It’s not clear to me that ZFC without regularity, replacement, infinity, choice, power set or foundation with a totally ordered field with the LUB property does allow you to talk about most things you want to do with the reals : without replacement or powerset you can’t prove that cartesian products exist, so there doesn’t seem to be any way of talking about the plane or higher-dimensional spaces as sets. If you add powerset back in you can carry out the Hartogs number construction to get a least uncountable ordinal
Hmm, that’s a good point. Lack of cartesian products is annoying. We don’t however need the full power set axiom to get them. We can simply have an axiom that states that cartesian products exist. Or even weaker do the following (ad hoc axioms) with a new property of being Cartesian: 1. The cartesian product of any two Cartesian sets exist. 2. Any subset of R is Cartesian. 3. The cartesian product of two Cartesian sets is Cartesian. 4. If A and B are Cartesian then A union B, A intersect B, and A\B are all Cartesian. That should be enough and is a lot weaker than general power set I think.
It’s not clear to me that ZFC without regularity, replacement, infinity, choice, power set or foundation with a totally ordered field with the LUB property does allow you to talk about most things you want to do with the reals : without replacement or powerset you can’t prove that cartesian products exist, so there doesn’t seem to be any way of talking about the plane or higher-dimensional spaces as sets. If you add powerset back in you can carry out the Hartogs number construction to get a least uncountable ordinal
Hmm, that’s a good point. Lack of cartesian products is annoying. We don’t however need the full power set axiom to get them. We can simply have an axiom that states that cartesian products exist. Or even weaker do the following (ad hoc axioms) with a new property of being Cartesian: 1. The cartesian product of any two Cartesian sets exist. 2. Any subset of R is Cartesian. 3. The cartesian product of two Cartesian sets is Cartesian. 4. If A and B are Cartesian then A union B, A intersect B, and A\B are all Cartesian. That should be enough and is a lot weaker than general power set I think.