You say: “One can axiomatize sciences as suggested by Hilbert long ago, but it has essentially two sets of axioms — math ones, taken from math, and empirical ones taken from assigning symbols to observations.”
Axiom means foundational proposition and because Euclid explicitly based geometry on axioms many people think the term refers alone to mathematics. This is not quite correct.
Newton famously put his PHYSICAL axioms on the first pages of Principia. In methodological analogy ECONOMIC axioms have to be laid down by economists. Axioms define the subject matter/paradigm.
The actual situation is this: Orthodoxy clings to a thoroughly refuted axiom set and Heterodoxy so far has failed to formulate a new one. As Keynes famously put it: “Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry. Something similar is required to-day in economics.” (1973, p. 16)
Only retarded economists still maintain that axioms apply exclusively to mathematics.
There is but ONE way to build up a valid theory: “The basic concepts and laws which are not logically further reducible constitute the indispensable and not rationally deducible part of the theory. It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” (Einstein, 1934, p. 165)
The methodological term for ‘basic concepts and laws which are not logically further reducible’ is axioms. As long as economic theory is not based on a formally and materially consistent set of axioms it is outside of science and cannot be taken seriously.
Einstein, A. (1934). On the Method of Theoretical Physics. Philosophy of Science, 1(2): 163–169. URL
Keynes, J. M. (1973). The General Theory of Employment Interest and Money. London, Basingstoke: Macmillan.
Related 'If it isn’t macro-axiomatized, it isn’t economics' and '10 steps to leave cargo cult economics behind for good' and 'Both orthodox and heterodox economists are cargo cult scientists' and 'Still in the proto-scientific wood' and 'Heterodoxy ― an axiomatic failure just like Orthodoxy' and 'Do first your macroeconomic homework!' and 'The canonical macroeconomic model' and 'Stop Recycling Dead Economic Theories, Start the Paradigm Shift'. For details of the big picture see cross-references Paradigm Shift and cross-references Axiomatization.
Immediately preceding There is no truth in political economics