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Wikipedia states "While we cannot construct a well-order for the set of real numbers R, AC guarantees that such an order exists."

Ok, so the Well-Ordering Theorem states that the reals R can be well-ordered. Yet no one has been able to find a well-ordering for R in 100 years since the Well-Ordering Theorem was proved based on the Axiom of Choice.

I just want to know if mathematicians are still looking for an explicit well-ordering of R right now, or has someone proven that an explicit construction is impossible (despite its existence)?

And what about other uncountable sets? Has it been proven no explicit contructed well-ordering is possible for any uncountable set (that does not already have an explicit well-ordering defined)?

Ok, so the Well-Ordering Theorem states that the reals R can be well-ordered. Yet no one has been able to find a well-ordering for R in 100 years since the Well-Ordering Theorem was proved based on the Axiom of Choice.

I just want to know if mathematicians are still looking for an explicit well-ordering of R right now, or has someone proven that an explicit construction is impossible (despite its existence)?

And what about other uncountable sets? Has it been proven no explicit contructed well-ordering is possible for any uncountable set (that does not already have an explicit well-ordering defined)?

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