formalizing the theory of real numbers

Question

Does anybody know what is the difference between the second order theory of the real numbers and the theory of the real numbers formalized in ZFC? Is any of them more expressive than the other? Since the real numbers can not be axiomatized in FOL we have to use one of the solutions above. Which method do mathematicians use?

Answer 1

Since you're curious, here's a curious fact. The computable reals have exactly the same first-order theory as the 'real' reals. And for any real-world (engineering, physics, ...) application one needs (and can manipulate) only computable reals. So arguably we don't need anything more than the first-order theory practically speaking. More mathematically... $ \def\eq{\leftrightarrow} $

Notice that if you look carefully at the categorical second-order axiomatization of the reals, it has just one second-order axiom $X$ that states that every bounded set of reals has a least upper bound, but this axiom is useless unless you also have axioms that permit you to construct sets of reals. All $X$ can do by itself is to force the meta-system (say ZFC) to 'see' that all models of the second-order axiomatization are isomorphic, simply because $X$ 'invokes' the meta-system's viewpoint (namely to 'know' what are sets of reals). The meta-system is certainly going to be equivalent to a first-order one (and ZFC already is), because it must have a recursive set of rules, and hence if it is consistent then it has a countable model. So (in the words of André Nicolas) the problem of categoricity just gets transferred upwards.

To make it clearer, suppose you believe that ZFC is meaningful. Then you clearly believe that ZFC is consistent. Then by a proof in ZFC you believe that there is a countable model $M$ of ZFC. In $M$ you can find the set $R$ corresponding to the reals as given by a construction (existential statement) in ZFC. $R$ satisfies the second-order axiomatization of reals from the viewpoint of $M$, but $R$ only has countably many elements from the viewpoint of ZFC. Do you consider $R$ to be the reals? No, but what are the reals? You can't just say "as constructed in ZFC", since $M$ is a model of ZFC and $R$ is a model of your chosen axiomatization according to $M$.

Next you may try using second-order logic with Henkin semantics to axiomatize the real numbers, so that it is more 'independent' of the foundations. But then as mentioned above you need to add set-existence axioms to even be able to use the second-order supremum axiom $X$. What could you add? The obvious choice would be to permit construction of any set $\{ x : P(x) \}$ where $P$ is some $1$-parameter sentence over the language of real arithmetic. But would you allow $P$ to contain only first-order quantifiers?

If so, then the whole thing ends up reducing to (being conservative over) the first-order theory of the reals, because such constructions are equivalent to definitorial expansions, and the existence of the supremum of definable bounded sets of reals is a first-order schema that is true in the reals and hence in any model of its (complete) first-order theory.

If no, then you can construct $N = \{ n : \forall S\ ( 0 \in S \land \forall k\ ( k \in S \to k+1 \in S ) \to n \in S ) \}$ in the resulting theory $R_2$. Note that $R_2$ easily proves that $0 \in N$ and also that $\forall k\ ( k \in N \to k+1 \in N )$, so $R_2$ can carry out induction over natural numbers as follows. Given any $1$-parameter sentence $P$ such that $P(0) \land \forall n \in N\ ( P(n) \to P(n+1) )$, we can in $R_2$ construct $Q = \{ n : n \in N \land P(n) \}$ and prove that $0 \in Q \land \forall k\ ( k \in Q \to k+1 \in Q )$, and then prove that $\forall n \in N\ ( n \in Q )$ (by the definition of $N$), which gives $\forall n \in N\ ( P(n) )$. Thus $R_2$ interprets arithmetic. Note that $R_2$ has a proof verifier program, and hence $R_2$ is essentially syntactically incomplete, unlike the first-order theory of the reals.

But $R_2$ has a subtle issue of impredicativity, in that it can construct a set of objects defined using quantification over all sets of objects, including the one being defined. This circularity is precisely what led to Russell's paradox in naive set theory. So one could question whether $R_2$ is meaningful or not. Of course, ZFC proves that the reals (as constructed in ZFC) satisfy $R_2$, but ZFC is itself impredicative, so if you wish you can transfer that question upwards...

Answer 2

If you mean to ask how mathematicians as opposed to logicians deal with the real numbers, the answer tends to be on the side of set theory. Thus, the classical construction using Dedekind cuts relies explicitly on subsets of the rationals, whereas the construction using equivalence classes of Cauchy sequences also involves set theory in justifying the quotient construction. In fact, a majority of mathematicians have trouble relating to the distinction between first and second order logic, and the fact that the familiar completeness axiom involves second-order quantification tends to surprise them.

Answer 3

Thank you for all of you for the answers. I have been thinking more about the problem of the different constructions of the real numbers lately. As we can see from the previous posts there are some very interesting possible constructions of the reals which are very similar to the "standard"(?) construction, in that they are the same from the perspective of first order logic. However some of the most important topics in the theory of reals are not first order. Namely the convergence of series. I think that the reason for the use of these second order axioms is that we can prove the existential theorems of analysis (e.g. existence of Lebesgue integral) for a wide set of objects and so we can do physics. Also some theories which use the reals become not-so-hard and not-so-ugly. (Theory of Hilbert spaces for example) Sorry for my broken English I'm still learning the language.

edit: In Physics one only needs the computable reals but if we have a convergent sequence of computable reals the lim of this sequence can be uncomputable. So the nice theorems of analysis may not be true.