In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of calculus are equivalent to the completeness axiom of the reals and which ones aren't?" and ended by speculating "Perhaps somebody wrote a beautiful Monthly article a few decades ago that explained things so clearly as to make the whole matter seem trivial, with the result that the article was forgotten? :-)"

Since the article I was looking for doesn't seem to exist, I decided to write one myself; the current draft can be found at http://jamespropp.org/reverse.pdf .

One issue I'm a little confused about is the relationship between truth and provability in this context. As I ask on the bottom of page 2 and the top of page 3, is saying "Every ordered ring $R$ satisfying property $P$ satisfies property $P'$" the same as saying "From the ordered field axioms plus the assumption that $P$ holds one can prove that $P'$ holds"?

I believe that they're not the same (because for instance the Riemann Hypothesis might be true but unprovable), but I'd like to hear from people who know more about foundations and model theory than I do.

All kinds of comments on the article are welcome, but comments on the truth-versus-provability issue are especially sought.

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