Not all statements in a given language can be given a truth value, in that they don't refer to anything that allows to determine a truth value — leo
If it can't be given a truth value, its not a statement would be a simple rule to adopt. — Devans99
We can say things such as "This sound is purple" or "This smell is true", but they don't refer to anything in the range of what we experience, or at least I can't form a mental image of whatever these statements may refer to — leo
Like the paradoxes in the theory of relativity, they are a consequence of the postulates at the basis of the theory, we can choose to ignore them and just "shut up and calculate" and make predictions that fit somewhat with observations, or we can change the framework (change the theory, pick different postulates) so that the paradoxes disappear while making similar observable predictions, in the end it depends whether we're looking for mathematical 'elegance' with symmetries and so on or if we're looking for intuitive simplicity. I'm a bit like you on this, I prefer intuitive simplicity that can be grasped by many over mathematical elegance that leads to complexity, paradoxes and confusion — leo
So in the case of the Godel statement, ‘this statement is not provable’… means 'it is not provable that this is a statement'. If you can’t prove its a statement then you can even start to prove it. — Devans99
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