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Getting rid of the 'exists' concept is more or less what I'm trying to do - or rather, show that if you have an 'exists' concept such that ¬exists(infinite sets) then your 'exists' concept is incoherent; moreover, that an 'exists' defined by exists(our Universe) and ¬exists(everything else) is not an important concept and should be detached from the connotations the savannah brain likes to associate with things that 'exist'.
"exist" doesn't have a referent. Any attempt to define it will either be special pleading (my universe is special, it "exists", because it's the one I live in!), or will give a definition that applies equally to all mathematical structures.
Surely can't be exactly what you mean, as exists(our Univese) and ¬exists(everything else) seems coherent if rather unlikely, and seems consistent at our present state of knowledge with ¬exists(infinite sets).
It seems your 'exists' concept is pretty much indistinguishable from 'logically coherent', and that that's the whole point you're trying to make - that we're in no position to distinguish these, and should simply abandon the 'exists'.
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.