In the ZF set theory ordinals are transitive sets which are well-ordered by $\in$. They are canonical representatives for well-orderings under order-isomorphism. In addition to the intriguing ordinal arithmetics, ordinals give a sturdy backbone to models of ZF and operate as a direct extension of ...

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14
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2answers
232 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
-1
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1answer
84 views

$X = [ 0,\omega_1 )$ is maximal countably compact

(a) : why $X = [ 0,\omega_1 )$ is a first countable space? ($\omega_1$ is the first uncountable ordinal number.) (b) : is $X = [ 0,\omega_1 )$ maximal countably compact?
6
votes
0answers
112 views

AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$ [duplicate]

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
6
votes
1answer
209 views

Ordinal arithmetic exercise in Kunen

I'm self studying Kunen (2013 edition) right now, and got stuck on this exercise - I generally lack the tools and/or intuition for these kind of ordinal arithmetic exercises. Exercise. Let ...
2
votes
2answers
135 views

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ? Note : I am an absolute novice, and I'm having a little trouble visualizing ...
0
votes
2answers
68 views

How can we describe different sorts of limits using ordinals, and what kind of “work” can they do in transfinite induction?

I recently encountered transfinite induction and ordinal numbers for the first time in a real analysis text (Bruckner, Bruckner, and Thomson) and I am still coming to grips with these tools and their ...
2
votes
2answers
84 views

Definition of ordinals in Grothendieck toposes

What is a useful definition of an ordinal in Grothendieck toposes. Useful in this context means that I would like to construct fixed points of monotone maps on complete lattices using iteration from ...
4
votes
1answer
94 views

If $(W,<)$ is a well-ordered set and $f : W \rightarrow W$ is an increasing function, then $f(x) ≥ x$ for each $x \in W$

I could use a hand understanding a proof from Jech's Set Theory. Firstly, note that Jech defines that an increasing function $f : P \rightarrow Q$ is a function that preserves strict inequalities ...
1
vote
2answers
105 views

Prove that if $x,y\in\alpha$, then $x<y,x>y$ or $x=y$.

I refer to pg.4, Lemma 12 of this article on ordinal numbers. It says "If $x,y\in \alpha$, then $x<y, x>y, \text { or }x=y$". My question: As $\alpha$ is an ordinal, we know $x<\alpha$ ...
7
votes
3answers
186 views

Is there any countable ordinal number which has a member undefinable?

Let $Q\colon=\{\alpha \in Ord \mid \exists \beta \in \alpha(\beta \text{ is undefinable in } \langle\alpha,<\rangle )\}$ denote the set of all ordinal numbers which has an undefinable member. ...
4
votes
2answers
131 views

If $\alpha$ is an ordinal, proving that $\alpha\cup\{\alpha\}$ is an ordinal.

I refer to pg.4 of this article. Assuming $\alpha$ is an ordinal, we have to prove $\alpha\cup \{\alpha\}$ or $\alpha +1$ is an ordinal. Isn't this obvious from the construction of ordinals? As ...
2
votes
1answer
144 views

Why $\omega$ can't be bijectively mapped to $\omega +1$

Let $\omega$ be the order type of the totally ordered set $\mathbb{N}$, and $\omega +1$ the set $\Bbb{N}\cup \{0\}$. $0$ is greater than all the natural numbers as per this ordering. My question ...
7
votes
3answers
248 views

How to prove the Milner-Rado Paradox?

For every ordinal $\alpha<\kappa^+$ there are sets $X_n\subset\alpha$ $(n\in\Bbb{N})$ such that $\alpha=\bigcup_n X_n$ and for each $n$ the order-type of $X_n$ is $\le\kappa^n$. [By ...
2
votes
0answers
73 views

Is there a nice characterization of posets induced by trees?

Define that a tree in $X$ is a set of ordinal-indexed sequences with codomain $X$ that is closed under the operations of restricting to an ordinal. (I do not know if this definition is standard.) ...
3
votes
1answer
91 views

Does every countable subset of the set of all countable limit ordinals have the least upper bound in it?

I'm sorry if the question is that kind of trivial, I just feel uncertain about these ordinals all the time. Is the answer to the following question "yes": Denote by A the set of all countable limit ...
0
votes
1answer
124 views

The limit elements of well-ordered proper classes

For any well-ordered class $W$ with no maximum element, we can define a successor function $f_W : W \rightarrow W$ by asserting that $f_W(w) = \mathrm{min}\{x > w \mid x \in W\}.$ This allows us to ...
-3
votes
1answer
79 views

Statements regarding ordinal numbers [duplicate]

Let $m$ and $n$ be infinite ordinal numbers Which of the following is true a) $m<n \Rightarrow |m|^{|m|}<n^{|n|}$ b) $m+n$= Max{$m,n$} c) $m=n \Rightarrow |m|=|n|$ d)$|m|=|n| \Rightarrow ...
2
votes
2answers
79 views

Why is $n+\omega=\omega$ for finite $n$?

