There are many questions about well-ordered sets on the main and I consider well-orders to be an important topic. So perhaps they could deserve their own tag. On the other hand, many questions about well-orders are at the same time questions about ordinals so tagging them by both ordinals and (w...

It is well-known (forgive the pun) that the axiom of choice (which states that the product of every non-empty family of non-empty sets is non-empty) implies that the proper class of all cardinal numbers is well-ordered, at least in the presence of the ZF axioms. Does the converse hold? If not, i...

Given a well-orderable infinite set $A$, can we always say that the set $$\left\{R\in A\times A:\langle A,R\rangle\, \text{is a well-ordering}\right\}$$ has cardinality $2^{|A|}$? How much Choice is required for the proof of this? I believe that where $A$ is countably infinite, we can proceed wi...

I tried to write a formal proof for the theorem: $A$ subset of $\mathbb R$ well ordered by the normal order $\implies A$ is at most of cardinality $\aleph_0$. Any suggestions? Thanks.

Is there an explicit well-ordering of $\mathbb{N}^{\mathbb{N}}:=\{g:\mathbb{N}\rightarrow \mathbb{N}\}$? I've been thinking about that for awhile but nothing is coming to my mind. My best idea is this: Denote by $<$ the usual "less than" relation on $\mathbb{N}$. Since $\mathbb{N}^{\mathbb{N}}$...

There is no known well-order for the reals. Is there a known well-order for any uncountable set? If not, is it known whether or not an axiom stating that only countable sets can be well-ordered is consistent with ZF?

So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: as anyone constructed a well ordering on the reals? First, I was going to ask this questi...

The tag math-jokes was created recently. The tag-wiki is empty, but based on the only question currently having this tag it seems that the intention is that this tag is to be used for questions related to mathematical jokes and questions asking for explanations of such jokes. I can imagine that ...

So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My question is: as anyone constructed a well ordering on the reals? First, I was going to ask this questi...