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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How to prove $cf(\aleph_{\omega_1})=\omega_1$?

I am reading Jech's & Hrbacek's book "Introduction to set theory" and this is question 2.2 from chaper 9: Prove $cf(\aleph_{\omega_1})=\omega_1$ Isn't it clear from the fact that $\aleph_{\...
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2answers
63 views

Theorem $($Transfinite Induction and Construction$)$ (James Dugundji - 5.1, Page - 40)

Let $W$ be a well ordered set, and let $Q \subset W$. If $[ W(x) \subset Q] \Rightarrow [ x \in Q]$ for each $x \in W$, then $Q = W$. For each $a \in W$, the set $W(a) = \{x \in W; (x\prec a) \...
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1answer
51 views

Do ordinal sets exist?

An ordinal is woset, where for all members $a$ in the set, $a$'s segment is $a$ itself. But, the definition of $a$'s segment is: $X_a=\{x\in X:x \subsetneq a\}$ But, given that definition, how can ...
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1answer
19 views

Given stationary set $A$ of $k$, there exists an ordinals $\alpha$ such that $A$ contains all limit ordinals greater then $\alpha$

Is it true that given stationary set $A$ of $k$, there exists an ordinal $\alpha < k$ such that $A$ contains all limit ordinals of $k$ greater then $\alpha$? I think it is true because the set of ...
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1answer
42 views

What is wrong with the following function $f:ON \rightarrow ON$?

Define the following function on ON (The class of ordinals): $f(0)=0$ and for every $\alpha >0$, if $\alpha$ is a successor then $f(\alpha) = \alpha -1$ and if $\alpha$ is limit then, $f(\alpha) = \...
2
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1answer
29 views

If $f\colon\kappa\rightarrow\kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded.

If $f\colon\kappa\rightarrow \kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded. This is from Jech's book (page 103) so ...
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0answers
33 views

Ordinals that are not cardinals [duplicate]

I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are ...
1
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1answer
44 views

A question about well-ordered subsets of partially ordered sets

Let $S$ be an infinite partially ordered set. Is there a name for the smallest ordinal number which is not ordinally similar to any subset of any chain of $S$? Assume that we are working in ZF. If the ...
3
votes
1answer
83 views

Does every countable sequence of ordinals converge?

Let $A=\{ \alpha_n | n \in \omega \}$ a monotonic increasing sequence. Can we say that there exists an $\alpha \in ON$ such that $\alpha = \lim_{n \in \omega} \alpha_n$? i am asking this because I am ...
4
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1answer
71 views

How do we know that $\omega_1$ exists in ZF? [duplicate]

In ZF, not all "collections of objects" are sets. For example, there is no set of all sets, and there is no set of all ordinals. So, how do we know that there is a set of all countable ordinals? In ...
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2answers
46 views

Lexicographic order on $\alpha^\beta$ is well-ordered

Suppose that $\alpha$ and $\beta$ are ordinals, and define the following order on $\alpha^\beta$, the set of all functions $\beta\to\alpha$ with finite support: $$f\,R\,g\iff\exists x\in\beta\,(f(x)&...
4
votes
2answers
35 views

Is $f: \mathcal{P}(\omega) \to \omega \cup \lbrace \omega \rbrace, \ x \mapsto \bigcup \lbrace n \in \omega: n \subset x \rbrace $ surjective?

Intro: I am currently preparing for an exam and I have found this particular question in a previous exam of the same class. It is a 'simple' question where you only have to write the answer without ...
1
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1answer
89 views

finite sequences,linear order, lexicographic order, well order, cardinal

Let $\lambda$ be an infinite cardinal.Consider the lexicographic order on $\lambda^{< \omega}$. Why this order is not a well-order: how may an infinite long descending chain look like? Why every ...
0
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2answers
48 views

Is there a strict superset of the reals with total order?

The ordinals immediately come to mind, but I am mainly interested if there are new elements bounded between two real numbers $[a,b]\subseteq\mathbb{R}$ that can be added to the reals while preserving ...
2
votes
1answer
100 views

Directed limits of topological spaces and embeddings

Let $(\{X_\alpha\}_\alpha,\{f_{\alpha\beta}:X_\alpha\rightarrow X_\beta\}_{\alpha\preceq\beta})$ be a directed system of topological spaces and $(X,\{g_\alpha:X_\alpha\rightarrow X\}_\alpha)$ its ...
2
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0answers
71 views

Compact, sequential spaces

A compact, Hausdorff space $X$ is sequential if each for each $A\subset X$ and $x\in \overline{A}$, there exists a countable set $A_0\subset A$ such that $x\in \overline{A}_0$. I am asked to show ...
1
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0answers
25 views

