I already demonstrated that the numbers of ideas that can be expressed is countable. So how come the number of real numbers is uncountable? In what sense are the real number real? Each real number can be considered as an idea. Can we express an infinite number of ideas in a finite number of words?
But I already said that it all depends on the language we use. Let’s start with checking our definition of numbers. Since we have the inductive definition of natural numbers, and we can define rational numbers by a ratio of two natural numbers – lets check our definition of irrational numbers. There are a few ways to define irrational numbers. Lets define irrational numbers (or more generally speaking, real numbers) as the limit of a known sequence of rational numbers, which is increasing and has an upper bound.
It can be argued that a limit of such a sequence might not be regarded as a number, if it’s not in itself a rational number. But this is just terminology. It doesn’t matter if we call it a limit or a number, as long as we know it exists. It exists in the sense that it is computable – we can calculate it to any desired precision by a finite, terminating algorithm. Or to be more accurate – it is computable if the original sequence of rational numbers can be generated by a known algorithm.
So in the language of computers and deterministic algorithms, we can define any computable number in such a way. Can we define numbers which are not computable? It all depends on our definition of “definition”. While there might exist languages in which such numbers can be defined, my view is that there also exist languages in which such numbers cannot be defined – for example, the language I’m using now. It can be claimed that an infinite definition is also a definition, and these numbers can be defined by the infinite sequence of rational numbers itself. But in reality, it will take us infinite time to express such a definition, and we would need infinite memory to remember it. Therefore, my view is that anything that requires an infinite definition is not real. So lets limit ourselves to definitions of finite length. If there exists a finite algorithm that can define a number (by defining a sequence of rational numbers that converges to it) then this number is computable and therefore definable. If there doesn’t exist such an algorithm – then we cannot define such a number.
So, if I conclude that numbers that can’t be defined don’t exist (because we are not able to express them in the language I’m using), then we come to a conclusion that the set of real numbers is countable. How does it get along with arguments such as Cantor’s diagonal argument, which claim that the set of real numbers is uncountable? Well, while Cantor’s diagonal argument claims that there exists a sequence of rational numbers, which converges to a real number, and is not computable (because the set of computable numbers is countable) – we are not able to express such a set in a finite way in the language I’m using. And since it can’t be expressed, I can claim that it doesn’t exist.
Why doesn’t it exist? Consider this – I can claim that there is a computer language, in which there is an algorithm, that is able to solve the halting problem (decide whether any given computer program will halt). Indeed, there is such a language and algorithm in the same sense that there are numbers which can’t be computed. But there is no computer who runs such an algorithm – it can’t be compiled into known computer languages. So in what sense does such a computer language exist? It doesn’t exist in reality – it exists only in our minds. And therefore, numbers which can’t be computed exist only in our minds, too. They are not real numbers, in the sense that they are not real. They are real in the same sense that a computer who solves any problem is real.