You have the explanation, but more precisely: the set of definable real numbers is countable, because a mathematical definition can be encoded as a finite sequence of mathematical symbols (of which thereare only finitely many), and so there are only countably many definitions.
Hence most real numbers are undefinable.
By the way, there is a simple proof that all natural numbers are definable: if not, then there is a smallest undefinable number. But “the smallest undefinable natural number” would then be a definition of that number :)
That is true. Naturals are explicitly constructible by definition anyway, but Russell’s paradox applies to the concept of “interesting numbers” and is why they can’t be well-defined. https://en.wikipedia.org/wiki/Interesting_number_paradox
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well real numbers are uncountable, but the set of numbers you can think of and describe is still countable
Is it? I could be convinced but I’m going to need a proof before I believe that
It’s obvious after the chapter and left unexplained as an exercise for the reader
the set of finite length natural language sentences is countable.
You have the explanation, but more precisely: the set of definable real numbers is countable, because a mathematical definition can be encoded as a finite sequence of mathematical symbols (of which thereare only finitely many), and so there are only countably many definitions.
Hence most real numbers are undefinable.
By the way, there is a simple proof that all natural numbers are definable: if not, then there is a smallest undefinable number. But “the smallest undefinable natural number” would then be a definition of that number :)
I thought that all self referencing proofs are trouble since Russell’s paradox
That is true. Naturals are explicitly constructible by definition anyway, but Russell’s paradox applies to the concept of “interesting numbers” and is why they can’t be well-defined. https://en.wikipedia.org/wiki/Interesting_number_paradox