Second-order logic says that, “if the domain is the set of all real numbers,” then “one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum.”
Do you mean to say we can somehow prove this in first-order logic for numbers between 0 and 1, but we can’t extend it to the real number line as a whole?
It’s not about what you can prove, it’s what you can state. The first-order theory of the reals doesn’t even have the concepts to state such a thing. If your base theory is the reals, then sets of reals are a second-order notion.
No. I meant that we can prove that some specific sets of numbers have least upper bounds. What we cannot prove is that every bounded, nonempty set has a least upper bound.
Second-order logic says that, “if the domain is the set of all real numbers,” then “one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum.”
Do you mean to say we can somehow prove this in first-order logic for numbers between 0 and 1, but we can’t extend it to the real number line as a whole?
It’s not about what you can prove, it’s what you can state. The first-order theory of the reals doesn’t even have the concepts to state such a thing. If your base theory is the reals, then sets of reals are a second-order notion.
No. I meant that we can prove that some specific sets of numbers have least upper bounds. What we cannot prove is that every bounded, nonempty set has a least upper bound.