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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Ordinal existence

Is there any ordinal $\alpha$ such that $\omega ^ {\omega ^ \alpha} = \alpha$? Could you please suggest me how to even try to solve this?
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1answer
57 views

Prove $\omega_1$ is first countable

Given a well order (W,$\le$), where W is uncountable, and $\omega_1$:= {x$\in$W: only countably many y$\in$X s.t. y $\le$ x}, prove $\omega_1$ is first countable. I saw a proof saying that {(a,x]: a ...
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1answer
60 views

why $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$ doesn't hold?

this rule doesn't hold for all ordinals $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$. I tested many examples but all of them holds for it ! does this hold ? $(B \cup ...
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1answer
34 views

How to show $A_1 \approx A_2 \iff \mathrm{card}(A_1)=\mathrm{card}(A_2)$

For any set $A_1$ and $A_2$, let us define the relation $\approx$ if there exists a bijection between $A_1$ and $A_2$. Then I want to show that $A_1 \approx A_2 \iff ...
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1answer
53 views

Insistence on limit ordinal in Kunen's proof of reflection.

I'm reading for an exam on set theory using Kunen's first edition, and we've more or less been told that the reflection theorem will come up. The proof constructively yields an ordinal ...
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0answers
40 views

$\sigma$-field and Uncountable ordinal

I have been trying to get my head around this question. Any help greatly appreciated. Let $\mathscr{C}$ be any class of subsets $\Omega$ with $\emptyset,\Omega\in \mathscr{C}$. Define ...
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1answer
75 views

The order-theoretic structure of the ordinal numbers in non-standard models

I have read the following. Proposition. Let $T$ denote the first-order theory of $\mathbb{N}$ in the language of arithmetic. Then any countable non-standard model of $T$ is order isomorphic to a ...
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63 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
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1answer
43 views

A set which is $\in$-transitive and well ordered by $\in$ is an ordinal

A set $x$ is said to be $\in$-transitive if $\forall y$ $\forall z$($ y \in x$ and $ z \in y \Rightarrow z \in x$). A set $x$ is said to be an ordinal if $x$ and every member of $x$ is ...
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1answer
49 views

Ordinal arithmetic question

The question is: find $\delta \le \omega_1$ and $\rho \lt \omega$ such as $\omega_1 = \omega \cdot \delta + \rho$. My guess is that $\delta = \omega_1$ and $\rho = 0$, abut I don't know how to prove ...
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1answer
55 views

Ordinal $10^\omega$

$10^\omega$ = $10 \cdot 10 \cdot 10 \cdot ...= \lim_{\alpha \lt \omega} (10^\alpha) = \omega$. Are my thoughts correct? Is this sufficient explanation, given the ordinal arithemtic proved from ZFC?
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1answer
468 views

Biggest countable ordinal number

I need to find biggest countable ordinal number. But I am sure there is no one, if I understand proofs and definitions correctly. So here are my idea: Suppose there is biggest countable ordinal number ...
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1answer
58 views

Proving a relation between Ordinal Exponentiation and its corresponding set

(Characterization of Ordinal Exponentiation) Let $\alpha$ and $\beta$ be ordinals. For f:$\alpha \rightarrow \beta$, let s(f)={$\epsilon < \alpha|f(\epsilon) \not= 0$}. Let S($\alpha, ...
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2answers
103 views

What do you need to know about ordinals to understand Zorn's Lemma's proof?

I'm trying to understand the proof of Zorn's Lemma but the one which does not use ordinals (Halmos' proof) is extremely long and I really feel I get lost somewhere along the way. On the other hand, ...
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1answer
20 views

Finite set of rank $\alpha$

Defining the rank function $\in$-recursively as $\text{rank}(x)=\bigcup\{\text{rank}(y)^{+}\mid y\in x\}$ for a set $x$, for which ordinals $\alpha$ does there exist a finite set of rank $\alpha$? And ...
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1answer
31 views

Different types of well-ordering of $\mathbb{N}$

The question is: "Find 3 non-isomorpic with each other well-orderings of $\mathbb{N}$. Define their ordinals and arrange them in magnitude." The question in ZFC, but I only know that such ...
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1answer
44 views

Are these ordinals cardinals?