Exponentiation on order types

How is exponentiation defined on order types? We know that $2^\omega=\omega$. What is $2^{\omega^*}$? Is it $\omega^*$? $\eta$? $\lambda$? I'm guessing $\eta$, but I'm not sure. $\omega$ is the ...
0
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0answers
29 views

If $x,y$ are ordinals and $x \cong y$ then $x = y$

If $x,y$ are ordinals and $x \cong y$ then $x = y$ The proof goes as follows; Suppose $f$ is an isomorphism between $x$ and $y$ then show $f$ is the identity map. Assume that $f$ is not the ...
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1answer
93 views

Is this the real definition of an ordinal number?

I have to prove that if $\alpha$ is ordinal and $\beta \in \alpha$ then $\beta$ is oridinal. To do this I wanted to prove that for every $x\in\beta$, $x\in\alpha$. This seems like it should be true, ...
1
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1answer
35 views

Show $\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$

$\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$ Sorry if this question has been asked already but I couldn't find it on this site. I assume by definition of a ...
2
votes
1answer
99 views

Prove $\alpha+\beta=\beta$ if $\alpha < \beta$

$\alpha$ and $\beta$ are ordinal numbers and $\beta$ is limit ordinal. I assume the proof is relatively simple because it's been omitted in my textbook. Could someone also provide an intuitive way to ...
1
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1answer
37 views

Difference between a criteria of well-ordered and transitive in terms of an ordinal?

From what I gather a set $x$ is $transitive$ if whenever $y \in z , z \in x \rightarrow y \in x$ And one of the properties of a well-ordered set is that $x \in y , y \in z \rightarrow x \in z$ Now ...
5
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4answers
323 views

Uses of ordinals

Are there any interesting theorems outside of set theory that use ordinals in their proofs? The only example I know of is Goodstein's theorem, and I haven't been able to find anything else. In other (...
2
votes
1answer
104 views

Order preserving injection from first uncountable ordinal to the real numbers

Can there be an ordering preserving (1-1) map $f \colon \langle\aleph_1,\in\rangle \rightarrow \langle\mathbb R, \lt\rangle$? I remember that if $f$ is order preserving then consider the following ...
3
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1answer
60 views

if $\alpha\leq\beta$ then there is $\gamma$ such as $\alpha+\gamma=\beta$

I'm trying to prove the statement in the title, a.e existence of ordinal substraction. I think it can be done with transfinite induction, but do any of you have a better/easier way? thanks
1
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1answer
65 views

Epsilon number cannot be a successor?

Let $\omega$ be the canonical representation of the natural numbers. A epsilon number $\alpha$ is a ordinal number such that $\alpha=\omega^\alpha$ holds. I suspect that a epsilon number cannot be a ...
4
votes
1answer
101 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...
2
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2answers
162 views

Subsets of the rationals order isomorphic to transfinite ordinals

Let $\langle\mathbb Q,\lt\rangle$ be the set of rational numbers with the usual $\lt$ ordering. Find subsets $A$ and $B$ of $\mathbb Q$ which, with the usual ordering, are order isomorphic to the ...
6
votes
1answer
75 views

For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
0
votes
1answer
42 views

Proof of the Principle of Transfinite Induction for On

Principle of Transfinite Induction for On: If $C\neq\varnothing, C\subseteq$ On then $\exists\alpha\in C\forall\beta\in C[\alpha\in\beta\vee\alpha=\beta]$ Proof: (1) As $C\neq\varnothing$, let $\...
0
votes
1answer
60 views

Take $S$ to be a set of ordinals, show $\cup S=\sup(S)$

Ok so firstly I can see how this makes sense. If you have say: $S=\{0,1,2\}$ with $0 = \emptyset, 1 = \{0\} = \{\emptyset\}, 2 = \{0,1\} = \{\emptyset,\{\emptyset\}\}$. $$\cup S = \{\emptyset,\...
0
votes
1answer
70 views

Can set containing not only all ordinal numbers exist?

I've always seen the same proof of non-existence of set of all ordianal numbers. It was based on a fact that such a set would fit into definition of ordinal number, and the set itself would be an ...
0
votes
0answers
109 views

What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
3
votes
2answers
56 views

Open neighborhoods in ordinals with the order topology

While learning about ordinals, my teacher made some remarks about any ordinal α being equipped with the order topology, and some facts, one of which was basically that the finite ordinals and ω are ...
3
votes
3answers
125 views

Is every countable ordinal homeomorphic to a subspace of $\mathbb R$?

