# 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).

211 questions
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### Trying to find faster growing variants of the successor function

The successor function $S : \alpha \mapsto \alpha \cup \{\alpha\}$ where $\alpha$ is an ordinal is notable for being a strictly increasing function that has no fixed points, because it is ...
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### The cartesian product of a well-ordered set with $[0,1)$ is a linear continuum in dict. order

Definition: A linear continuum is a simply ordered set $L$ such that: (1): $L$ has the least upper bound property; (2): For every $x<y$ in $L$ there is a $z$ sucht that $x<z<y$. I ...
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### Show that the lexicographic ordering $≤_l$ on $A\times B$ for $A$ well ordered by $≤$ and $B$ well-ordered by $≤'$ is well-ordered.

Q: Show that the lexicographic ordering $≤_l$ on $A\times B$ for $A$ well ordered by $≤$ and $B$ well-ordered by $≤'$ is well-ordered. A: I have previously shown that this lexicographic ordering is ...
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### Let $≤$ be a total order on $A$. For each $u\in A$, let $s(u)=\{x\in A:x<u\}$…Prove that $≤$ is a well-ordering on $A$.

Q: Let $≤$ be a total order on $A$. For each $u\in A$, let $s(u)=\{x\in A:x<u\}$. Suppose that for every $u\in A$ and $X$, if $X\subseteq s(u)$ is nonempty, then $X$ has a $≤$-least element. ...
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### Provable well-founded ordering

Quote from Sieg/Parsons commentary on Gödel's Zilsel lecture: ...it has to be clarified how to grasp ... specific countable ordinals ... That can be archieved in the system $S_1$ with function ...
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### Prove that for every natural number $a$ there are integers $t \geq 0$ and $r$ such that $a = 3^t r$ and 3 does not divide r.

Prove that for every natural number $a$ there are integers $t \geq 0$ and $r$ such that $a = 3^tr$ and $3 \not | r$. Would I use the well ordering principle for this? There’s so many variables so I’m ...
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### Zermelo's Proof of the Well Ordering Theorem

I am trying to find the paper by Zermelo in the early 1900's in which he proved the Well Ordering Theorem (implicitly using the axiom of choice). Does anyone know where I can find this?
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### Directed set and partially/totally ordered sets.

I am barely new to order theory and this motivates if the question is trivial. I understood the definitions of preorder, partially and totally ordered sets and well ordered sets. In particular there ...
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### Indexing of uncountable sets and uncountable collections of sets, uncountable intersections containing a point

Definitions Let $\mathcal{A}$ be an uncountable collection of sets so that if $I_{\mathcal{A}}$ is the index set of elements of $\mathcal{A}$ then $|I_{\mathcal{A}}|\not=\aleph_{0}$ (I mean this to ...
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### Why is the Axiom of Well-ordered Choice not strong enough to prove Zorn's Lemma?

This is based on this question: How strong is the axiom of well-ordered choice? The "axiom of well-ordered choice" says that any transfinitely-indexed family of sets has a choice function. The ...
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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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### Help understanding Well-Ordering Theorem [duplicate]

Forgive me for my lack of formal notation, I haven't taken any classes on set theory, or any advanced math topics for that matter. From my understanding based on the wikipedia entries, a well-ordered ...
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### Proof by contradiction and the well ordering principle

I have a question regarding proof by contradiction and the well ordering principle on integers. Okay lets says for example an arbitrary value $p$ is a positive integer and which I assume is the ...
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### Help solving this proof using Well Ordering Principle

Let $a_1,a_2, ... , a_n ∈ \Bbb{N}$ . Prove that there exists $l ∈ \Bbb{N}$ such that $a_i | l$ for all $i ∈ \{1,2,...,n\}$ and if $x ∈ \Bbb{N}$ is such that each $a_i$ divides $x$, then $l | x$. ...
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### Set or list compression

Apologies for poor use of terms, I do not understand enough of the problem to even ask the right questions. My main question is, what domain of mathematics is this problem and is it solved problem ...
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### Well order of naturals [closed]

I have an exercise that asks me for 15 non-isomorphic well order types of natural numbers, I have some, can you help me with other well uncommon orders? Thank you
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### Linearly ordering the power set of a well ordered set with ZF (without AC)

As the title says, my question is, how one can use only ZF-theory to prove that the power set of A, whereby (A, <) is a well-ordering, can be linearly ordered?
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### Infinite descendant sequences

"Show that the order $(A,<)$ is well ordering if and only if there no exists infinite descendant sequences in $A$". Can you help me whit this problem, please, what happens is that my professor isn'...
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### proof of Well Ordering Principle over positive integers

Theorem If $A$ is any nonempty subset of $\mathbb{Z}^+$, there is some element $m \in A$ such that $m \leq a$, for all $a \in A$ where such $m$ is the minimal element of $A$. (AA: Dummit and Foote) ...
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### There exists a minimal uncountable well ordered set. [duplicate]

There exists a well-ordered set $A$ having a largest element $\Omega$ such that the section $S_\Omega$ of $A$ by $\Omega$ is uncountable but every other section is countable. Can anyone make me ...
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### Well ordering and maximal chains in power set

Let $M$ be a set and "$\le$" a well-ordering of $M$. For $x \in M$ define: $$M_{\le x} := \{ y \in M \ \vert\ y \le x \}$$ The map $$f : M \to \mathcal{P}(M) \ ,\ x \mapsto M_{\le x}$$ is injective ...
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### For- and backwards well-ordered set is finite.

I'm working on the following problem and can't seem to come up with a satisfactory proof. Let $X$ be a totally ordered set which is well-ordered forwards and backwards. Show that $X$ is finite. I ...
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### Could every Hausdorff space be induced by a total order relation [closed]

Let $(H,\mathcal T)$ is a Hausdorff space then is there any total order relation on $H$ such that the topological space induced by the order relation be the same $(H,\mathcal T)$? Thanks a lots ...
2answers
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### Showing that $[0, \omega_1]$ is compact. [duplicate]

I want to show that $[0, \omega_1]$ is compact, where $\omega_1$ is the least uncountable ordinal, and I have just been introduced to the concepts of ordinals. The tips I have seen to showing this ...
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### Prove the comparability theorem for well ordered sets using transfinite induction

My question is about the same proof discussed here, but I am confused about more than this question addresses. In Naive Set Theory, Halmos phrases the comparability theorem as follows: The ...
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### Well ordering principle question

Is every non empty subset of the integers well ordered and does this mean that every subset contains a least element? Are the positive rationals well ordered? i believe not. Is this because of the ...
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### Showing that the class of finite subsets of a well-ordered class can be well-ordered

We know that if $(X, <)$ is a well-ordered class, then $\mathcal{P}(X)$ can be totally ordered by the following: $$A<^*B\ \text{iff min}(A\Delta B) \in A$$ where $\Delta$ is the symmetric ...
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### Showing a Group Can Be Totally Ordered

If every finitely generated subgroup of a group $G$ is totally ordered, what approach should I take to prove that $G$ is also totally ordered? Should I define a map that takes elements from the ...
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### Prove that every finitely generated subgroup of a totally ordered group can be totally ordered.

Prove that every finitely generated subgroup of a totally ordered group can be totally ordered. A group can be totally ordered if there is a total order $\leq$ on that group. Is the proof of this ...
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### Equivalence Induction and Well-Ordering

I need some help please. I am trying to get to grips with proving the equivalence between mathematical induction (MI) and well-ordering principle (WOP). As a theorem, I have Principle of ...