elements have multiple names, you can easily well order a finite number symbols by building a lexicographic order and only consider the smallest name).
Or, you can also easily build a bijection between a finite number of symbols and N. That's just an encoding like ASCII or UTF-8. Therefore, the set of names is countably infinite.
Yet R is uncountable (see Cantor's diagonal argument).
Therefore, by contradiction, there has to exist elements in R which can't be named by a finite number of symbols.
You've confused two different senses of "a finite number of symbols". The requirement is that it takes a finite number of symbols to write down any given number, not that it takes a finite number of symbols to write down every number at once. (Though in fact that also takes a finite number of symbols; it is the precise meaning of ℝ.)
Symbols are, in their simplest interpretation, bounded two-dimensional curves; there is no danger of running out of them before we run out of numbers.
This is false. In some sense, there exist numbers that can't be referred to. We can refer to the set of real numbers as a whole, but not some of the elements. https://en.wikipedia.org/wiki/Definable_real_number
But then there are issues with defining "definable numbers", which complicates things by a lot. https://mathoverflow.net/questions/44102/is-the-analysis-as-...
Now if you fix a particular formal language for defining real numbers, with a finite alphabet, then the language only has countably many words, hence there are reals not definable in the language. But the notion of "definability" here is not independent of the choice of formal language.
So "choosing an undefinable real number" amounts to "choosing a real number not definable in L", where L is some fixed formal language---and this isn't particularly hard to imagine; given a specific L, you can probably quite concretely construct a real number not definable in L.