To me, it’s all about reflective equilibrium. In our minds, we have this idea about what a “natural number” is. We care about this idea, and want to do something using it. However, when we realize that we can’t describe it, perhaps we worry that we are just completely confused, and thinking of something that has no connection to reality, or maybe doesn’t even make sense in theory.
However, it turns out that we do have a good idea of what the natural numbers are, in theory...a system described by the Peano axioms. These are given in second-order logic, but, like first-order logic, the semantics are built in to how we think about collections of things. If you can’t pin down what you’re talking about with axioms, it’s a pretty clear sign that you’re confused.
However, finding actual (syntactic) inference rules to deal with collections of things proved tricky. Many people’s innate idea of how collections of things work come out to something like naive set theory. In 1901, Bertrand Russell found a flaw in the lens...Russell’s paradox. So, to keep reflective equilibrium, mathematicians had to think of new ways of dealing with collections. And this project yielded the ZFC axioms. These can be translated into inference rules / axioms for second-order logic, and personally that’s what I’d prefer to do (but I haven’t had the time yet to formalize all of my own mathematical intuitions).
Now, as a consequence of Gödel’s first incompleteness theorem (there are actually two, only the first is discussed in the original post), no system of second-order logic can prove all truths about arithmetic. Since we are able to pin down exactly one model of the natural numbers using second-order logic, that means that no computable set of valid inference rules for second-order logic is complete. So we pick out as many inference rules as we need for a certain problem, check with our intuition that they are valid, and press on.
If someone found an inconsistency in ZFC, we’d know that our intuition wasn’t good enough, and we’d update to different axioms. And this is why, sometimes, we should definitely use first-order Peano arithmetic...because it’s provably weaker than ZFC, so we see it as more likely to be consistent, since our intuitions about numbers are stronger than our intuitions about groups of things, especially infinite groups of things.
All of this talk can be better formalized with ordinal analysis, but I’m still just beginning to learn about that myself, and, as I hinted at before, I don’t think I could formally describe a system of second-order logic yet. I’m busy with a lot of things right now, and, as much as I enjoy studying math, I don’t have the time.
I guess I was confused about your question.
To me, it’s all about reflective equilibrium. In our minds, we have this idea about what a “natural number” is. We care about this idea, and want to do something using it. However, when we realize that we can’t describe it, perhaps we worry that we are just completely confused, and thinking of something that has no connection to reality, or maybe doesn’t even make sense in theory.
However, it turns out that we do have a good idea of what the natural numbers are, in theory...a system described by the Peano axioms. These are given in second-order logic, but, like first-order logic, the semantics are built in to how we think about collections of things. If you can’t pin down what you’re talking about with axioms, it’s a pretty clear sign that you’re confused.
However, finding actual (syntactic) inference rules to deal with collections of things proved tricky. Many people’s innate idea of how collections of things work come out to something like naive set theory. In 1901, Bertrand Russell found a flaw in the lens...Russell’s paradox. So, to keep reflective equilibrium, mathematicians had to think of new ways of dealing with collections. And this project yielded the ZFC axioms. These can be translated into inference rules / axioms for second-order logic, and personally that’s what I’d prefer to do (but I haven’t had the time yet to formalize all of my own mathematical intuitions).
Now, as a consequence of Gödel’s first incompleteness theorem (there are actually two, only the first is discussed in the original post), no system of second-order logic can prove all truths about arithmetic. Since we are able to pin down exactly one model of the natural numbers using second-order logic, that means that no computable set of valid inference rules for second-order logic is complete. So we pick out as many inference rules as we need for a certain problem, check with our intuition that they are valid, and press on.
If someone found an inconsistency in ZFC, we’d know that our intuition wasn’t good enough, and we’d update to different axioms. And this is why, sometimes, we should definitely use first-order Peano arithmetic...because it’s provably weaker than ZFC, so we see it as more likely to be consistent, since our intuitions about numbers are stronger than our intuitions about groups of things, especially infinite groups of things.
All of this talk can be better formalized with ordinal analysis, but I’m still just beginning to learn about that myself, and, as I hinted at before, I don’t think I could formally describe a system of second-order logic yet. I’m busy with a lot of things right now, and, as much as I enjoy studying math, I don’t have the time.