> 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.
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.
It is a function from objects onto booleans.
> what's the definition of the implication symbol
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.
Next question?
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.