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 ...

learn more… | top users | synonyms

1
vote
1answer
53 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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
86 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 ...
4
votes
2answers
98 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 ...
1
vote
1answer
38 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 $n\in\mathbb{N}$?
0
votes
1answer
89 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 ...
-1
votes
1answer
106 views

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

Let $ω$ be the ordinal of the natural numbers. Simplify $(ω+1) ω²$
4
votes
1answer
65 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 ...
2
votes
1answer
164 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
votes
1answer
123 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 ...
1
vote
0answers
55 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 ...
0
votes
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
votes
1answer
165 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
votes
2answers
36 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
votes
1answer
77 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 ...
1
vote
1answer
46 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
vote
1answer
48 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
votes
3answers
175 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
votes
1answer
79 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
votes
3answers
146 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
vote
1answer
46 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 ...
1
vote
1answer
73 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
votes
3answers
79 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
vote
1answer
78 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
vote
1answer
32 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
votes
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" ...
2
votes
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?
1
vote
1answer
196 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
vote
1answer
93 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
votes
2answers
60 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
votes
2answers
99 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. ...
1
vote
1answer
234 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 ...
0
votes
2answers
82 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
90 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
29 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 $\alpha$ ...
7
votes
1answer
110 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 ...
3
votes
2answers
71 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
61 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
95 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 ...
1
vote
1answer
59 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
1
vote
1answer
47 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
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 ...
2
votes
3answers
139 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
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 ...
2
votes
1answer
85 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
73 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
97 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
108 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
132 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
88 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 ...