Consider $\omega_2 \times \omega$ and $\omega \times \omega_2$ with ordinal arithmetic. Then $| \omega_2 \times \omega | =\omega_2$ and $| \omega \times \omega_2 | =\omega_2$ Does this imply ...
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1k views

Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems ...
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1answer
28 views

Countable limit ordinal as limit of $\omega$ ordinals

I've got a most probably silly question, but I can't find the answer to it: If $\alpha$ is a countable limit ordinal, how can we be sure that there exist ordinals $\alpha_n$ such that $\alpha = ...
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2answers
110 views

Proof that $\omega_1$ is countable (where is the flaw?)

I have discovered a "proof" of the fact, that $\omega_1$ can be countable, contradicting its definition. It mustn't assume axiom of choice, so I guess there are some parts of this construction which ...
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2answers
47 views

Element of ordinal a subset of the same ordinal

I've got a very short question in set theory. I am currently reading P. T. Johnstone's book Notes on Logic and set theory, and the proof of the fact that Every element of an ordinal is a subset of ...
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2answers
58 views

Is a limit ordinal necessarily a cardinal?

Maybe this is a trivial question. I see that every infinite cardinal is necessarily a limit ordinal, but is the converse true ?
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1answer
44 views

Is the cardinality of an ordinal, $\alpha$ less than $\alpha$?

Let $k^+=\{\alpha : \alpha \hspace{2mm} \text{is an ordinal,and} \hspace{2mm} |\alpha|<\kappa\}$ where $k^+$ is an infinite cardinal. I am trying to understand the proof the the following ...
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1answer
28 views

Prove: Suppose $\alpha > 1$ and $\beta, \gamma$ are ordinals with $\beta < \gamma$. Then $\alpha^\beta < \alpha^\gamma$.

In this question, $\alpha$, $\beta$ and $\gamma$ are ordinals. I want to prove this by transfinite induction on $\gamma$, which typically has two or three cases. I'm considering three cases: the base ...
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1answer
52 views

On the proof that every ordinal is either the initial ordinal, a successor ordinal or a limit ordinal.

I was reading this paper http://www.math.umn.edu/~garrett/m/algebra/notes/14.pdf (p. 206, or p. 8 on a pdf) on Naive Set Theory on the proof that every ordinal is either the initial ordinal (empty ...
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2answers
76 views

Proof of Hartogs' Lemma

I'm reading "Sets, Models and Proofs" by I. Moerdijk and J. van Oosten to have some basic understanding of set theory. There's one step in "Hartogs' Lemma" proof that I don't see how to it works: I ...
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1answer
32 views

Conjecture about ordinal exponentiation: $n^ω=ω$ $\forall n\in\mathbb{N}$-{0,1}

Let $ω$ be the ordinal of the natural numbers. I think this is true: $n^ω=ω$ $\forall n\in\mathbb{N}-${0,1} Am i right? If I am wrong, is it true for any $n\in\mathbb{N}$?
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1answer
73 views

Tarski's fixed point theorem

My professor stated the following "For every normal function $f \colon ON \to ON$ and for every ordinal $\beta$ there exist an ordinal $\gamma \ge \beta$ such that $f(\gamma)=\gamma$". (normal ...
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1answer
88 views

Simplify this expression of ordinal numbers: $(ω+1) ω²$ [closed]

Let $ω$ be the ordinal of the natural numbers. Simplify $(ω+1) ω²$
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1answer
48 views

Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
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1answer
99 views

Ordinal addition is associative

We've been asked to teach ourselves a unit on ordinals for our final exam tomorrow, I grasp how to prove that certain ordinals are distinct but I am having trouble figuring out a proof to show ordinal ...
3
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1answer
111 views

When does $f_{\omega+1}$ catch up the $G_n$-sequence?

Which is the minimal number k, so that $f_{\omega+1}(n) > G_n$ is true for all $n\ge k$ ? For the definition of $f_{\omega+1}$ look at wikipedia fast growing hierarchy $G_n$ is defined by ...
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0answers
49 views

How many omegas are there in $\large f_{\epsilon_0}$?

For a description look at fast growing hierarchy at wikipedia. $\large f_{\epsilon_0}$ is not defined any more, it is a power tower of omegas, but how many omegas ? I found a defition $$\large ...
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1answer
25 views

An alternative succinct proof needed for trivial cardinality fact

Let $|X|$ denote the cardinality of a set, i.e. the least ordinal $\alpha$ such that there is a bijection between X and $\alpha$. For any sets $X$ and $Y$ we write $X\preccurlyeq Y$ if the exists an ...
2
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1answer
148 views

Definition of the function $f_{\epsilon_0}$ in the fast-growing hierachy

I found an article where the growth of $$f_{\epsilon_0}$$ in the fast-growing-hierachy is described, but the text is very long and difficult to understand. Is ...
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2answers
32 views

