> Maybe it's the idea of a completed infinity that's my problem; maybe it's the fact I don't understand how to define (or forgot cauchy sequences in detail) an arbitrary real.
As someone who also has never fully made his peace with the diagonality argument, but just chosen to accept it as true, as a given, this kind of bumps up against an interesting implication of different cardinalities of infinity.
To precisely define an arbitrary real you'd need some kind of finite string that uniquely identifies that real number. Finite strings can be mapped, 1 to 1, to natural numbers. Therefore there can't be a finite string for any real number that uniquely identifies it. Otherwise we'd have a mapping between natural numbers and real numbers.
In fact, the set of uniquely identifiable real numbers is a countable subset of real numbers. [1]
Somehow, this realization has helped me make peace with the uncountability of real numbers.
[1] Sorry if use words like "unique", "identify", "define" in not quite the right way. I hope the meaning I'm going for comes across.
As someone who also has never fully made his peace with the diagonality argument, but just chosen to accept it as true, as a given, this kind of bumps up against an interesting implication of different cardinalities of infinity.
To precisely define an arbitrary real you'd need some kind of finite string that uniquely identifies that real number. Finite strings can be mapped, 1 to 1, to natural numbers. Therefore there can't be a finite string for any real number that uniquely identifies it. Otherwise we'd have a mapping between natural numbers and real numbers.
In fact, the set of uniquely identifiable real numbers is a countable subset of real numbers. [1]
Somehow, this realization has helped me make peace with the uncountability of real numbers.
[1] Sorry if use words like "unique", "identify", "define" in not quite the right way. I hope the meaning I'm going for comes across.