CRGreathouse said:You're essentially asking if ZF + there exists a well-ordering of the reals is weaker than ZFC, right?
dmuthuk said:Yes, I believe I am. So, I guess what I wanted to know is if there exists a proof that the reals can be well-ordered without AC.
The well-ordering principle states that every non-empty set of positive integers has a smallest element. This means that any set of positive integers can be arranged in ascending order, and there is always a first element.
The well-ordering principle can also be extended to the real numbers. This means that any non-empty set of real numbers has a smallest element. However, unlike the positive integers, the real numbers are uncountable, so they cannot be arranged in a list.
The well-ordering principle is important because it helps to establish the existence of a smallest element in any non-empty set. This is useful in many mathematical proofs and helps to show that certain sets are well-behaved and have a clear structure.
The well-ordering principle is closely related to mathematical induction. In fact, the well-ordering principle is often used as a basis for mathematical induction. This is because the principle allows us to prove that a statement holds for the smallest element in a set, and then show that if it holds for any element in the set, it also holds for the next element.
The well-ordering principle is an axiom, which means it is accepted as a fundamental truth without proof. It cannot be proven using the standard rules of mathematics. However, many mathematical proofs rely on the well-ordering principle as an assumption or starting point.