Surely can't be exactly what you mean, as exists(our Univese) and ¬exists(everything else) seems coherent if rather unlikely

I would dispute this, on the grounds that my deductions in formal systems come from somewhere that has a causal relation to my brain - the formal system causes me to be more likely to deduce the things which are valid deductions than the things that aren't. So, if I 'exist', I maintain that the formal systems have to 'exist' too, unless you're happy with 'existing' things being causally influenced by 'non-existing' things - in which case there's not a lot of point in asserting that ¬exists(infinite sets). A definition of 'exists' which doesn't satisfy my coherence requirements is, I am attempting to argue, simply a means of sneaking in connotations.