I am trying to understand the following claim; $n+\omega=\omega$ where $n$ is order type of a finite set and $\omega$ is the order type of $\left\{ 1,2,\dots, \right\}$ with the usual meaning of ...
2
votes
1answer
84 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
1
vote
1answer
124 views

Cardinal numbers

Suppose $m, n$ are infinite ordinal numbers. $$a) m=n → |m|=|n|$$ $$b)|m|=|n| →m=n$$ $$c)m<n→ |m|<|n|$$ $$d)|\max{(m,n)}|< |m|+|n|$$ $$e)|m|<|n| →|m|^{|n|}<|n|^m$$ Which of the above ...
3
votes
3answers
123 views

Set $A$ has the cardinality of $\aleph_0$. Prove some properties of partial order $ \langle \mathcal P (A), \subseteq \rangle$.

I try to solve the following task: Set $A$ has the cardinality of $\aleph_0$. The truth is that in the partial order $ \langle \mathcal P (A), \subseteq \rangle$: (answer true or false) a) ...
2
votes
2answers
70 views

Why are the ordinal operations defined that way?

If $\mu$ is a limit ordinal, then we define that $$\alpha + \mu = \sup_{\beta < \mu} (\alpha + \beta).$$ Why don't we similarly define that if $\lambda$ is a limit ordinal, then $$\lambda + \beta ...
2
votes
3answers
377 views

show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant

show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant.Where $S_{\mathbb{\Omega}}$ denote the first uncountable ordinal. There is a hint that for each ...
3
votes
2answers
137 views

The concept of ordinals

I am trying to understand this concept and have some difficulties. For example, can I say that $\alpha$ is the cardinality of $\{1,2,3,...\}=\Bbb N$? And if so, what is $\alpha +1$? I guess it is ...
1
vote
1answer
82 views

This set of ordinals is $\in$-transitive?

I was reading this problema about ordinals. I'm following the Z.F theory. Let $A$ be a nonempty set of ordinals. Prove that if $\cup A \notin A$ then $\cup A$ it's a limit ordinal, i.e $\cup A $ is ...
1
vote
3answers
59 views

Let $\alpha$ be an ordinal then $\alpha=\cup\alpha$ or $\alpha=s(\cup\alpha)$.

Let $\alpha$ be an ordinal then $\alpha=\cup\alpha$ or $\alpha=s(\cup\alpha)$. Attempt: Since $\alpha$ is transitive $\alpha=\cup\alpha$ (here we are done) or $\alpha\supsetneq\cup\alpha$. If ...
2
votes
2answers
122 views

Is $\alpha + \omega$ limit ordinal

If $\alpha$ is any ordinal number does $\alpha + \omega$ have to be limit ordina, where $\omega$ is ordinal number of $\mathbb{N}$? I know that $\omega+\alpha$ is not always limit ordinal since ...
1
vote
2answers
39 views

If $\gamma$ is an uncountable ordinal, then $\gamma$ contains uncountably many successor ordinals.

Suppose $\gamma$ is an uncountable ordinal, i.e. $\gamma$ has uncountably many elements. We write $$ \gamma = \{0\} \cup\{\alpha \in \gamma: \alpha \text{ is a successor ordinal}\} \cup\{\alpha \in ...
7
votes
1answer
171 views

Continuous surjective functions $\omega_1 \to \omega_1$.

I am looking for nontrivial examples of surjective continuous functions from $\omega_1$ onto $\omega_1$ (with both $\omega_1$'s in the order topology). What sorts of properties must these functions ...
1
vote
1answer
87 views

Proving $\alpha+(\beta+\gamma) = (\alpha+\beta)+\gamma$ for ordinals

I am following Jech's construction, by definition $\alpha+0 = \alpha, \alpha+(\beta+1)=(\alpha+\beta)+1$, and for limit $\beta$ we define $\alpha+\beta = \cup\{\alpha+\xi: \xi<\beta\}$. Jech's ...
3
votes
1answer
101 views

A club in $\omega_1$ induced by a bijection

I'm trying to prove that if $f:\omega_1\to\omega_1\times\omega_1$ is a bijection, then the set $X_f=\{\alpha\in\omega_1:f[\alpha]=\alpha\times\alpha\}$ is a club in $\omega_1.$ So I think I see that ...
3
votes
3answers
126 views

Which ordinals embed in $2^\omega$ ordered by inclusion?

