The phrase "axiomatic theory" is correct and usable in written English.
It can be used in contexts related to mathematics, logic, or philosophy, where a set of axioms is foundational to a theoretical framework.
Example: "In his paper, he proposed an axiomatic theory that could unify the various branches of mathematics."
Alternatives: "axiomatic system" or "axiomatic framework".
Thus, in an axiomatic theory of sets, set and the membership relation ∊ are undefined terms.
The method of models, however, offers no hope for proving the consistency of an axiomatic theory of sets.
(Gödel showed that any consistent axiomatic theory that comprises a certain amount of elementary arithmetic is incapable of being completely axiomatized).
The mathematical theory (developed by the formalists) to cope with proofs about an axiomatic theory T is called proof theory, or metamathematics.
Thus, the existence of undecidable sentences in each such theory points out an inevitable gap between the sentences that are true in mathematics and sentences that are provable within a single axiomatic theory.
There was considerable progress in this direction, and there emerged both a powerful school of mathematical logicians (notably in Poland) and an axiomatic theory of sets that avoided Russell's paradoxes and the others that had sprung up.
In the case of set theory and, indeed, of axiomatic theories generally, the alternative is a direct approach to the problem.
It's set in a pre-culture wars America, where anti-intellectualism was prevalent but two articulate, self-indulgent and arrogant debaters could pull in huge television audience while lamenting the other's inability to grasp "axiomatic" theories.
Modern axiomatic theories of measurement derive from the work of two German scientists, Hermann von Helmholtz and Otto Hölder, and contemporary work on the application of measurement theory to psychology and economics derives in large part from the work of Oskar Morgenstern and John von Neumann.
They concern the limits of provability in formal axiomatic theories.
Some axiomatic theories include Zalta 1983 and Jubien 1988.
