All Questions
Tagged with ordinals well-orders
90
questions
2
votes
1
answer
127
views
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 $On$, ...
- 21.5k
3
votes
1
answer
80
views
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
5
votes
1
answer
75
views
Exercise 7.1.6 Introduction to Set Theory by Hrbacek and Jech
This is exercise 7.1.6 of the book Introduction to Set Theory 3rd ed. by Hrbacek and Jech.
Let $h^{*}(A)$ be the least ordinal $\alpha$ such that there exists no function with domain $A$ and range $\...
- 4,827
2
votes
1
answer
42
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If $X_1$,$X_2$ are wosets isomorphic to ordinals $\alpha_1,\alpha_2$ then $X_1\times X_2$ is isomorphic to $\alpha_2\cdot \alpha_1$
I want to prove the following:
Let $X_1$,$X_2$ be wosets, isomorphic to ordinals $\alpha_1,\alpha_2$ respectively. Then $X_1\times X_2$, with the lexicographic order, is isomorphic to $\alpha_2\cdot\...
- 4,827
4
votes
0
answers
105
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Why does proof of Zorn's lemma need to use the fact about ordinals being too large to be a set?
I'm not understanding why its necessary to invoke the knowledge about ordinals in order to prove Zorn's lemma.
The Hypothesis in Zorn's lemma is
Every chain in the set Z has an upper bound in Z
Then ...
- 41
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
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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
1
vote
1
answer
126
views
a proof idea: Every well-ordered set has an order-preserving bijection to exactly one ordinal.
I have seen a proof of the statement, and its usually by transfinite induction. And I'm trying to find out why my proof doesn't work, it seems too simple:
Let $X$ be a well-ordered set. Define $X^{<...
- 438
0
votes
1
answer
67
views
A couple of well ordering proofs.
I'm having trouble understanding a couple of things when studying well orderings and ordinals.
I know that given a well ordering $(A,<)$ there is no $a\in A$ s.t. $(A,<)\cong (A_a,<)$ where $...
- 391
1
vote
1
answer
95
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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
13
votes
1
answer
670
views
Why do we need "canonical" well-orders?
(I asked this question on MO, https://mathoverflow.net/questions/443117/why-do-we-need-canonical-well-orders)
Von-Neumann ordinals can be thought of "canonical" well-orders, Indeed every ...
- 5,704
2
votes
2
answers
145
views
What is the meaning of "induction up to a given ordinal"?
Given an ordinal $\alpha$, what does it mean: "induction up to $\alpha$"? When $\alpha=\omega$, is this is ordinary mathematical induction? Also, Goodstein's Theorem is equivalent to "...
- 4,432
1
vote
1
answer
137
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Question about a proof that any well-ordered set is isomorphic to a unique ordinal
I am studying a proof that every well-ordered set is isomorphic to a unique ordinal. However, I don't understand why $A = pred(\omega)$ (see yellow).
One direction is clear:
Let $x \in pred(\omega)$, ...
- 1,089
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
0
votes
1
answer
100
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Description of largest possible countable set / number
I am looking for an elegant / standard (if any) description of the largest countable set.
A first naive approach would be to construct this set, X, by taking the integers (0 to, but not including, ω_0,...
