Zermelo's axiomatization of set theory (1908; for more details see entries on set theory, Alternative Axiomatic Set Theories) supplied a sensible tool for blocking contradictions.
Von Neumann's system of 1925 deals with an alternative axiomatic foundation of set theory.
"What Francis said was that 'everybody has to take responsibility for what they were part of' - that's axiomatic and the alternative is no responsibility and no accountability".
It needs to be said that several authors refer to (Wang 2002) for an alternative axiomatic approach to time consistency of dynamic risk measures.
(Certainly they are not in the most standard set theory, ZFC. However, the SEP entry on Alternative Axiomatic Set Theories does mention quite a few theories which treat classes as first-class objects).
In the case of set theory and, indeed, of axiomatic theories generally, the alternative is a direct approach to the problem.
The proper definition of the alternatives is axiomatic, and altogether various combinations of some 53 axiom schemes are introduced, with none examined in detail.
Axiomatic now as the only alternative to doing nothing is the use of lethal force backed by the occasional choice of diplomacy as clean-up job.
The second valuable contribution of recent research in this field is the development of an alternative ethical approach through the axiomatic study of the properties of indices.
Then, we replace the weak continuity of the involved operators by the concepts of ww-compactness and ws-compactness and interchange the weak compactness by some alternative conditions formulated using the axiomatic measure of weak noncompactness.
