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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53 views

Prove: $2^{\aleph_0}\not=\aleph_{\epsilon_0}$

Prove: $2^{\aleph_0}\not=\aleph_{\epsilon_0}$ I would like a hint for this problem
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1answer
44 views

Ordinal numbers and onto function

So, another one in set theory (I think I am falling inlove with the subject). The question itself as presented: Given $\Bbb Z$ is ordered by $<'$, where $a<'b$ iff $a\ge 0, a<b$, ...
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1answer
32 views

Is there a difference between the order type of Q·ω and Q·Q?

From what I understand, the expression is "a countable amount of the order type of Q", which intuitively should be equal to the second expression. Is this true? How do I explain this formally? Thanks ...
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104 views

Countable subset bounded in uncountable set

Let $(B, \prec)$ be a well ordered set with the ordinal $\omega_1$. Show that every countable subset of B is bounded in $(B, \prec)$. Let A be such a subset. A is a subset of a well ordered set and ...
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45 views

Set of Cardinals

Let $A$ be a set of cardinals. Prove that there is a cardinal that that is greater than every cardinal in $A$. Assume that there isn't such a cardinal. Then for any cardinal $x$ there is $y\in A$ such ...
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1answer
64 views

How to prove by induction that the set of all natural numbers is an ordinal

I have seen alternative methods of this proof, with one being: let $n$ be the set of all natural numbers. Then (1) $\omega$ is an ordinal, (2) If $\alpha$ is an ordinal and $\beta \in \alpha$, then ...
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1answer
60 views

Ordering ordinals by size [closed]

Well define $\omega,\omega_1, \omega_2$ to be the first three infinite ordinals. Order them according to their size: $2\cdot\omega_1+\omega\cdot3+3,$ $\omega\cdot3+\omega_1+3,$ ...
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1answer
80 views

Prove that $\omega + \omega_1 = \omega \cdot \omega_1 = \omega^{\omega_1} = \omega_1$

I am assuming already that a) the union of countably many countable sets is countable and b) $\omega_1$ is the least uncountable ordinal, so $x < \omega_1$ if and only if $x$ is a countable ...
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2answers
98 views

What is the cardinality of the open ordinal space $[0,\Omega)$ if we remove it's limit ordinals?

Let $\Omega$ be the first uncountable ordinal. and for any limit ordinal $\lambda < \Omega$ Let $U_\lambda$ an open neighborhood of $\lambda$ with the order topology. what is the cardinality of the ...
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1answer
110 views

Is any subset of the open ordinal space $[0,\Omega)$ $G_\delta$?

Consider the open ordinal space $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal. Can I say that every subset of $[0,\Omega)$ is $G_\delta$? If yes, does this imply that $[0,\Omega)$ is ...
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3answers
66 views

What is the cardinality of all limit ordinals $\alpha$ s.t. $\alpha < 2^\mathfrak c$

Let $\Omega$ be the first ordinal with cardinality $2^\mathfrak c$. Take now the set of all ordinals $\alpha < \Omega$ which are limit ordinals. Is the cardinality of this set countable or is it ...
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2answers
31 views

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
37 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
102 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 ...
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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
87 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
48 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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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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244 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
81 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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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 ...
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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
54 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
118 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 ...
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1answer
83 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: ...
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1answer
42 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 ...
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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 ...
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1answer
130 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
158 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 ...
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138 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
87 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
51 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
194 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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219 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 ...
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1answer
86 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 ...
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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
152 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
60 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
109 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> ...