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).

A well-order is a linear order where every non-empty set has a minimal element. Equivalently, it is a partial ordering where every non-empty set has a minimum.

We say that a set $A$ is well-ordered if it comes with a well-order of $A$; and that it is well-orderable if there is a well-ordering of $A$. Assuming the axiom of choice (or equivalently, Zorn's lemma) every set is well-orderable. That is Zermelo's theorem.

Well-ordered sets are exactly the linear orders on which we can perform recursive definitions and inductive proofs. The simplest examples include all the finite linear orders, and the natural numbers.

On the other hand, the rational numbers, and the real numbers, are all examples of linear orders which are not well-ordered.

The study of well-orders goes hand in hand with the study of ordinals, which are order types of well-ordered sets, and the von Neumann ordinal assignment, which gives us a specific well-ordered set isomorphic to a given well-order.

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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 question is: Has anyone constructed a well ordering on…
Seamus
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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 consistent with ZF?
Harry Johnston
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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 set property? Equivalently, if $W \subset…
Trevor Wilson
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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 $S$ to construct a total ordering of…
Kangin
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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," I don't mean that the sets within the family are…
Mike Battaglia
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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 integer between $0$ and $1$. My question is how can you even…
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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 this: Denote by $<$ the usual "less than" relation on…
ragrigg
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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-orderable, then so too is $2^X$? Proof of…
goblin GONE
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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 advantage of the well ordering principle is that it…
Yuan
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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 related proofs are beyond me. From what I gather,…
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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 : We start with the natural numbers, $N$. We take…
Sohan Biswas
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Interlacing addition of ordinals

Given ordinals $\alpha,\beta$, one definition of $\alpha+\beta$ is as the order type of the disjoint union $\alpha\sqcup\beta$ ordered with all the elements of $\alpha$ before the elements of $\beta$. But this looks like only one point on a spectrum…
Mario Carneiro
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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 1904, Zermelo's in 1908), all involving choosing a…
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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 doesn't have a minimal element. My next thought was…
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Direct proof of principle of transfinite induction

This is a problem from the book Set theory by You-Feng Lin. Principle of Transfinite Induction Let $(A,\le)$ be a well-ordered set. For each $x \in A$, let $p(x)$ be a statement about $x$. If for each $x \in A$, the hypothesis "$p(y)$ is true for…
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