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I can see no justification whatever for that method. What's so special about you, that you imbue things with "existing-ness" by sensing them? Surely that's an egregious Mind Projection Fallacy? "It's in my map, so it must be in some territory somewhere"?
I wasn't actually justifying the move - merely saying this is how the concept of existence comes about. My final sentence sums up what I actually think - that 'exists' is a deeply misleading concept in situations such as this, and should be confined to folk philosophy, where it's commonly used meaning is sufficient.
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.