Consider the following statement
Every real number must have a definition in order to be discussed.
What this statement doesn't specify is how that loose-specific that definition is.
Some examples of definitions include:
"the smallest number that takes minimally 100 syllables to express in English" (which is indeed a paradox)
"the natural number after one" (2)
"the limiting value of the sequence $(1 + 1/n)^n$ as $n$ is moved towards infinity, whereas a limit is defined as … (epsilon-delta definition) … whereas addition is defined as … (breaking down all the way to the basic set theoretic axioms) " (the answer to this being of course e)
Now here is something to consider
The set of all statements using all the characters in the English in English language is a countable set. That means that every possible mathematical expression can eventually be reduced to an expression in English (that could be absurdly long if it is to remain formal) and therefore every mathematical expression including that of every possible real number that can be discussed is within this countable set.
The only numbers that are not contained in this countable set are…
That's a poor question to ask since the act of answering it is a violation of the initial assumption that the numbers exist outside of the expressions of our language.
Which brings up an interesting point. If EVERY REAL number that can be discussed is included here, then what exactly is it that is not included?
In other words, why are the real numbers actually considered to be uncountable?
Best Answer
"The real numbers are uncountable" means that, in the set-theoretic universe where we have defined "the set of natural numbers" and "the set of real numbers", there is not a function that is a bijection between these two sets.
It means nothing more, and nothing less than that.
There are all sorts of traps, mistakes, and really subtle misunderstandings one can run into by trying to ascribe more meaning to this statement than it actually has.
You may find Skolem's paradox an interesting topic to read about, given that it involves a rigorous and precise way to see the real numbers as countable in a sense different from what is meant by "the real numbers are uncountable", and the consequent difficulties people have trying to unravel what's going on.