Is there a known axiom for ZFC that:
Fixes a total ordering on the cardinal numbers appearing in Cichon's diagram.
Is "natural looking" - in particular, its not allowed to be a conjunction of statements asserting that one particular cardinal is less than another, or anything else artificial like that.
Does not allow us to prove any equalities between the cardinal numbers in Cichon's diagram, except for an equality corresponding to the requirement that $\mathrm{add}(\mathcal{K}) = \mathrm{min}\{\mathrm{cov}(\mathcal{K}),\mathfrak{b}\},$ and another corresponding to $\mathrm{cof}(\mathcal{K}) = \mathrm{max}\{\mathrm{non}(\mathcal{K}),\mathfrak{d}\}.$