The phrase "a arithmetical" is not correct in English.
It should be "an arithmetical" due to the vowel sound that follows. You can use it when referring to something related to arithmetic or mathematics, typically in a descriptive context.
Example: "The problem requires an arithmetical approach to find the solution."
Alternatives: "a mathematical" or "an arithmetic".
Note that the format of the competition is a bit like an arithmetical version of a spelling bee.
"Logical operations are performed with propositions, arithmetical ones with numbers," says Wittgenstein (WVC 218); "[t]he result of a logical operation is a proposition, the result of an arithmetical one is a number".
The history of mathematics has shown that making a "detour" through higher mathematics can sometimes lead to a proof of an arithmetical statement that is much shorter and that provides more insight than any purely arithmetical proof of the same statement.
Napier discovered that the basis for this computation was a relationship between an arithmetical progression a sequence of numbers in which each number is obtained, following a geometric progression, from the one immediately preceding it by multiplying by a constant factor, which may be greater than unity (e.g., the sequence 2, 4, 8, 16... ) or less than unity (e.g., 8, 4, 2, 1, 1/2... )...
One example is the "modal structuralism" of Hellman 1989, where an arithmetical sentence A is analyzed as ☐∀X∀f∀x[PA2 X/ℕ, f/s, x/0 A(X/ℕ, f/s, x/0)], where PA2 is the conjunction of the axioms of second-order Peano Arithmetic.
He even hints (though, in true Lacanian fashion, he waits until we've been talking for two hours and he really must be going) that he wants to be something he resembles, at first glance, even less than an arithmetical horse: namely, a shaman.
The first incompleteness theorem tells us that in any consistent formal theory whose theorems are recursively enumerable and which entails a certain (rather limited) amount of arithmetic, there will be an arithmetical sentence such that neither it nor its negation is provable.
Jevons distinguishes between a 'mean' (the approximation of a definite existing quantity) and an 'average' or 'fictitious mean' (an arithmetical average).
He said, in 1798, "Population, when unchecked, increased in a geometrical ratio and subsistence for man in an arithmetical ratio". So, the population growth follows an exponential growth curve.
It's a cumulative process but never merely an arithmetical one.
