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
I hoped someone would make that connection! This one is actually sound but there is a closely related limitative result, the undefinability of truth (attributed to tarski) which uses a “liar sentence” like the “liar set” of Russell’s paradox: “this sentence is not true”. Of course, liar sentence have been known since ancient times, but it was only in the 20th century when we could give them a mathematical interpretation, rather than a purely logical one.
This means that there is no mathematical definition of what is true about the natural numbers, but there are still definitions of other things, and we can still quantify over those definitions.
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Is it? I could be convinced but I’m going to need a proof before I believe that
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
I hoped someone would make that connection! This one is actually sound but there is a closely related limitative result, the undefinability of truth (attributed to tarski) which uses a “liar sentence” like the “liar set” of Russell’s paradox: “this sentence is not true”. Of course, liar sentence have been known since ancient times, but it was only in the 20th century when we could give them a mathematical interpretation, rather than a purely logical one.
This means that there is no mathematical definition of what is true about the natural numbers, but there are still definitions of other things, and we can still quantify over those definitions.
It’s obvious after the chapter and left unexplained as an exercise for the reader