NJW posits that Real Numbers and Set Theory are based on the notion that it is possible to do an infinite number of calculations which he considers disingenuous.
He highlights in some of his Youtube videos that in respected Math textbooks the definition of real numbers is left vauge.
In his opinion set theory has the same kind of holes the we are expected to accept that we can add an infinite quantity of things to a Set by describing a function or simply having a desciption of the elements of the Set.
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.
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.
ZFC models the set of real numbers, but only provides a model for a measure-zero amount of specific individual real numbers. It just says "yeah they exist".
People like Wildberger believe that anything that exists in math should have some way of determining its exact value, otherwise, what is "it"?
That I understand. I objected to the idea that the theory was vague rather than that the theory contains objects that are vague. Even so, if one thinks the theory contains vague objects then we run into considerations such as the following:
He highlights in some of his Youtube videos that in respected Math textbooks the definition of real numbers is left vauge.
In his opinion set theory has the same kind of holes the we are expected to accept that we can add an infinite quantity of things to a Set by describing a function or simply having a desciption of the elements of the Set.