Any given body of truths is necessarily internally consistent. Therefore mathematics, setting aside the odd screw-up, is consistent.
People make the mistake of assuming that ‘consistent’ means ‘capable of being axiomatized.’ Being able to axiomatize a given body of truths is sufficient but not necessary for establishing its consistency.
And this method of establishing the consistency of the class of mathematical truths is not a possibility, simply because any given axiomatization is itself a mathematical result and is therefore itself what is described by other mathematical results.
Setting aside a few Mickey Mouse domains, consistency is either established in a local, block-chain fashion, by comparing statements with other statements in their immediate dialectical vicinity, or by establishing truth. And the idea that consistency is a prerequisite for mathematical truth is a by-product of the mistaken idea that mathematics describes nothing and is simply the closure under the consequence of relation of arbitrary chosen first principles. The truth is that, in mathematics as in other domains, consistency is a consequence of truth, not vice versa.