[bweitzman]

https://www.youtube.com/watch?v=jlnBo3APRlU

The gist of the argument is that addition + other operations on non-computable numbers (which the real numbers contain) require infinite algorithms or something similar (unlike addition on computable irrational numbers which may require infinite work, but the algorithms are finite). You can therefore get situations where, say, the tenths digit in a sum of non-computable numbers is not defined because of potentially infinite carries, and there's no way to determine if the sequence of carries terminates or not. He discusses the problem in the context of different representations of real numbers, including infinite decimals, cauchy sequences, and dedekind cuts etc. This is just the gist of it.

[syzarian]

This is different than saying the definition of the reals is “vague”. The models and various definitions are not vague. They are as precise as the axioms of Euclidean geometry or any other axiomatic system. His objections are reasons why he doesn’t like the axioms. One can either accept or reject the axiomatic system but it isn’t vague.