The phrase "arithmetic can be" is correct and usable in written English.
You can use it when discussing the nature or characteristics of arithmetic in various contexts, such as education or mathematics.
Example: "Arithmetic can be challenging for some students, especially when they encounter complex problems."
Alternatives: "arithmetic may be" or "arithmetic is capable of".
However, Lucas draws an invidious comparison: a human being with a full command of arithmetic can be consistent (even if he is actually inconsistent due to inattention or wishful thinking).
If one accepts a certain open-endedness of the collection of arithmetical predicates, then a categoricity theorem of sorts for arithmetic can be obtained without overstepping the bounds of first-order logic and without appealing to an informal concept of computability.
Because, in both ZFC and NBG, elementary arithmetic can be developed, Gödel's theorem applies to these two theories.
Moreover, given suitable axioms, standard postulates for natural-number arithmetic can be derived as theorems within set theory.
Justification for this terminology rests with the fact that the Peano postulates (five axioms published in 1889 by the Italian mathematician Giuseppe Peano), which can serve as a base for arithmetic, can be proved as theorems in set theory.
If so, then wouldn't that vindicate the suggestion that arithmetic can be known a priori?
Taken together, our modeling results show that arithmetic division (as well as other arithmetic operations) can be implemented simply using analog post-translational chemical kinetics.
The method is based on the concept of bounding hyperplane arithmetic, which can be viewed as a generalization of interval arithmetic.
One version of Gödel's first incompleteness theorem states that for any consistent axiomatic theory of arithmetic, which can be recognised to be sound, there will be an arithmetic truth viz., its Gödel sentence not provable in it, but which can be established as true by intuitively correct reasoning.
That is, there are sentences of first-order arithmetic that can be deduced from the second-order induction axiom (together with the other axioms of arithmetic, which are common to first-order and second-order arithmetic) but not from the instances of the first-order induction schema (see Shapiro 1991: 110).
Arithmetic codes can be viewed as tree codes.
