The phrase "arithmetic statement" is correct and usable in written English.
It can be used in mathematical contexts to refer to a statement that involves arithmetic operations or expressions.
Example: "The equation 2 + 2 = 4 is a simple arithmetic statement."
Alternatives: "mathematical expression" or "numerical statement".
(On the other hand, Kant famously rejects the idea that the arithmetic statement given above (or any arithmetic statement) is analytic.
The First Incompleteness Theorem provides a counterexample to completeness by exhibiting an arithmetic statement which is neither provable nor refutable in Peano arithmetic, though true in the standard model.
For example, where + is the binary addition function of arithmetic, we may represent the arithmetic statement 2+3=5 in our language as a claim of the form +(2,3,5), where + is taken to be a 3-place functional relation that obeys the condition: + x,y,z) & + x,y,w) → z=w.
Participants were asked to correct a false arithmetic statement, presented in Roman numerals constructed from matchsticks, by moving one stick from one position to another position without adding or discarding a stick (see figure 1).
For example, pure arithmetic statements such as 6 + 7 = 13 are not counted as formulas.
And he thought that every arithmetical statement that can be proved by making a detour through higher mathematics, can also be proved directly in Peano Arithmetic.
Then Kurt Gödel proved that there exist arithmetical statements that are undecidable in Peano Arithmetic (Gödel 1931).
Thus, we have here a case of a statement of arithmetic (and, in fact, a very simple one) that cannot be settled on the basis of the standard axioms.
Here Con(PA) is a statement of arithmetic that expresses the informal statement that PA is consistent.[1] Under slightly stronger assumptions (for example, that PA is Σ01-sound[2]) one can strengthen the conclusion by adding that PA does not prove ¬Con(PA); in other words, under this stronger assumption, Con(PA) is independent of PA.
Gödel's ingenious argument was based on the observation that syntactical statements about the language of mathematics can be translated into statements of arithmetic, hence into the language of mathematics.
Boolos 1989 exploits a Berry-type argument to prove incompleteness in the form "there is no algorithm whose output contains all true statements of arithmetic and no false ones" without appealing to diagonalization.
