Very cool short story. One of the lessons I got from it is that you should never consider somebody crazy for having a crazy idea. Some of our most brilliant scientists, mathematicians, entrepreneurs, etc have been called crazy fools for having radical ideas (That ended up being right). Our society needs radical thinkers to shake things up, and revolutionize the world. Next time somebody tells you about their own "bleem" just consider it, and don't throw them to the street as a heretic.
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?
I am reminded of Ramanujan's proof that the sum of all positive integers is -1 / 12. The Wikipedia article has a fascinating excerpt from his letter to Dr. Hardy:
