There’s also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that it’s not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities).
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/​2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/​morality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/​2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/​morality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.