All Questions
Tagged with ordinals order-theory
159
questions
2
votes
0
answers
73
views
Set of ordinals isomorphic to subsets of total orders
Background. Given a poset $(S,<)$ we'll indicate with $\tau(S,<)$ the set of all the ordinals which are isomorphic to a well ordered subset of $(S,<)$. We're in $\mathsf{ZFC}$.
Questions.
...
- 509
-2
votes
1
answer
57
views
Prove that the order type of $\alpha\cdot\beta$ is the antilexicographic order in $\alpha\times\beta$. [closed]
This question is related to this one, but not a duplicate, since I am struggling with injectivity and monotonicity, rather than proving that $\{\alpha\cdot\eta + \xi:\eta<\beta\textrm{ and }\xi<\...
- 4,906
0
votes
0
answers
39
views
Find necessary and sufficient conditions for ordinal monotonicity.
First of all let's we remember the following result.
Theorem
Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ ...
- 4,906
8
votes
1
answer
109
views
What does the cardinality alone of a totally ordered set say about the ordinals that can be mapped strictly monotonically to it?
For any cardinal $\kappa$ and any totally ordered set $(S,\le)$ such that $|S| > 2^\kappa$, does $S$ necessarily have at least one subset $T$ such that either $\le$ or its opposite order $\ge$ well-...
1
vote
0
answers
46
views
Limit Countable Ordinal - is it a limit of a intuitive sequence of ordinals?
I am studying set theory, ordinal part.
Set theory is new to me.
I know that commutativity of addition and multiplication
can be false in infinite ordinal world.
$ \omega $ = limit of sequence $\, 1,2,...
- 295
2
votes
0
answers
40
views
The set of predecessors of a node in a tree is finite?
I came across the following definitions of tree and transitive tree, respectively, in the book Modal Logic by P. Blackburn:
Definition 1. A tree is a relational structure $(T, R)$, where:
$T$, the ...
3
votes
2
answers
196
views
The intersection of a class of ordinals belongs to the class
Let $C$ be a non-empty class of ordinals. I want to prove that $\bigcap C \in C.$ What I managed to prove so far :
That $\bigcap C$ is an ordinal.
If $\alpha \in C$, that $\bigcap (\alpha^+ \cap C) = ...
- 576
1
vote
1
answer
95
views
What are the order types of computable pseudo-ordinals with no c.e. descending chains?
The notion of a “computable pseudo-ordinal”, i.e. a computable linearly ordered set with no hyperarithmetical descending chains, is an old one going back to Stephen Kleene. Joe Harrison wrote the ...
- 10.3k
0
votes
1
answer
78
views
Special Aronszajn tree actually has continuum cardinality?
Wikipedia gives the following construction of a special Aronszajn tree. Supposedly, this tree has $\aleph_1$ nodes, as each level and each branch is countable. However, it seems to me that this ...
- 5,830
2
votes
1
answer
185
views
Exercise 5.11 - Ch.6 in "Introduction to Set Theory" by Hrbacek and Jech
I am self-learning about ordinal numbers using Hrbacek and Jech's book. On page 123 of the second edition, exercise 5.11 asks:
Find a set $A$ of rational numbers such that $(A,\leq_{\mathbb{Q}})$ is ...
- 4,432
0
votes
0
answers
169
views
Cantor-Bendixson derivative sets
I try to show that a compact subset $A\subset \mathbb{C}$ is at most countable if and only if there exists a countable ordinal number $\alpha$ (i.e $\alpha <\omega_{1},$ where $\omega_{1}$ is ...
- 51
1
vote
0
answers
25
views
Minimal "real-orderable" ordinal, under natural ordering [duplicate]
I've been learning about and playing around with ordinals, and I've come across the following question:
Suppose we take a subset of the real numbers $S\subseteq\mathbb{R},$ under the ordering assigned ...
3
votes
1
answer
80
views
Partially ordered sets and ordinals
Could you please help me with this problem?
Given the partially ordered set $(A,\le_A)$, where $A=\omega \times \omega$ and $$(a,b)\le_A(a',b')\iff a\le a' \land b \le b'$$ with $\le$ being the usual ...
- 192
0
votes
1
answer
72
views
Is there an order-preserving injection from $\alpha\cdot\alpha+\omega$ to $\mathcal{P}(|\alpha|)$?
Let $\alpha$ be an infinite ordinal. Is there an order-preserving injection from $\langle\alpha\cdot\alpha+\omega,\in\rangle$ to $\langle\mathcal{P}(|\alpha|),\subsetneq\rangle$?
I tried to construct ...
- 1,371
2
votes
1
answer
111
views
Infinite Lexicographic Order on Bijections is Well-Order
Problem. Prove, without using $\mathsf{AC}$ if possible , that if $\alpha$ and $\beta$ are ordinals such that $\alpha$ is countable and $\beta>1$, then $\alpha^\beta$ is countable.
The induction ...
- 1,371