@MoziburUllah is correct, but it is not so ambiguous as all that. At least not lately.
Since starting to dwell on its roots, math really has accepted a slightly less ambiguous formulation of than coherence we have had historically. The current model of math (along with the main forms in which formal logic is taught) is dual. It really only involves two proof positions -- deduction of implication via grammar of combination, and construction of 'worlds' or models via a grammar of description.
(These two positions, along with the distinction between them, go back to Euclid. So people who want to limit or minimize the assumption of coherence can easily just ignore the intermediate positions, and claim they were just spurious nonsense. This is what modern formalists do.)
Consistency may not be available via formal construction, but people's real intuition of validity or consistency is the one based upon models, not the one based on formal proofs. Then from a point of view focussed mainly on mathematics as the psychology of shared intuition, we really do not believe, deep down, that formal language is what really proves things. It is only a way of checking ourselves when we act upon our other intuitions, which we presume reliable, as long as we do not pursue them too far.
From that point of view, we can look at the stripped-down 'universes' of Von Neumann's V, or Goedel's L, (or even Conway's "Surreal Numbers") and agree that our system has a model. Then even if our formal manipulations ultimately do not really apply directly to the objects we are studying, our chosen base model does contain something isomorphic to them.
So, in any case where we are explicit enough about what we are saying to define isomorphism clearly, we can validate that what we are doing would not have a contradiction in it, by mapping its interpretation back onto our chosen base model.
This is the formalist escape from the ultimate weakness of formalism: to claim mathematical claims "only exist up to isomorphism" and to ignore the reliance of the existence of the base models upon intuition.