The phrase "arithmetic sentences" is correct and usable in written English.
It can be used in contexts related to mathematics, particularly when discussing expressions or equations that involve arithmetic operations.
Example: "In our math class, we learned how to create and solve various arithmetic sentences involving addition and subtraction."
Alternatives: "mathematical expressions" or "numerical statements".
Intuitively, TRUEN is the set of those natural numbers that are Gödel numbers of true arithmetic sentences of vocabulary τ; and ψ(x) says that x is a Gödel number in the extension of the predicate TRUE.
Skolem's definable ultrapower construction from 1933 (see Skolem 1933) gives a direct construction of a non-standard model of True Arithmetic (which extends Peano arithmetic, being the set of arithmetic sentences true in the natural numbers).
Using code numbers for arithmetic sentences, he was able to demonstrate a correspondence between sentences of mathematics and facts about which sentences are and are not provable in PA.
Hence, if the arithmetic sentence were provable, p would also be provable contradicting the previous result.
The first half leads to Gödel's theorem on consistency proofs, which says that if a system is consistent, then the arithmetic sentence expressing the consistency of the system cannot be proved in the system.
The proof of this theorem consists essentially of a formalization in arithmetic of the arithmetized version of the proof of the statement, "If a system is consistent, then p is not provable"; i.e., it consists of a derivation within number theory of p itself from the arithmetic sentence that says that the system is consistent.
In this way he can deny, for arithmetic at least, that there are any non-determinate sentences since every true arithmetic sentence is provable using the ω-rule (relative to a fairly weak finitary logic, considerably weaker than classical logic).
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.
Solovay's completeness theorem provides an alternative way to construct many arithmetical sentences that are not provable in Peano Arithmetic.
It says that if first-order arithmetic is ω-consistent (which it is believed to be), then there must be arithmetical sentences that can neither be proved nor disproved by the formal procedures of first-order arithmetic.
Higher mathematics can prove arithmetical sentences, such as consistency statements, that are beyond the reach of Peano Arithmetic.
