# Is there a combined theory of the Reals and the Naturals that has a model where the Naturals and Reals have the same cardinality

The Upward Lowenheim-Skolem theory decrees that there must be a (non-standard) model of the naturals of cardinality the same as that of the standard model of the Reals.

For any combined theory of the Reals and the Naturals, a proof similar to that of Upward Lowenheim-Skolem means that there must be a model with any infinite cardinality of Naturals. But, is there a model where both the Reals and the Naturals have the same cardinality?

Equivalently, is the statement "There is a bijection between the reals and the naturals" satisfiable in some combined theory of naturals and reals where the standard Peano axoims and the axioms of the reals hold / can be encoded.

Or, is $\forall r, \exists i, \textrm{isInteger}(i) \land f(i) = r$ first-order provable in some minimal axiom scheme that can prove every thing that Peano axioms and the real axioms can prove?

• What do you call "the reals axioms"? – Régis Jun 15 '18 at 7:09
• Didn't realise there was debate regarding that. I'm going to say, dense linear order without endpoints and.. umm... Aaaah, did not realise that the least-upper-bound property is not FOL axiomatizable. – nishantjr Jun 15 '18 at 7:23
• You might like to look at real closed fields. – Régis Jun 15 '18 at 19:13
• Anyway, as long as your language vocabulary is countable, whatever theory you consider, it will have countable models. – Régis Jun 15 '18 at 20:45
• Questions like this are very hard to ask accurately and the answer often depends subtly on what you ask. How are the naturals and reals related? Certainly upward L-S guarantees there are models of PA with cardinality $\mathfrak c$. Those have elements that are not part of our standard reals. If that isn't a problem, we are done. If you want the PA model to be a subset of the real model, I don't know what the reals based on that sort of PA model look like. – Ross Millikan Jun 16 '18 at 3:21