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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If $n$ is a finite ordinal, either $n=\emptyset$, or there exists a finite ordinal $m$ such that $n=m \cup\{m\}$

I have a difficulty with this statement : If $n$ is a finite ordinal, either $n=\emptyset$, or there exists a finite ordinal $m$ such that $n=m \cup\{m\}$. My professor's proof is totally unclear. I ...
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0answers
35 views

Find ordinal of a strange ordered set

I was asked the following question: for every $x\in \mathbb R$ we define the set $Q(x)=\{q \in \mathbb Q|q\leq x\}$, the set of all rational numbers less or equal to $x$. Let $M=\{Q(x)|x\in \mathbb ...
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1answer
36 views

Union of limit ordianl [closed]

Let $\alpha$ be a limit ordinal. Show that $\cup\alpha=\alpha$. My definition of a limit ordinal is an ordinal number that is not 0 and not a successor ordinal.
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2answers
98 views

Some questions about elementary set theory (cardinal and ordinal numbers)

I have three questions about elementary set teory and i don't figure out how to solve them: 1)Let $X$ a subset of the cardinal number $2^{\aleph_0}$ (seen as an initial ordinal). Is true or false ...
3
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1answer
40 views

Ordinal inequality - simple question

We are given 2 ordinals: $\alpha$ and $\beta$ where $\beta$ does not have a maximal number (So it's transfinite, right?) We are asked to find $\alpha,\beta$ such that: $\alpha+\beta > ...
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2answers
86 views

What does $\alpha+\gamma$ mean when $\alpha$ and $\gamma$ are well-ordered sets?

I was asked to prove the following: let $\gamma$ be a well ordered set with the following property: for any $\alpha$ and $\beta$ well ordered sets, if $\alpha+\gamma=\beta+\gamma$ then ...
3
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1answer
45 views

Well ordered Set with $\le$ order type

Let $(x,\le)$ be well-ordered set and let $f: \ x \rightarrow x$ be monotonically increasing function. Prove that $\forall a \in x$ $$a \le f(a)$$ Find an example of set x linearly ...
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1answer
40 views

Examples of $\omega_{1}<\alpha<\omega_{2}$ definable

Can you think of any examples of definable ordinals between $\omega_{1}<\alpha<\omega_{2}$? I am trying to show that countable $M\prec (H(\aleph_{2},\in)$ contains ordinals $>\omega_{1}$. ...
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1answer
44 views

Are the following sets isomorphic or have the same order type?

Are the following sets isomorphic or have the same order type ? $(\mathbb R, \le) ,\ (\mathbb R,\ge)$ $(\mathbb Q, \le), \ (\mathbb R, \le) $ $(\mathbb N,\ge ), \ (\mathbb N,\le)$ ...
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239 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
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1answer
78 views

Sets and ordinals - homework questions

I have two questions which I don't even know how to start. I would like you to give some hints. I know it would be better that I show some work, but I really don't know where to begin... The question ...
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2answers
73 views

Proof of Proposition 0.17 in Folland wrong?

Proposition 0.17 in Folland's Real Analysis (2e) is If $X$ and $Y$ are well ordered, then either X is order isomorphic to $Y$, or $X$ is order isomorphic to an initial segment in $Y$, or $Y$ ...
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0answers
59 views

Generalization of simple and transfinite induction

Definition For a set of ordinals $\boldsymbol\alpha$ and ordinals $\gamma$, $\beta$, let $$\boldsymbol\alpha \xrightarrow[\gamma]{}\beta$$ symbolize the proposition that ...
2
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0answers
27 views

what is the third term of the $\omega^{\omega}$-accelerated squares?

let $\{s_n\}_{n=0,1,2...}$ be a strictly increasing sequence of positive integers, indexed by the finite ordinals. if $S$ denotes the space of all such sequences, let the accelerator function, $\alpha ...
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0answers
47 views

Well ordering of type epsilon one

I have been very interested in the countable ordinals for awhile now, but one thing has eluded me despite my research into the subject. What is a well-ordering of the natural numbers corresponding to ...
4
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1answer
115 views

Bijection between countable ordinals and reals

The set of all countable ordinals is $\omega_1$, which has a cardinality of $\aleph_1$. When accepting the continuum hypothesis, $2^{\aleph_0} = \aleph_1$, so a bijection between countable ordinals ...
4
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1answer
82 views

Is the ordinal $\omega \uparrow^\omega \omega$ still recursive?