Which ordinals can be embedded in the power set of $\omega$ ordered by inclusion? I see that $\omega\cdot\omega$ can (and therefore anything less than that): we can partition $\omega$ into $\omega$ ...
2
votes
2answers
75 views

Compactifications of limit ordinals

I thought I knew that but it seems I don't. Let $\alpha$ be a countable, limit ordinal $\alpha>\omega$. Give $\alpha$ its order topology. What is the Stone-Čech compactification of $\alpha$? Is ...
4
votes
2answers
101 views

Stationary sets

Currently learning about stationary sets, came across this problem: Let $\{S_\alpha : \alpha<\omega_1\}$ be disjoint pairwise sets with $S_\alpha \subseteq \omega_1$ non-stationary for each ...
3
votes
1answer
64 views

Generalizing the central series to ordinal length

One can generalize the ascending and descending central series by transfinite induction, setting $G _{\alpha +1}=[G_\alpha, G]$ and $ G_\beta= \cap _{\alpha <\beta} G_\alpha $ (and analogously for ...
1
vote
1answer
50 views

Existence of arbitrarily large ordinal subgroups in a group structure on a regular cardinal [duplicate]

Suppose $\kappa$ is an uncountable regular cardinal, and $(\kappa, \cdot, ^{-1}, e$) is a group. Prove that that $C = \{\alpha < \kappa: \alpha\, \textrm{is a subgroup of}\, \kappa)$ is unbounded ...
8
votes
1answer
224 views

Infinite combinatorial games

Hercules vs. Hydra: Recall the story where every time Hercules cuts of a head, two more heads grow instead. Now suppose the following: The hydra starts off with one head, but every time Hercules cuts ...
6
votes
1answer
100 views

Basic examples of ordinals

I'm reading a book on stochastic games which apparently needs ordinals somewhere. This is the first time I meet a concept, and although the definition is clear to me, the lack of practice makes it ...
0
votes
1answer
63 views

Definition of ordinal exponentiation

I found that the usual ordinal exponentiation $\alpha^{\beta}$ is the set of functions from $\beta$ to $\alpha$ with finite support, ordered by antilexicographic order. (least significant position ...
3
votes
1answer
71 views

Base change and ordinals

Problem. Define the operation base change from $k$ to $m$: to make the operation for natural number $n$ we should write $n$ in the base-$k$ numeral system and read this in the base-$m$ numeral ...
6
votes
0answers
186 views

Largest provably existing ordinal in ZF without power set

If I remove the power set axiom from ZF, but retain the axiom of infinity, what will be the largest ordinal that can still be proven to exist. Or, if no largest such ordinal exist, what is the limit ...
4
votes
3answers
259 views

Countable ordinals are embeddable in the rationals $\Bbb Q$ — proofs and their use of AC

Yesterday, Asaf Karagila's answer to my question sparked an extensive discussion on ways of proving that all countable ordinals are embeddable in $\Bbb Q$, and whether particular solutions to this use ...
5
votes
1answer
61 views

Equivalence between these definitions of ordinal numbers

Von Neumann defines an ordinal $\alpha$ as a transitive set whose elements are well-ordered with respect to the membership relation $\in$. Meanwhile, in Naive set Theory, Halmos defines an ordinal ...
8
votes
1answer
146 views

Do there exist totally ordered sets with the 'distinct order type' property that are not well-ordered?

Define that the order type of an element $x$ in a totally ordered set $X$ is the order type of $\{w \in X\mid w < x\}$. Under this definition, distinct elements of a well-ordered set have distinct ...
4
votes
1answer
222 views

Ordinal arithmetic $3+\omega^2 = \omega^2$

Which one of the following equalities is false: a) $2\cdot \omega = 3\cdot\omega$ b) $3+\omega+\omega^2 = \omega+3+\omega^2$ c) $\omega^2 + 3 = 3+\omega^2$ d) $12\cdot(5+\omega)=60 \cdot\omega$ I ...
10
votes
3answers
201 views

How many epsilon numbers $<\omega_1$ are there?

An epsilon number is an ordinal $\epsilon$ such that $\epsilon=\omega^\epsilon.$ What is the cardinality of the set of all epsilon numbers less than $\omega_1$? I'm asking this because of a proof ...
5
votes
1answer
265 views

What set theory axioms do I need to believe in uncountable ordinals?

Math people: The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof. It is not intuitively clear to me that uncountable ordinals or cardinals should ...
2
votes
1answer
125 views

Prove $\omega^{\omega_1}=\omega_1$ and $2^{\omega_1}=\omega_1$

Prove that $\omega^{\omega_1}=\omega_1$ and $2^{\omega_1}=\omega_1$. I also found an exercise asking to compute the ordinal number $\omega_1^{\omega}$, but I do not even understand what I am supposed ...
5
votes
3answers
157 views

How to show that $(\Bbb Q, <)$ contains an order isomorphic copy of $\epsilon_0$ which is the least ordinal satisfies $\omega^\epsilon = \epsilon$?

We define $\epsilon_0 = \sup\{\omega, \omega^\omega,\omega^{\omega^\omega},\omega^{\omega^{\omega^\omega}},\ldots\}$. How to find a subset of $ \Bbb Q$, $(A, <_{\Bbb Q})$ that is isomorphic to ...
5
votes
1answer
106 views

$\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal

I am trying to understand this proof of the existence of an uncountable ordinal. I don't see why $\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal as it is said. For ...