Questions tagged [well-orders]
For questions about well-orderings and well-ordered sets. Depending on the question, consider adding also some of the tags (elementary-set-theory), (set-theory), (order-theory), (ordinals).
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Is there a known well ordering of the reals?
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 ...
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Is there any known uncountable set with an explicit well-order?
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 ...
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Why isn't this a well ordering of $\{A\subseteq\mathbb N\mid A\text{ is infinite}\}$?
So, to explain the title, I'm referring to the necessity of the axiom of choice in the existence of a well ordering on reals, or any uncountable set.
Now, while tweaking some sets, I came across this ...
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How strong is the axiom of well-ordered choice?
I sometimes see references to the "Axiom of Well-Ordered Choice," but I'm not sure how strong it is. It states that every well-ordered family of sets has a choice function.
By "well-ordered family," ...
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Well-orderings and the perfect set property
From a wellordering of an uncountable set of reals, Bernstein constructed a set of reals without the perfect set property. My question is, does an uncountable well-ordering itself violate the perfect ...
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Prove that there is no positive integer between 0 and 1
In my textbook "Elementary Number Theory with Applications" by Thomas Koshy on pg. 16, there is an example given just after the well ordering principle:
Prove that there is no positive ...
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Why do we need "canonical" well-orders?
(I asked this question on MO, https://mathoverflow.net/questions/443117/why-do-we-need-canonical-well-orders)
Von-Neumann ordinals can be thought of "canonical" well-orders, Indeed every ...
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Totally ordering the power set of a well ordered set.
Let's say I take a set $S$, where $S$ can be well ordered. From what I understand, one can use that well ordering to totally order $\mathscr{P}(S)$.
How does a body actually use the well ordering of $...
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Can Well Ordering Theorem Be Proved Without the Axiom of Power Set?
Can it be proved in ZFC - Pow (ZFC excluding the Axiom of Power Set) that Well Ordering Theorem holds? I have seen several proofs of Well Ordering Theorem in ZFC (including Zermelo's original one in ...
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Without appealing to choice, can we prove that if $X$ is well-orderable, then so too is $2^X$?
Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable.
Question: can we show the stronger result that if $X$ is well-...
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The well ordering principle
Here is the statement of The Well Ordering Principle: If $A$ is a nonempty set, then there exists a linear ordering of A such that the set is well ordered.
In the book, it says that the chief ...
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Explicit well-ordering of $\mathbb{N}^{\mathbb{N}}$
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 ...
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"There is no well-ordered uncountable set of real numbers"
I recently learned (from Munkres) about the axiom of choice, and how it implies the well-ordering theorem.
I've looked through various posts about how to well-order the reals (e.g. this one) but the ...
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How to define a well-order on $\mathbb R$?
I would like to define a well-order on $\mathbb R$. My first thought was, of course, to use $\leq$. Unfortunately, the result isn't well-founded, since $(-\infty,0)$ is an example of a subset that ...
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Every well-ordered set is isomorphic to a unique ordinal
I'm following a proof in Jech's book that every well ordered set is isomorphic to a unique ordinal and hitting a point where I'm not sure why a certain move is justified. He writes
Proof. The ...
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