Existence of a function from ordinal to limit ordinal of the same cardinality

Let $\alpha$ be a limit ordinal, such that $\big | \alpha \big | = \omega$. Prove that there is a strictly increasing function $f: \omega \to \alpha$ such that for all $\zeta < \alpha$ there is $n ...
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1answer
66 views

Prove $\bigcup \omega_1 = \omega_1$

This is a question regarding ordinals and probably requires some background knowledge of ordinal arithmetic. I will list what I hope is the relevant information for this question: $\omega_1$ is ...
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1answer
36 views

Ordinal Arithmetic sufficient condition that a + c < b + c

I know that in general ordinal addition is not strictly increasing in the left argument (as in $0+ \omega = n+\omega$). Now I have a fixed countable ordinal $\delta$ and a natural number $k$. My ...
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1answer
33 views

Weak monotonicity of ordinal addition

I'm trying to prove the weak monotonicity of ordinal addition, i.e. if $\alpha \leq \beta$, then $\alpha + \gamma \leq \beta + \gamma$. The proof is not all that difficult, but I want to make sure I ...
3
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3answers
133 views

Why is it that $\omega + \omega_1 = \omega_1$?

I have seen some other questions about the proofs around ordinals and $\omega$ and $\omega_1$, but some of the answers have confused me. An ordinal $\alpha$ is countable if $\alpha < \omega_1$. ...
2
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1answer
62 views

A canonical injection $f:\alpha\to[0,1]$ for an arbitrary countable ordinal $\alpha$

Is there a canonical injection $f:\alpha\to[0,1]$ for an arbitrary countable ordinal $\alpha$? As long as $\alpha$ is of the form $m\cdot\omega + n$ for $m,n<\omega$, this is fairly simple, but if ...
2
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2answers
93 views

Prove using transfinite induction that if ordinals $\alpha$ and $\beta$ are countable, then so is $\alpha + \beta$.

This question requires using transfinite induction. I plan to fix $\alpha$ and then do transfinite induction on $\beta$. Recall that an ordinal $\alpha$ is countable if $\alpha < \omega_1$, where ...
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1answer
45 views

Cardinality of Orderings of $\mathbb{R}$

For a finite set $S$ there are $\vert S\vert!$ orderings of its elements. What is the cardinality of all orderings of $\mathbb{N}$? What would $$\vert \mathbb{N}\vert!$$ mean? Is it ...
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1answer
62 views

Prove that for every three ordinals $\beta \lt \gamma \Rightarrow \alpha+\beta \lt \alpha+\gamma$

Prove that for every three ordinals $\alpha,\beta,\gamma$ we have $\beta \lt \gamma \Rightarrow \alpha+\beta \lt \alpha+\gamma$ It's obvious if all of them are finite, also if only alpha is ...
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3answers
67 views

What is the usual definition of “ordinal number”?

My definition for the ordinal number is "von-Neumann ordinal". I thought this is the only definition for the ordinal, but i found some other definitions in wikipedia. What is the usual definition of ...
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1answer
58 views

Short question about ordinals multiplication definition from wiki

Why does $2\cdot\omega$ looks like this: $0_0 < 1_0 <0_1 <1_1 ...$ ? Is that another way to represent $1_0 < 1_1 < 2_0 <2_1 <3_0<3_1 ...$ ? Edit: also, why does it equal to ...
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1answer
25 views

Prove or disprove - order types and constants

Let $k$ and $l$ be natural numbers and let $\omega=[(\mathbb N, \le)], \ \eta=[(\mathbb Q, \le)]$ be order types (or ordinals). Prove or disprove the following: if $k+\eta=l+\eta$ ...
3
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2answers
71 views

Embedding $\omega_1$ into the direct sum/product of $\Bbb R$'s and $\Bbb N$'s

I'm trying to find a way to visualize the first uncountable ordinal $\omega_1$. This is rather difficult, as the visualization tactic that I often use for countable ordinals - namely, the "matchstick" ...
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0answers
30 views

Existence of countable ordinal [duplicate]

Prove that there exist countable ordinal $\xi$ such that $\xi=\omega^\xi$. The $\xi= \sup \{b^i \mid b_1=w, b_{i+1}=w^{b^i}\}$ should work. But how to prove that $\xi$ is countable?
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1answer
114 views

Ordinality of a Set

What is the difference between Ordinal number and cardinal number of a set?....I have a confusion in understanding the difference between the two.Can anyone help me to understand these two things? ...