Take a look at nonstandard calculus: https://en.wikipedia.org/wiki/Nonstandard_calculus https://en.wikipedia.org/wiki/Nonstandard_calculus It is based on the

Take a look at nonstandard calculus: https://en.wikipedia.org/wiki/Nonstandard_calculus https://en.wikipedia.org/wiki/Nonstandard_calculus It is based on the hyperreal numbers: https://en.wikipedia.org/wiki/Hyperreal_number https://en.wikipedia.org/wiki/Hyperreal_number Practically speaking, I don't think it buys you anything over traditional calculus/analysis. It's just pointing out that there are alternative approaches to formalizing calculus.