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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Motivation of inventing concept of well-ordered set?
I've started studying set theory for a while. I understand what is an ordered sets but i still fail to see what motivated mathematicians to invent these concept.
Could you please enlightment me ?
...
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Is every strict ordering by inclusion a well-ordering?
Given a set $s$ which is transitive and completely ordered by inclusion, that is, such that
$z \in s \rightarrow z \subset s$ and $\left( x \in s \wedge y \in s \wedge x \neq y \right) \rightarrow
\...
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Is there something missing in Jech's proof of Zermelo's Well-Ordering Theorem?
Here is the proof from p. 48 of the Millennium Edition, corrected 4th printing 2006:
My question: how do we know that there is any ordinal $\theta$ such that $A=\{a_\xi\,\colon\xi<\alpha\}$? ...
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Proving the shortlex ordering is a well-ordering
Let $(A,<)$ be a nonempty linearly ordered set, and let $\operatorname{Seq}(A)$ denote the set of finite sequences of elements of $A$. That is, $f\in\operatorname{Seq}(A)$ is a function $f:n\to A$, ...
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Iterative argument in math proofs
I am reading proof of subspace of $\mathbb{R^n}$ has a basis. And most of them like this(classic proof): let $S\subset\mathbb{R^n}$ ,if $v_1\neq0$ and $<v_1>=S$, we finished proof, otherwise, we ...
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When does a set of real numbers have a minimal element?
I Believe the answer is when the set is closed in the sense of standard topology (we exclude $\mathbb{R}$) itself.
Examples:
Point sets have a minimum, and they are closed.
Closed interval also have ...
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How does well-ordering theorem apply to real numbers if real numbers are not countable?
From what I understand, the well-ordering theorem assumes ANY set can be made well-ordered depending on how you define the ordering.
For example, $\mathbb{Z}$ is not well-ordered according to the ...
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Given two well-orders $\langle A,R \rangle$ and $\langle B,S \rangle$, one of the following holds.
Let $\langle A,R \rangle$ and $\langle B,S \rangle$ be two well-orders, and let $\text{pred}(A,x,R) := \{y \in A \;|\; yRx\}$ and similarly for $\text{pred}(B,z,S)$.
It is claimed that one of the ...
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Definition of limit using ordered variables
Many years ago, when I was at the university, I learned all the theory of limits of a real function of real variable by means of ordered variables. Precisely I was taught to consider a set $\mathrm{O}$...
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How does one prove without the axiom of choice that the product of a collection of nonempty well-ordered sets is nonempty?
Suppose $\{X_{\alpha}\}_{\alpha\in\mathcal A}$ is an indexed family of nonempty well-ordered sets, where $X_{\alpha}=(E_{\alpha},\le_{\alpha})$ for each $\alpha$. It seems intuitively obvious that we ...
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Complete iff Compact in Well-Ordered space
Let $T=(S, \leq, \tau)$ a well-ordered set equipped with order topology (defined here).
Definition 1: $T$ is called complete iff every non-empty subset of $T$ has a greatest lower bound (inferior) and ...
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Is the first-order theory of the class of well-ordered sets the same as the first-order theory of the class of ordinals?
Consider the class $K$ of well-ordered sets $(W,\leq)$. Although that class is not first-order axiomatizable, it has an associated first-order theory $Th(K)$. Now consider the class of ordinals $On$, ...
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Partial order on sets and application of Zorn's Lemma to construct well-ordered subset
I would appreciate help with the following question:
Let $(A,<)$ a linear ordered set.
a. Let $F\subseteq P(A)$. Prove that the following relation is a partial order in $F$: $X\lhd Y$ for $X,Y\in F$...
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Any subset of a well-ordered set is isomorphic to an initial segment of this well-ordered set.
I wanted to prove the fact for which I have a sketch of proof: Let $(W,\leq )$ be a well-ordered set and $U$ be a subset of $W$. Then considering the restriction of $\leq $ to $U\times U$, we have ...
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Are these two notions of weak well-foundedness equivalent?
Background (optional): I have a state transition system $Q$ with two "kinds" of transitions: progress-making ($\delta_P : Q \times \Sigma \rightarrow Q$) and non-progress making ($\delta_N : ...
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