No one shall be able to drive us from the paradise that Cantor Conway created for us. - Hilbert
From Cantor $\rightarrow$ Conway
You are free to do whatever you want in mathematics. You are! - Antonsen
The essence of mathematics lies in its freedom. - Cantor
It seems to us, however, that mathematics has now reached the stage where formalisation within some particular axiomatic set theory is irrelevant, even for foundational studies.
What is proposed is instead that we give ourselves the freedom to create arbitrary mathematical theories of these kinds, but prove a metatheorem which ensures once and for all that any such theory could be formalised in terms of any of the standard foundational theories.
(i) Objects may be created from earlier objects in any reasonably constructive fashion.
(ii) Equality among the created objects can be any desired equivalence relation.