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.