Do you mean the underlying math we need for probability theory does not allow us to construct or derive the part of number theory that Gödel’s theorem needs?
Essentially. The specifics are explained in more detail in the comments above.
Going by Wikipedia, it means that we don’t know a non-empty set of probabilities has a least upper bound.
That is not true. What Wikipedia article are you referring to?
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.
Essentially. The specifics are explained in more detail in the comments above.
That is not true. What Wikipedia article are you referring to?
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.