The phrase "arithmetical language" is correct and usable in written English.
It can be used when discussing the specific terminology or symbols used in mathematics or arithmetic.
Example: "In order to solve the equation, we must first understand the arithmetical language that defines the operations involved."
Alternatives: "mathematical language" or "numerical language".
What Gödel showed is that, for any consistent, recursively axiomatized formal system, F, strong enough for arithmetic, there are truths expressible in purely arithmetical language which are not provable in F.
Once again numbers, arithmetical language.
(Carnap sought to accommodate Gödel's incompleteness results by separating analyticity from effective provability and by postulating that arithmetic consisted of an infinite series of ever richer arithmetical languages; see the discussion and references in section 3.2 below).
In first-order formalizations of arithmetic, this is formulated as a scheme: for each first-order arithmetical formula of the language of arithmetic with one free variable, one instance of the induction principle is included in the formalization of arithmetic.
(See also Moltmann (2013) for some challenges concerned with arithmetical vocabulary in natural language).
Then one can extend the language with arithmetical operations such as addition and multiplication, and with operators such as equality and inequalities to compare probability terms.
This also easily yields a weak version of the incompleteness result: the set of sentences provable in arithmetic can be defined in the language of arithmetic, but the set of true arithmetical sentences cannot; therefore the two cannot coincide.
The association of stored arithmetical properties with left hemispheric language areas also drives the assumption of this store being verbally mediated.
If the only ways of proving the consistency of arithmetic make essential use of notions which arguably belong to higher-order mathematics, then the consistency of arithmetic, even though it can be expressed in the language of Peano Arithmetic, is a non-arithmetical problem.
Of course solving arithmetical problems in arithmetic is in some cases practically impossible.
Axiom 1 says that an atomic sentence of the language of Peano arithmetic is true if and only if it is true according to the arithmetical truth predicate for this language (Tr0 was defined in Section 3.1).
