"All publicly funded pharma research," he said, as though correcting a mistake in elementary arithmetic.
I mentioned this to my lunchmate, finishing with: "And that's why all pharma research must be open access". "All publicly funded pharma research," he said, as though correcting a mistake in elementary arithmetic.
In sum, when it is said that "a certain amount of elementary arithmetic can be carried out within a system", what is meant is either that the system is an axiomatizable extension of Q, or that Q can be interpreted in it.
For a proof in a formal system of higher mathematics or of elementary arithmetic is a finite combinatorial object which can, modulo coding, be considered to be a natural number.
(In Quine's well-known metaphor, they stand near the centre of the 'web of belief'.) The truths of elementary arithmetic are an example: they play a role in almost every branch of systematic knowledge.
(Gödel showed that any consistent axiomatic theory that comprises a certain amount of elementary arithmetic is incapable of being completely axiomatized).
A coarse analogy with elementary arithmetic operations, perhaps, can shed some light on the situation.
In the first and loose statements of the incompleteness theorems given above, the vague requirement that "a certain amount of elementary arithmetic can be carried out" occurred.
Then, of course, the unexpected happened when Gödel proved the impossibility of a complete formalization of elementary arithmetic, and, as it was soon interpreted, the impossibility of proving the consistency of arithmetic by finitary means, the only ones judged "absolutely reliable" by Hilbert.
The former is an introductory mathematics textbook, proceeding from elementary arithmetic to algebraic calculations.
