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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first countability space

We know that $ [0,\omega_{1}‎)‎ $‎ is a first countable space. To show this we take $ ‎‎\mathfrak{B}(\alpha )‎ =‎ ‎\{ (\zeta,‎\alpha ] :‎ ‎‎\zeta <‎ ‎‎\alpha ‎ ‎\}‎ $ as local base. But I ...
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14 views

$\Sigma$-product and Tychonoff product [duplicate]

I want to say that the below example is countable compact but not closed, $X$ a Tychonoff product $[0,1]^{ω_1}$ and as $Y$ the $\Sigma$-product $$ \{ x ∈[0,1]^{ω_1}:|\{α<ω_{1} :x(α)≠0 \}|≤ω ...
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90 views

What classification of countable ordinals above $\omega_1^{CK}$ exists?

What classification of countable ordinals above the Church–Kleene ordinal $\omega_1^{CK}$ exists? Are there such things as $\omega_2^{CK}$, $\omega_{\omega_\omega^{CK}}^{CK}$ or ...
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1answer
78 views

How to prove that $\text{pred}_{(A,\prec_A)}(y)=\{x\in A \,|\, x\prec_A y\}$

In Schimmerling's book on set theory, question 3.1 reads: "Let $(A,\prec_A)$ be a wellordering such that $A\neq\emptyset$. for each $y$ define $\text{pred}_{(A,\prec_A)}(y)=\{x\in A \,|\, x\prec_A ...
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88 views

The classification of order types

In response to a previous question which I asked concerning the order type of the Rationals, a reference was made (by MDJ) to a theorem of Cantor stating that any two countable, dense, linearly ...
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248 views

The order type of the rationals.

Herewith another mind-numbingly naive question from a reader of philosophy. My question concerns the order type of the rational numbers. Omega squared seems a natural first choice, but obviously ...
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1answer
277 views

Cardinal Arithmetic versus Ordinal Arithmetic

I am reading Philosophy, not Set Theory, so please excuse the naivety of my question. My question concerns the wildly different character of ordinal arithmetic versus cardinal arithmetic. The ...
14
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1answer
153 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 ...
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1answer
79 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
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110 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
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1answer
199 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 ...
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2answers
120 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 ...
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2answers
64 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 ...
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2answers
80 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 ...
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1answer
81 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 ...
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2answers
101 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 $x<\alpha$ ...
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173 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. ...
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114 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
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1answer
135 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 ...
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226 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 ...
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65 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.) ...
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1answer
83 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 ...
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116 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 ...
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76 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 ...
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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 ...
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1answer
82 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 ...
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1answer
113 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 ...
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117 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) ...
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66 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 ...
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3answers
289 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
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2answers
128 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 ...
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75 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 ...
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3answers
55 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 ...
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2answers
107 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 ...
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2answers
38 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
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1answer
160 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 ...
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1answer
83 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
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1answer
98 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
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3answers
120 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 $\omega$ ...
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2answers
73 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 ...
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2answers
96 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
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1answer
62 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 ...
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1answer
47 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 ...
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1answer
213 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 ...
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1answer
91 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 ...
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1answer
56 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 ...
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1answer
69 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 ...
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175 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
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3answers
220 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
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1answer
56 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 ...