I know that every countable ordinal is isomorphic to some subset of $\mathbb R$ as ordered sets. Is it also the case that every countable ordinal (with the order topology) is homeomorphic to some ...
7
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2answers
126 views

Is the limit of a recursive sequence of recursive ordinals itself a recursive ordinal?

Is the limit of a recursive sequence of recursive ordinals itself a recursive ordinal? If so, is there a nice proof of this?
0
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1answer
53 views

A Question about cardinal numbers and homeomorphism

I'm studying my set theory lessons and I don't know if the following statement about homeomorphism between ordinal spaces is true or not and why. "If $\kappa$ is an infinite cardinal number then $[0,\...
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votes
1answer
70 views

Lindelöf subspaces in the ordinal space [closed]

How can we describe all Lindelöf subspaces of $$X = \omega_1 \times ( \omega_1+1).$$ Where ω1 is the first uncountable ordinal.
4
votes
1answer
152 views

$H(\kappa)$ a model of all the axioms of ZFC for $\kappa$ not inaccessible

Is this possible? For each cardinal $\kappa$, we can define $H(\kappa) = \{ x \mid |trclx| < \kappa\}$, where trcl is the transitive closure, and $|A|$ is the cardinality of $A$. For every regular ...
0
votes
1answer
63 views

Compactness of second-countability of $\omega$X$\omega_1$

Please discuss the following properties of the product space consisting of $\omega$X$\omega_1$: Is it compact? Is it 2nd countable? $\omega$ is the first infinite ordinal and $\omega_1$ is the ...
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2answers
271 views

Showing that a Transitive Set of Transitive Sets is an Ordinal

My definition of an ordinal is a transitive set that's well ordered by $\in$. Let $\alpha$ be a transitive set all of whose elements are transitive sets. Since every element of $\alpha$ is transitive, ...
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3answers
157 views

Confused about transfinite induction

QUESTION: I seem to be confused about how transfinite induction is carried out. I have looked at several examples and they seem to follow a procedure consisting of grounding the induction, proving the ...
2
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2answers
71 views

unbounded class of ordinals not a set

A class $C$ of ordinals is unbounded just in case ∀α∈ORD (the class of all ordinals), there exists a β ∈ $C$ with α ∈ β. How would I show that no unbounded class of ordinals is a set? Do I need to ...
2
votes
1answer
62 views

Countable transitive model of ZFC and $\mathcal{P}(\omega)$

Let $\mathbb{M}$ be a countable transitive model of ZFC. I understand that $\omega^\mathbb{M} = \omega$ but $\mathcal{P}(\omega)^\mathbb{M} \neq \mathcal{P}(\omega)$, for $\mathcal{P}(\omega)^\mathbb{...
2
votes
1answer
51 views

for $x\in V_{\omega}$ is then $\operatorname{rank}(\operatorname{tc}(x))=\operatorname{rank}(x)$?

Let $x\in V_{\omega}$, is $\operatorname{rank}(\operatorname{tc}(x))=\operatorname{rank}(x)$? while $\operatorname{tc}(x)$ is the minimal transitive set ${a}$ such that $x\subseteq a$ i what to show ...
5
votes
2answers
193 views

Ordinal exponentiation identity with natural sum of exponents

This is related to a previous question on How to think about ordinal exponentiation? One possible definition for the natural product $\alpha\otimes\beta$ of ordinals is based on Cantor Normal Forms ...
6
votes
2answers
116 views

Embedding $\omega_1$ in the hyperreals

I've been trying to show that there exists a subset of the hyperreals which has order type $\omega_1$, that is, that there exists a subset $A \subseteq {}^*\mathbb{R}$ for which there is an order ...
0
votes
1answer
96 views

How to deal with probability problems in proper class sample spaces?

Here my focus is mainly on $Ord$ and questions such as: An ordinal is chosen by random. What is the probability of the event that it is a cardinal number? A coin is tossed proper class many ...
8
votes
4answers
163 views

Is $0^\omega=1$?

According to a definition of ordinal exponentiation defined in Kunen's Set Theory: An Introduction to Independence Proofs (pp. 26), we define $$\begin{align} \alpha^0&=1\\ \alpha^{(\beta+1)}&...
2
votes
1answer
38 views

to show that there is no injection from a finite successor of finite ordinal to itself

im trying to prove this proposition let $\alpha \in \omega$ so i want to show that there is no injection (not necessarily keeps order) from $\alpha +1$ to $\alpha$ without using cardinality or the ...