Next time somebody tells you about their own "bleem" just consider it, and don't throw them to the street as a heretic.
Mathematicians are aware of bleem. It's what you get when you remove the induction axiom from Peano arithmetic: A number which is not in the set {0, 1, 2, ...}.
In general the higher up you go in academia the more open people are to "variant realities". Most of our analysis of curved space-time comes to us thanks to mathematicians who thought they were playing with entirely theoretical constructs ("what happens if we remove the parallel postulate?").
You actually do not have to remove any axioms from Peano arithmetic to prove that a model of PA with such a 'bleem' exists. It is a straightforward consequence of the compactness theorem: just enrich PA with a new constant c and an infinite series of axioms 'c != N' for every numeral N.
For any finite set of these new axioms a model exists (just set c to a large enough number), therefore by the compactness theorem there exists a model which satisfies all our new axioms.
Except the number you get this way is not between 3 and 4. It's essentially infinite with no functions in the model that separate it from the finite numbers (and this can be proven in PA: if a number isn't 0..N, then it's greater than N).
Generally, I don't think it's fair to characterize this story as something studied by mathematicians. Really, it's an exercise in reasoning about nonsense.
Inequality can be defined in Peano arithmetic in a total way. Maybe you're talking about defining some relation on the model? But what justifies calling such a relation an inequality?
Mathematicians are aware of bleem. It's what you get when you remove the induction axiom from Peano arithmetic: A number which is not in the set {0, 1, 2, ...}.
In general the higher up you go in academia the more open people are to "variant realities". Most of our analysis of curved space-time comes to us thanks to mathematicians who thought they were playing with entirely theoretical constructs ("what happens if we remove the parallel postulate?").