I'm really trying to sweep out the cobwebs in my memory here...

And I'm trying to make sure I don't misuse words like "model" and "true". (If any of you who know this stuff better than I do think I'm misusing terminology or making some other sort of mistake, please let me know!)

The hardest thing to characterize in a theory of the integers is the "finite # of steps between any two integers". I.e., "finiteness" is the hard property, not infiniteness. This well-ordering property requires a _second order_ axiom, and hence integers require a "second order" theory.

I think I see what you mean. If one is going to quantify over objects called "subsets of Z", one will need a bunch of set-theoretic axioms to capture or partially capture what this means, and the answer to my question might depend on what bunch of axioms one chooses. I'm not sure what bunch of axioms corresponds most closely to what I meant to ask when I posed the question, but I will say that I think there might be some fun models in which, for every real x, the set of ALL integers bounded below by x has a least element, but in which there exists a set S of integers such that S is bounded below by some real number x but S doesn't have a least element. Can anyone come up with a model like this?

I think you may have to formally axiomatize what the properties of "ceil(x)" are.

I use just one axiom for this purpose: "for every real number x there exists an integer m such that for all integers n that are greater than or equal to x, n is greater than or equal to m." It is easy to prove that this m is unique, and we call it ceil(x).

Presumably the integers are the range of ceil()

That follows as an easy consequence: for every integer n, ceil(n) = n. (Henry's email suggests an alternative way to try to ask my question: Axiomatize the ceiling function directly as a function on the reals, and then *define* Z as the image of this function. That's not what I had in mind, though it's interesting too.) Jim