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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28 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 ...
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0answers
14 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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23 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
22 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
116 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 ...
2
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2answers
24 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 ...
2
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1answer
52 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
24 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 ...
1
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1answer
15 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
93 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
53 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
34 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 ...
1
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1answer
38 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
52 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 ...
0
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3answers
42 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 ...
1
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1answer
32 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 ...
1
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1answer
23 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 $k+\eta=l+\eta$ ...
2
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2answers
48 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" ...
1
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0answers
29 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
54 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? ...
1
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1answer
22 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 = ...
0
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2answers
50 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\} \\ & ...
3
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2answers
78 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
79 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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2answers
47 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 ...
2
votes
1answer
55 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 + ...
1
vote
1answer
23 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 $\alpha$ ...
6
votes
1answer
65 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 ...
1
vote
1answer
25 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 ...
0
votes
1answer
33 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?
2
votes
1answer
53 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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1answer
51 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
37 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 $a\ge 0, a<b$, ...
0
votes
1answer
30 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 ...
2
votes
3answers
59 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 ...
2
votes
1answer
26 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
42 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 ...
0
votes
1answer
51 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: $2\cdot\omega_1+\omega\cdot3+3,$ $\omega\cdot3+\omega_1+3,$ ...
1
vote
1answer
60 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 ...
2
votes
2answers
92 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 ...
1
vote
1answer
71 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 ...
2
votes
3answers
53 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 ...
0
votes
2answers
25 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
30 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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votes
1answer
32 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.
0
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2answers
78 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
votes
1answer
33 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 > ...
1
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2answers
76 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
votes
1answer
39 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 ...
0
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1answer
39 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 $>\omega_{1}$. ...