> even if you think it is built on shaky philosophical foundations.
What does this mean? What shaky philosophical foundations?
What does this mean? What shaky philosophical foundations?
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.
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"?
https://mathoverflow.net/questions/44102/is-the-analysis-as-...
I suppose being a finitist allows one the get around what Hankins wrote.