In this question, a very large countable ordinal $\omega \uparrow^\omega \omega$ is defined. Is this ordinal still recursive?
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1answer
79 views

Order type of standard models of arithmetic

The standard model of PA has order type $\omega$. By compactness PA has a model of order type $\omega+n$ for any $n$, since every finite subset of the following set of statements is provable: ...
2
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1answer
38 views

Every woset is an ordinal.

I use this notation: a well ordered set $Y$ is an ordinal if for every $a\in Y$, $Y_a=a$, where $Y_a=\{y\in Y|y< a\}$. Now, I know that for every woset there is an isomorphism from that woset to a ...
2
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2answers
50 views

Noninital ordinal of a set

If $\mathbb R$ is well ordered by an order $\ll$ such that the ordinal of R and that order $\alpha$ is not initial. How can I show that there is a closed interval $[a,b]$ ($a<b$) such that the ...
3
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1answer
119 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...
3
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1answer
138 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
3
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136 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
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3answers
91 views

Why $\omega+1$ and $\omega^2$ are not cardinal numbers?

I see the following definition of cardinal number in notes: An ordinal $\alpha$ is a cardinal number if $|\beta|<|\alpha|$ for all $\beta\in\alpha$. Why $\omega+1$ and $\omega^2$ are not cardinal ...
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2answers
83 views

For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

I don't understand the proof to the a/m claim. How we know that $\eta < \kappa$ and $(\alpha,\beta)<_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone ...
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1answer
50 views

Naturally definable order relations

Each and every function $f:X\rightarrow Y$ between two totally ordered sets $(X,\leq_X), (Y,\leq_Y)$ induces a relation $\preceq$ on $X$ by $$x \preceq x' :\equiv \begin{cases} x &\leq_X x' ...
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3answers
190 views

How uncountable is the set of countable ordinals?

From the answers (and comments) to this question on the uncountability of countable ordinals I don't get a lucid picture: How can I see that the uncountability of the set of countable ordinals ...
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1answer
46 views

If $\alpha,\beta$ are at most countable, so are $\alpha+\beta,\alpha\cdot\beta,\alpha^\beta$

If $\alpha$ and $\beta$ are at most countable ordinals, prove the following are at most countable: $\alpha + \beta$ $\alpha \cdot \beta$ $\alpha^{\beta}$
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1answer
34 views

A problem on ordinal arithmetic.

Let $0 < \alpha \leq \beta$ be ordinals. Could anyone advise me on how to prove there exists $\delta $ and $\xi$ such that $\xi < \alpha$ and $\alpha \ . \delta + \xi = \beta?$(ordinal ...
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217 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
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1answer
55 views

Reducing ordinals in the representation of a limit ordinal as $\bigcup_{\beta < \alpha} \beta$

Let $\alpha$ be a limit ordinal such that $\kappa$ < $\alpha$ < $\kappa^{+}$ where $\kappa$ is initial ordinal (so $|\alpha| = \kappa$). I want to know whether or not I can find a sequence ...
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1answer
47 views

Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$

I am getting repeatedly lost trying to approach this question: Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$ I can fairly well grasp the idea that the ...
2
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1answer
79 views

Uncountable compact sets in ordinal topology

The question I am interested in is the following: Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied with order topology. Prove ...
2
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1answer
64 views

A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
2
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1answer
149 views

Let $\gamma $ be an ordinal. Prove there is an ordering preserving $f:\gamma \to \mathbb{R}$ iff $\gamma < \omega_1.$

Let $<$ be the usual ordering on $\mathbb{R}.$ If $\gamma$ is an ordinal, then $f:\gamma \to \mathbb{R}$ is ordering preserving if $\forall \alpha \in\gamma \forall \beta \in \gamma [\alpha \in ...
2
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2answers
38 views

