Well-ordering of the Reals

  • Thread starter dmuthuk
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In summary, the conversation discusses the relationship between the axiom of choice and the existence of a well-ordering for any set, particularly the set of real numbers. It is questioned whether the axiom of choice is necessary for the reals to be well-ordered and if it is needed for all uncountable sets. The possibility of proving the well-ordering of the reals without the axiom of choice is also discussed. However, it is stated that there is no such proof and that the independence of AC has been shown using a model where the reals cannot be well-ordered.
  • #1
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We all know that the axiom of choice is equivalent to the existence of a well-ordering for any set. And, this of course implies that [tex]\mathbb{R}[/tex] can be well-ordered, in particular. However, how do we know that the axiom of choice is actually needed in the case of the reals? That is, if we remove the axiom of choice, do the reals become a set that cannot be well-ordered? Furthermore, is the axiom of choice needed for every uncountable set?
 
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  • #2
You're essentially asking if ZF + there exists a well-ordering of the reals is weaker than ZFC, right?
 
  • #3
CRGreathouse said:
You're essentially asking if ZF + there exists a well-ordering of the reals is weaker than ZFC, right?

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.
 
  • #4
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.

No there isn't. When Cohen proved the independence of AC he used a model in which there was no well-ordering of the reals.
 
  • #5
Does that actually prove what dmuthuk asked? I know that ZF + "there is no well-ordering of the reals" is consistent*, but what about ZF + ¬C + "there is a well-ordering of the reals"?


* By "consistent", I mean "equiconsistent with ZFC".
 

1. What is the well-ordering principle?

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.

2. How does the well-ordering principle apply to the real numbers?

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.

3. Why is the well-ordering principle important in mathematics?

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.

4. What is the connection between the well-ordering principle and mathematical induction?

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.

5. Can the well-ordering principle be proven?

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.

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