There is also the reverse case, where it is often common practice in math or logic to ignore bizarre and unnatural counterexamples. For example, first-order Peano arithmetic is often identified with Peano arithmetic in general, even though the first order theory allows the existence of highly “unnatural” numbers which are certainly not natural numbers, which are the subject of Peano arithmetic.
Another example is the power set axiom in set theory. It is usually assumed to imply the existence of the power set of each infinite set. But the axiom only implies that the existence of such power sets is possible, i.e. that they can exist (in some models), not that they exist full stop. In general, non-categorical theories are often tacitly assumed to talk about some intuitive standard model, even though the axioms don’t specify it.
There is also the reverse case, where it is often common practice in math or logic to ignore bizarre and unnatural counterexamples. For example, first-order Peano arithmetic is often identified with Peano arithmetic in general, even though the first order theory allows the existence of highly “unnatural” numbers which are certainly not natural numbers, which are the subject of Peano arithmetic.
Another example is the power set axiom in set theory. It is usually assumed to imply the existence of the power set of each infinite set. But the axiom only implies that the existence of such power sets is possible, i.e. that they can exist (in some models), not that they exist full stop. In general, non-categorical theories are often tacitly assumed to talk about some intuitive standard model, even though the axioms don’t specify it.
Eliezer talks about both cases in his Highly Advanced Epistemology 101 for Beginners sequence.