I recall being told  at tea, once upon a time  that there exist models of the real numbers which have no unmeasurable sets. This seems a bit bizarre; since any two models of the reals are isomorphic, you'd expect any two models to have the same collection of subsets. Can anyone tell me exactly what the story here is? Have I misremembered something? Is this some subtlety involving how strong a choice axiom you use to define your set theory?

As John Goodrick is asking in a few places, you have to be careful in stating what you mean by "a model of the reals". If you're going to talk about sets of reals, then you need to have variables ranging over reals, and also variables ranging over sets of reals. You also of course want symbols in your language for the field operations and ordering, and possibly more. Three Options One way to do this is to use the language of secondorder analysis, which is biinterpretable with the language of thirdorder number theory. (It's straightforward to translate between real numbers and sets of natural numbers, and then between sets of real numbers and sets of sets of naturals.) Another way to do this is to use ZF, which talks about the reals and sets of reals, but also many many other things. (Far more than any mathematician who's not a logician (or perhaps category theorist?) ever uses.) There's also an intermediate strategy, which is basically what Russell and Whitehead did in Principia Mathematica, where you have some variables ranging over objects at the bottom (which might be real numbers, or anything else), and then variables ranging over sets of objects, and then variables ranging over sets of sets of objects, and so on to arbitrarily high levels. This is still far weaker than ZF, because you don't get sets that mix levels, and you also can't make sense of infinitely high levels. Firstorder and Higherorder logic If you take the first or third option, then you have two more choices, which correspond to what David Speyer was saying. You can require that variables that range over sets of things range over "honest subsets" of the collection of things they're supposed to be sets of. Or you can interpret the set variables in a "whacked model". (The technical term is a "Henkin model".) On this interpretation, the "sets" are just further objects in your domain, and "membership" is just interpreted as some arbitrary relation between the objects of one type and the objects of the "set" type, and you interpret all your axioms in firstorder logic. The difference is that the honest interpretation uses secondorder logic, while the Henkin interpretation just uses firstorder logic. Secondorder logic (and higherorder logic) is nice in that it lets you prove all sorts of uniqueness results  there is a unique model of honest second order Peano arithmetic, and if you require honest sethood then this means there will be unique models at the third order level and higher, giving you one result that you remember. But firstorder logic is nice because there's actually a proof system  that is, there is a set of rules for manipulating sentences such that any sentence true in every firstorder model can actually be reached by doing these manipulations. That is, Gödel's Completeness Theorem applies. However, his Incompleteness Theorems also apply  thus, there are lots of models of firstorder Peano arithmetic, and then there are even more Henkin models of "secondorder" Peano arithmetic, and far far more Henkin models of "thirdorder" Peano arithmetic, which is the theory you're interested in. Unfortunately, I don't know what these Henkin models look like. It all depends on what set existence axioms you use. There's a lot of discussion of this stuff for "secondorder" Peano arithmetic in Steven Simpson's book Subsystems of SecondOrder Arithmetic, which is the canonical text of the field known as reverse mathematics. However, none of that talks about arbitrary sets of reals, which is what you're interested in. Solovay's results The other result you mention, which is cited in one of the other answers here, takes the other option from above. That is, we do everything in ZF and see what different models of ZF are like. (Note that I don't say ZFC  of course if you have choice, then you have nonmeasurable sets of reals.) Every model of ZF has a set it calls ω, which is the set it thinks of as "the natural numbers". Set theorists then talk about the powerset of this set as "the real numbers"  you might prefer to think of this set as "the Cantor set", and some other object in the model of ZF as its "real numbers", but there will be some nice translation between the Cantor set and your set, that gives the relevant topological and measuretheoretic properties. Of course, since we're just talking about models of ZF, none of this is going to be the real real numbers. After all, since ZF is a firstorder theory, the LöwenheimSkolem theorem guarantees that it has a countable model. This model thinks that its "real numbers" are uncountable, but that's just because the model doesn't know what uncountable really means. (This is called Skolem's Paradox  http://en.wikipedia.org/wiki/Skolem%27s_paradox>wikipedia, http://plato.stanford.edu/entries/paradoxskolem/>Stanford Encyclopedia of Philosophy.) What Solovay showed is that if you start with a countable model of ZFC that has an inaccessible cardinal (assuming that inaccessibles are consistent, then there is such a model, and we have almost as much reason to believe that inaccessibles are consistent as we do to believe that ZFC is consistent) then you can use Cohen's method of forcing to construct a different (countable) model of ZF where there are no unmeasurable sets of "reals". Of course, the first result you stated (that any two models of the reals are isomorphic) holds within any model of set theory, assuming you're talking about "honest" secondorder models (that is, models of reals that are "honest" with respect to the notion of "subset" that you get from the ambient model of ZF). But the notion of "honest" secondorder model doesn't even translate when you move from one model of set theory to another. So Solovay's model of ZF has the property that every "honest" model of secondorder analysis (or thirdorder number theory) has no nonmeasurable sets, while any model of ZFC has the property that every "honest" model of secondorder analysis (or thirdorder number theory) does have nonmeasurable sets. That's how your two results are consistent. 


It could certainly be true if you take your set theoretic axioms to be something other than ZFC. For instance I'm pretty sure that in ZF + AD (Axiom of Determinacy), every set of the real line is measurable. EDIT: In fact, whether or not you like AD, it is the case that "every set is measurable" is consistent with ZF. Thus ZF (without Choice) certainly supports two models of R, one in which every set is measurable and another in which there exist nonmeasurable sets. I'm not actually sure how to interpret your comment "any two models of the reals are isomorphic". 


I think if you assume the Axiom of Choice, then unmeasurable sets are inevitable. Was your teatime companion perhaps entertaining you with glimpses of the worlds that are possible without AC? 





Be warned: this is pretty far from my expertise; I may be making stupid errors. There is a subtlety which shows up in theorems like "All the models of the reals are isomorphic." In any axiomatization of the reals, you will have some sort of completeness axiom. For example: Dedekind's axiom: If L and R are subsets of the reals, such that l <= r for any l in L and r in R, then there is some x such that l <= x <= r for any l in L and any r in R. This axiom will always make reference to set theory; in more technical terms, it is not a first order axiom. Now, there are two things that are both true. (1) Define a "straightforward model of the reals" to be a set R, with relations +, *, 0, 1, =, < obeying the axioms of the reals. Here the word "subset" in Dedekind's axiom is to be interpreted as an honest subset of R. Any two straightforward models of R are isomorphic. Whether or not they have nonmeasurable sets depends on which set theory you use in the preceding definition; if you use ZFC, then they have nonmeasurable sets. (2) Define a "whacked model of the reals" to be a set R, with relations +, *, 0, 1, =, <, and a set S, with relations \subset and \in. These obey the axioms of "reals and sets of reals". So R is an honest ordered field with respect to (+, *, 0, 1, =, <); the Dedekind axiom holds if we interpret "set" as "element of S"; and (S, \subset, \in) has all the properties that the subsets of R should have according to ZF (no C). However, the elements of S do not have to be all the subsets of R; they just have to be all the subsets whose existence can be proven from ZF and the axioms of the reals. Not all whacked models are isomorphic; and some of them don't have nonmeasurable sets. 

