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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
by PureParadigm 7y ago
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.
- dchyrdvh 7y agoWe 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.
- PureParadigm 7y agoEven 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.
- dchyrdvh 7y agoIm 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.