All Questions
Tagged with well-orders set-theory
185
questions
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55
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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
\...
- 41.9k
5
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3
answers
488
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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 ?
...
- 1,350
1
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1
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72
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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 $A=\{a_\xi\,\colon\xi<\alpha\}$? ...
- 307
3
votes
2
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98
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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 $f:n\to A$, ...
- 4,442
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106
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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 ...
- 20.8k
0
votes
1
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64
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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 $X,Y\in F$...
- 35
1
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1
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81
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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 ...
- 375
2
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1
answer
42
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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 : ...
- 685
1
vote
1
answer
121
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Partition of $\mathbb R$ in convex subsets/badly ordered sets
Background: These questions come from two different exercises, but since the first is much shorter and of the same kind of one of the others, I preferred to put everything in only one thread. (We work ...
- 509
6
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149
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Existence of uncountable set of functions on natural numbers
For $f,g:\mathbb{N}\rightarrow \mathbb{N}$ we write $f\leq g$ iff $f(n)\leq g(n)$ for all $n\in \mathbb{N}$. Let $\mathcal{S}\subseteq \{f\vert f:\mathbb{N}\rightarrow \mathbb{N}\}$ be a set of ...
- 113
2
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0
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75
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Law of Trichotomy for Well-Orderings
Often in beginning set-theory courses, and in particular in Jech's book Set Theory, it is proved from scratch that given any two well-orderings, they are isomorphic or one is isomorphic to an initial ...
- 97
2
votes
1
answer
85
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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 ...
- 1,307
3
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1
answer
73
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Is it consistent with ZC that a well-order of type $\omega_\omega$ does not exist?
Working in Zermelo's set theory (with choice for simplicity) - the construction in Hartogs' theorem shows that starting with a set $X$, there is a set $X'$ in at most $\mathcal{P}^4(X)$ (where $\...
- 4,963
1
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3
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Shouldn't the Well Ordering Principle apply only to sets with at least two elements?
From what I've been taught in school, the well-ordering principle states that every non-empty set must have a least element. To me, the least element of some set $X$ is an element $a$ such that, for ...
- 927
3
votes
1
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80
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Hessenberg sum/natural sum of ordinals definition
I was given the following definition of Hessenberg sum:
Definition. Given $\alpha,\beta \in \text{Ord}$ their Hessenberg sum $\alpha \oplus \beta$ is defined as the least ordinal greater than all ...
- 509