The premise of this idea - that anything describable can be written in a binary firm and is thus countable - seems wrong. It's wrong because we easily invent new concepts and put them into a symbolic form. We could invent a new concept, agree on a new symbol for it and add it to our alphabet. The set of ideas isn't countable and so our alphabet isn't countable. This alphabet can't be translated into some binary form either.
Why is the alphabet not countable? If each time you think of a new idea and make a symbol for it, I can also assign it to an integer (because there is always a next integer like there is always a new symbol you can come up with).
When you come up with a new concept, it should also be possible to write out a definition of it. If you can write down your definition (in English, math notation, etc.), then it comes from a countable set, since there are countably many things that you can write down.
We don't "come up" with ideas from other ideas using some closed form rules of logic, like in Coq or some Turing machine. Instead, we discover new ideas.
There is a world of ideas and the real world. People live in both worlds. When they discover a new idea, often by accident, they label it with a symbol and use it in the real world. Other people can see the same idea and since they can't fully describe it with words, they agree to use the new symbol.
We describe new concepts with words, but those definitions are underspecified: they refer to things with vague or non existent descriptions, or just common sense. What is "set" for example? The same words often mean different things in different contexts. This extra meaning that's always attached to words is what makes these definitions non countable.
Even if ideas come from an uncountable set (not convinced yet), there are still only countably many ideas people will ever have. Each time anyone comes up with an idea, I can assign it a new integer.
Im merely trying to drag the concept of separating ideas and reality as two different but very real worlds under the spotlight of everyone's attention. This concept is fundamental and very old. I won't be able to defend this idea with formal proofs.
So long as every real number exists, has properties and so on. Every such number is a separate idea. They exist, no matter whether we know about them or not.
Ah. Personally, I distinguish between potential ideas and actual ideas. To be an actual idea, it has to reside in someone's brain (or a computer, or some other data-processing system). The reals correspond to the set of potential ideas, but the set of actual ideas is not only countable, but almost certainly finite.
We can call it a materialized idea, like an implemented software algorithm. I'm indeed talking about the world of ideas that's not real, i.e. non material. The proof of the Fermat's theorem has always existed, but only recently it's been discovered by Wales.
The same word can have infinitely many meanings. So even if we restrict the length of proofs to 140 chars and restrict the alphabet to Latin, there will be infinitely many proofs there: well just start inventing new meanings for the same words.
> The same word can have infinitely many meanings.
But only countably many because definitions have to be finite too. The combination of proof + definitions must also be finite, so there can only be countably many of them.
What's the definition of "set"? Or what's the definition of the implication symbol, i.e. when someone says that something obviously follows from the previous theorems? We don't bother to define a lot of foundational things in math.
The implication symbol doesn't have a definition, it's part of a completely different kind of reasoning process. Formal symbolic reasoning is a completely different animal than informal arguments involving words that have definitions.
Definitional recursion has to bottom out somewhere. (OK, it can also be circular, but I'm guessing you would not find that satisfactory.) Whatever words I use to define "function" you can always turn around and insist that I define those words. It's a never-ending game. It ultimately boils down to the definitions of words like "true" and "false, "same" and "different", whose meanings can only be communicated by way of examples: X and X are the same, X and Y are different.
But none of this has anything to do with the matter at hand. There are a finite number of atoms in the universe. Those atoms can only arrange themselves into a finite number of sentient creatures (or computers), each of which has only a finite brain in which can reside only a finite number of thoughts. So no matter how you slice it, the number of realized ideas in this universe is going to be not only countable but actually finite because there is only a finite amount of time before heat death.
