I'm aware that >90% will outright reject this, so feel free to ignore it. I'd much appreciate those trying to figure out in which way this question (or rather its eventual answer) would make sense.

**Is there a way to construct the foundations of mathematics from just the real numbers?
If not, why not?**

Background: The real world (space, time) is arguably best described as a continuum. So it would make sense to take that as a foundation. Integers would only occur later e.g. as winding numbers, not as fundamental. All discrete things would be secondary. A continuous version of sets would probably be part of the foundation.

Rant: This construction from set theory to integers to rational numbers only to find that real numbers are then kind of problematic in the framework. I just can't see any sense in that.

just the real numbersthen Book 5 of Euclid (or similar) might do. If you want $\mathbf R$-based “foundation for all mathematics” your problem would be to translate from the language (sets) most of them are already written in. $\endgroup$eliminate sets? $\endgroup$"real numbers are then kind of problematic in the framework"What do you mean? $\endgroup$uniqueDedekind-complete ordered field, and this is what we mean when we talk about the real numbers. Dedekind cuts and Cauchy sequences show that one canconstruct(or “implement”) such a field starting from the rationals. $\endgroup$4more comments