Example of the property of rational numbers that must be proved using the axioms of real numbers?

Question

According to Godel's incompleteness theorem, some statements about rational numbers are independent from the axioms of the rational numbers given in Wikipedia--an ordered field generated by 1. So can anyone give a concrete example of such a proposition about rational numbers that is undeducible from the axioms of $\mathbb{Q}$ but provable in the axioms of $\mathbb{R}$? Or if such a preposition doesn't exists?

Answer 1

There is some vagueness in the question, but I want to write an answer about Julia Robinson's work in the area, particularly her 1949 paper Definability and Decision Problems in Arithmetic, Journal of Symbolic Logic 14(2). This paper proves some undecidability results for theories about the rational numbers, which are similar to the undecidability results that the incompleteness theorem gives for theories about natural numbers.

Robinson first proves some lemmas about the interdefinability of arithmetical operations. For example, addition of natural numbers is first-order definable from the multiplication and successor operations.

The next theorem, one of the main results, is that the set of integers and the set of positive integers are first-order definable in the field of rational numbers, using only addition and multiplication operations on the rationals as well as quantifiers over the rationals. This is a remarkable result, because we might think that the rationals are somehow "better behaved" than the integers, or at least "differently behaved" from the integers. Robinson's result uses a mixture of logic and number theory, much like her later contributions to Hilbert's 10th problem.

Robinson then turns to the undecidability of the theory of $(\mathbb{Q}, 0,1,+, \times,=)$. A theory is decidable if the set of sentences provable from the theory is decidable by an algorithm, and undecidable otherwise.

By using her definition of the integers into the rationals, along with logical methods, Robinson shows that the theory consisting of the set $T$ of true sentences in the rationals is undecidable, as is the theory generated from any subset of $T$. This is not quite an "incompleteness" result, because the theory consisting of all true sentences in the rationals is a complete theory.

Robinson obtains a nice corollary from her theorem about the rationals: the theory of fields (i.e. the theory generated by axioms for a field) is undecidable. The parallel result for rings instead of fields was proved earlier by Tarski.

The formula that defines the integers within the rationals is not too hard. Let $\phi(A,B,K)$ be $$(\exists X,Y,Z \in \mathbb{Q})[2+ABK^2 + BZ^2 = X^2 + AY^2].$$ Then a rational number $N$ is an integer if and only if $$ (\forall A,B \in\mathbb{Q})[ ( \phi(A,B,0) \land (\forall M \in \mathbb{Q})[\phi(A,B,M)\to\phi(A,B,M+1)] ) \to \phi(A,B,N)].$$ The proof that this is correct is not trivial.

Answer 2

The set of rational numbers $\mathbb{Q}$ (as well as the set of nonnegative rational numbers $\mathbb{Q}^{>0}$) does not satisfies any kind of principle of induction, which is a key ingredient in (the proof of) Gödel's incompleteness theorem. So, I don't see how Gödel's incompleteness theorem could say something about $\mathbb{Q}$ or $\mathbb{Q}^{>0}$.

By the way, what do you mean by giving an axiomatic characterization of $\mathbb{Q}$? First of all, you have to fix a language for this axiomatization. To maintain the analogy with Peano arithmetic, let us say that the language is first order logic with the symbols for an order relation, addition, multiplication and their neutral elements. If you want to characterize $\mathbb{Q}$, you would like to say that $\mathbb{Q}$ is an ordered field, isn't it? In such a language you can easily axiomatize the fact of being an ordered field.

According to Löwenheim–Skolem theorem, there is no axiomatic characterization of $\mathbb{Q}$ in such a language. Indeed, $\mathbb{Q}$ is a countably infinite ordered field and by Löwenheim–Skolem theorem for any infinite cardinality $\kappa$ there exists an ordered field $F$ of cardinality $\kappa$ which is elementary equivalent to $\mathbb{Q}$ (i.e. it satisfies the same formulas as $\mathbb{Q}$ in such a language). In particular, if $\kappa$ is greater than the cardinality of $\mathbb{Q}$ (and $\mathbb{N}$), then $F$ is not isomorphic to $\mathbb{Q}$ because there is no bijection between $F$ and $\mathbb{Q}$.

Summing up, your question presumes (when you say "axioms of the rational numbers") the existence of an axiomatic characterization of $\mathbb{Q}$, but unfortunately, this axiomatic characterization does not exist.