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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55 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 ...
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183 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 $\alpha < \omega_1$. ...
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1answer
86 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 ...
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3answers
152 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
47 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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77 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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89 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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90 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
34 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 $k+\eta=l+\eta$ ...
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2answers
77 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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31 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
257 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? ...
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1answer
96 views

Prove every nonzero ordinal is either successor or limit

I am trying to prove the statement in the title. Here is my partial proof: Let $n$ be a nonzero ordinal. It is obvious that $\cup_{\beta<n}\beta \subseteq n$. If $\cup_{\beta<n}\beta = ...
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2answers
61 views

Definition of $\omega_1$, comparing it to $2^\mathbb{N}$?

I'm taking an Intro to Topology class, and we just started defining ordinals. We defined finite ordinals as: \begin{align*} 0 & = \varnothing \\ 1 & = \{0\} \\ 2 & = \{0,1\} \\ & ...
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101 views

Do the ordinals exist before the universe of sets is constructed?

Should I worry about the following appeIarance of circularity in ZFC set theory? In constructing the universe of sets, you start with the empty set and then keep taking the power set over and over. ...
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1answer
257 views

If $\alpha$ and $\beta$ are ordinals, prove that $\alpha ^ \beta$ is a countable ordinal.

In this question I am supposing that both $\alpha$ and $\beta$ are ordinals. My definition of an ordinal is that: $x$ is an ordinal if $x$ is well-ordered by $\in$ and $x$ is $\in$-transitive. So ...
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87 views

Prove $\epsilon_0$ < $\omega_1$ [duplicate]

This is a question in ordinal arithmetic. (If anyone has read 'Classic Set Theory' by Derek Goldrei, this question comes from page 252.) $\epsilon_0$ = sup {$\omega$, $\omega^\omega$, ... } and ...
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1answer
103 views

Prove $\omega + \omega_1 = \omega_1$ [duplicate]

I am assuming that $\omega_1$ is the first uncountable ordinal and I'm using ordinal arithmetic. I have so far that if $\alpha$ and $\beta$ are ordinals, then $\alpha + \beta$ = sup{$\alpha + ...
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1answer
30 views

Equivalence between two definitions of successor ordinal

So, here am I again struggling with Schimmerling. Now I just want to check if understood correctly how to relate the two definitions of a successor ordinal. We can define the successor of $\alpha$ ...
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114 views

Order-preserving injections of ordinals into $[0,1]$

The usual "matchstick" representation of an ordinal number can be thought of as an order-preserving injection of that ordinal into the interval [0,1]. For example, here's a representation of ...
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2answers
80 views

Axiom of regularity and ordinal ranks

I am trying to prove that the following two statements are equivalent: Axiom of regularity $\forall x \exists \alpha (\alpha $ is an ordinal and $ x \in V_\alpha)$ I believe I understand how to ...
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1answer
62 views

Transitive sets and the Mostowski collapse

I was wondering if every set can be "transitized" - that is, made into a transitive version of itself. Is this basically what the Mostowski collapse says?
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1answer
100 views

Initial Segment Order Isomorphic to the Ordinal Numbers

Prove that every well-ordered proper class has an initial segment order isomorphic to the ordinal numbers, ON. I have a plan to prove this but it uses a recursive definition and induction which I do ...
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60 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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49 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
33 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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3answers
150 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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1answer
54 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
99 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
75 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
100 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
112 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
137 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
101 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
33 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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37 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
43 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
171 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
53 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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96 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
58 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
45 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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49 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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287 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
88 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
76 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
65 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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29 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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1answer
88 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 ...
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1answer
180 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 ...