Existence of certain $\left\langle{\alpha_n | n \in \omega}\right\rangle$

Let $\beta$ be a countable limit ordinal. Prove $\exists$ sequence $\left\langle{\alpha_n | n \in \omega}\right\rangle$ with the following properties: $(1): \alpha_0 = 0\;;$ $ (2): \forall n \in ...
2
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1answer
57 views

Functions from ordinals to ordinals

I'm trying to solve this problem, which appears in Schimmerling's "A Course in Set Theory." Problem. Find two functions $$f:\omega\rightarrow\omega\cdot2$$ and ...
2
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2answers
106 views

Proving, for every three ordinals $\alpha,\beta,$ and $\gamma$, that $\alpha^{\beta+\gamma}=\alpha^{\beta}\alpha^{\gamma}$

I am trying to prove, for every three ordinals $\alpha,\beta,$ and $\gamma$, that $\alpha^{\beta+\gamma}=\alpha^{\beta}\alpha^{\gamma}$. Proof is by induction on $\gamma$. Basis: If $\gamma = 0$, ...
2
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1answer
88 views

$\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$

Prove: $\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$. From left to right: if $\omega >\beta: \alpha=\omega^\alpha> ...
3
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1answer
113 views

Ordinals: if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$

I'm trying to proof that if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$, where $\omega$ is the least infinite ordinal. I started with transfinite induction on ...
0
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1answer
58 views

How to compute a Von Neumann Successor - ∪∪7?

I wonder if I have taken the correct approach to demonstrating what ∪∪7 is, using Von Neumann's proposal? And if I am indeed correct? My answer appears a little odd to me! Question: What is ∪∪7? By ...
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0answers
44 views

Additively indecomposable ordinals

As a continuation to my question here: Ordinal inequalities. I have tried to write down a proof for this claim: If $\alpha$ is of the form $\omega^\delta$ for some ordinal $\delta$, then, For every ...
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1answer
76 views

Ordinal inequalities

I have a problem with the following exercise: First a definition: $ \mathord{\in} = \{\langle x,y \rangle; x \in y \}$ Exercise: Show that the following conditions are equivalent: a. For every ...
3
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1answer
199 views

Showing ordinal addition and ordinal multiplication are defined

I've been reading absoluteless results in Kunens't latest Set Theory text. After talking about $\Delta_0$ formulas and absoluteness, he mentions that certain concepts are absolute, but not $\Delta_0$. ...
2
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1answer
34 views

Can ordinals multiplication be defined by recursion?

I am trying to figure out weather ordinals multiplication be defined by transfinite recursion. It seems a bit problematic to me since multiplication is a function $f: \alpha \times \beta \rightarrow ...
0
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2answers
86 views

the Class of Ordinals is a set?

Why the class of ordinals On is not a set? Please dont tell me it's well-ordered till you tell me what you mean by an ordering on something that you dont know in first place if it is a set or not.
3
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2answers
70 views

$f: \omega_1 \to \omega_1$. $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $.

Prove: if $f: \omega_1 \to \omega_1$ is an increasing, continuous, unbounded, function, then $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $. Can anyone give me a tip?
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4answers
41 views

If $A$ is a countable subset of $\omega_1$, then there is an $ \alpha < \omega_1$ such that $A \subseteq \alpha$

Prove : if $A$ is a countable subset of $\omega_1$. Then there exists $ \alpha < \omega_1$ with $ A \subseteq \alpha$. I don't really know where to start, can anyone give a tip first?
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2answers
63 views

An ordinal between an ordinal and its cardinality

Suppose $\alpha, \beta$ are ordinals (the first is not a cardinal) and $|\alpha| \subset \beta \subseteq \alpha$. Then why $|\alpha| \sim \beta?$ I know Cantor–Bernstein theorem does the job ...
2
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3answers
46 views

about definition of a cardinal

Definition: the cardinality of a set $A, |A|$ is the least ordinal s.t. $A \sim \alpha$ Definition: We define a cardinal to be an ordinal $\alpha$ s.t. $\alpha = |\alpha|.$ i.e, an ordinal s.th. ...