> Why do we accept that there are real numbers that cannot be defined by a finite string of symbols We don't accept it. It's easy to prove that it's the case.

> Why do we accept that there are real numbers that cannot be defined by a finite string of symbols We don't accept it. It's easy to prove that it's the case. I think you are unfamiliar with how R is defined. The traditional definition of R is a completion of Q with respect to the metric |x-y| (getting all Cauchy sequences to converge). That's larger than infinite sequence of decimals.