About Tarski decision method and the integers

Question

Roughly speaking, An elementary expression in the algebra of the real numbers, is an expression build with the following objects:

Variables over the real numbers.

Constants $c\in \mathbb{N}$.

The symbols $+,-,\cdot, \div$ which denote sum, subtraction, multiplication and division of real numbers respectively.

The symbols $>, =$ that denote the relations "greater that" and "equal to" respectively, of real numbers.

The logic connectives $\vee$ (disjunction), $\wedge$ (conjunction), $\neg$ (negation) and $\Rightarrow$ (implication).

The universal ($\forall$) and existential ($\exists$) quantifiers.

By Tarsky result, we know that there exists an algorithm to decide, given an elementary expression in the algebra of real numbers, whether it is true or false.

However, we also know that it is impossible to write down with an elementary expression the statement ``$x$ is an integer'', that is, the expression $x\in \mathbb{Z}$ is not elementary, neither is the following expression about integer equations

$(\exists x, y, z \in \mathbb{Z})(x^3 + y^3 = z^3)$

My question is this: How can we prove that the previous expressions are not elementary ? Does anyone know a formal proof or at least an intuitive explanation ?

Thanks in advance !!

Greetings...

Answer 1

There are various ways to prove the result. One relies on a detailed examination of the structure of the sets of reals definable in the first-order language for the reals.

There is another way that uses some basic theorems. First is the theorem of Tarski that you referred to, which says there is an algorithm which, when you feed a sentence of the appropriate language into it, will tell you whether or not that sentence is true in $\mathbb{R}$.

The second theorem that we use is the well-known theorem to the effect that there is no algorithm which, when you feed a sentence into it, will tell you whether that sentence is true in $\mathbb{Z}$. (The classical version involves truth in $\mathbb{N}$ but it can be easily extended to the integers.)

If the set of reals that are integers were definable by a formula, then that formula could be used to translate mechanically any sentence $\varphi$ in which variables range over the integers into an equivalent formula $\varphi'$ in which variables range over the reals. Then Tarski's decision procedure could be used on $\varphi'$ to determine whether $\varphi$ is true in $\mathbb{Z}$, contradicting the fact there is no algorithm for determining truth in $\mathbb{Z}$.

Remark: Ordinarily, for Tarski's theorem the language uses the ordinary symbols of first-order logic with equality, a constant symbol $0$, and binary function symbols for addition and multiplication. A binary relation symbol $\lt$ can be used, but need not be. A separate function symbol for subtraction is not needed, and using a function symbol for division leads to technical problems because division by $0$ cannot be allowed.

Note that every individual integer is easily definable in the first-order theory of the reals. It is the set of integers that is not definable. That means that there is no formula $\varphi(x)$ of the first-order theory of the reals such that for for any real number $a$, $\varphi(a)$ is true in the reals if and only if $a$ is an integer.

One should perhaps in this context mention the following result of Julia Robinson. The integers are definable in the first-order theory of the rationals. The result, though accessible, is not easy.