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

0
votes
0answers
25 views

'Popular mathematics' resource about infinite ordinals

The 'popular mathematics' literature (think Martin Gardner, William Dunham, Hofstadter, and the like) abounds with material on the mathematics of infinite cardinals, starting - and quite often ending ...
0
votes
1answer
17 views

For every ordinal $\alpha$, there is a cardinal number greater then $\alpha$.

I am trying to prove that for every ordinal $\alpha$, there is a cardinal number greater then $\alpha$. Proof: $\alpha$ is a (well ordered) set. Take the set $P(\alpha)$. Assume that we know that for ...
1
vote
1answer
16 views

A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ for some $\alpha$. [duplicate]

I am trying to prove ex 2.13 from Jech's book on set theory: Ex 2.13: A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ ...
3
votes
0answers
32 views

If a linearly ordered set $L$ has the property that every order-preserving injection $L \rightarrow L$ is expansive, is $L$ necessarily well-ordered?

Given a poset $P$, call a function $f : P \rightarrow P$ expansive iff $f(x) \geq x,$ for all $x \in P$. Now suppose a linearly ordered set $L$ has the property that every order-preserving injection ...
1
vote
1answer
32 views

Introduction to Set Theory(Ordinal Numbers)

Lemma: If $\alpha$ is an ordinal number, then $\alpha \not \in \alpha$ Proof: If $\alpha \in \alpha$, then the linearly ordered set $(\alpha, \in_{\alpha})$ has an element $x=\alpha$ such that $x \in ...
0
votes
1answer
38 views

Question regarding addition of order types

I have been reading through Herbert Endertons introductory book on set theory as I have stumbled upon a claim that baffled me through the day. As anyone I need sleep so I ask here for help. Namely ...
1
vote
4answers
123 views

Are $1+ω$ and $ω+1$ isomorphic?

In set theory, $1+ω$ is defined as the ordinal number of ordinal sum of sets $\{a\}$ and ℕ. Also $ω+1$ is defined as the ordinal number of ordinal sum of sets $\Bbb N$ and $\{a\}$. You know what the ...
0
votes
1answer
28 views

Ordinal multiplication property: $\alpha<\beta$ implies $\alpha\gamma\le\beta\gamma$

I'm having trouble proving the following two ordinal multiplication properties. If $\alpha, \beta$, and $\gamma$ are such that $\alpha \lt \beta$ and $\gamma \gt 0$, then $\alpha\gamma \le ...
0
votes
2answers
36 views

the smallest transfinite ordinal number

My guess is that $\omega$ is the smallest transfinite ordinal number. To prove this, let $\beta$ be any transfinite ordinal number and let $\operatorname{ord}(B, \le)= \beta$. And I need to show that ...
2
votes
2answers
49 views

Ordinal numbers addition property: $b<c$ implies $b+a \le c+a$

I'm having trouble proving the following property of ordinal numbers. If $a, b, c$ are ordinal numbers such that $b \lt c$, then $b+a \le c+a$. I first started by assuming $g$ as an order ...
1
vote
2answers
44 views

$\kappa\cdot\kappa= \kappa,$ for infinite cardinals

I am trying to understand the proof that uses a maximal-lexicographic ordering. For an infinite ordinal $\kappa,$ the canonical well-ordering of $\kappa \times \kappa,$ denoted by $<_{cw}$ is ...
1
vote
1answer
59 views

Is it possible to iterate a function transfinite times?

Let $f:A\rightarrow A$ be a function. We can simply define $f\circ f$, $f\circ f\circ f$, etc., for each given natural number inductively. $f^{(0)}=id_A$ $\forall n\in\omega \qquad f^{(n+1)}=f\circ ...
1
vote
2answers
30 views

How does cf(cf $\alpha$) = cf $\alpha$?

Assume $\alpha > 0$ is a limit ordinal, and cf $\alpha=$ the least ordinal $\beta$ such that there is an increasing $\beta$-sequence $\langle \alpha_\xi \, \colon\, \, \xi < \beta\rangle$ that ...
0
votes
0answers
10 views

How to compare two ordinals given in finite variable Veblen function?

The two-variable Veblen function can be generalized to finitely many variables. Wikipedia describes this generalization as: $$\gamma \mapsto \varphi(\alpha_n,\cdots,\alpha_{i + ...
1
vote
1answer
29 views

cardinals and ordinals, problem with proper class

A while ago I wanted to educate myself about cardinals and ordinals and stubled across this pdf (enjoyed it a great deal!). Visit www.math.ksu.edu/~nagy/real-an/ap-c-ord.pdf to see the pdf! It was ...
1
vote
1answer
29 views

Question about the proof of this lemma: If $\alpha$, $\beta$ are ordinals, then either $\alpha \subset \beta$ or $\beta \subset \alpha.$

Proof: Clearly $\alpha \cap \beta$ is an ordinal, $\alpha \cap \beta = \gamma.$ Then $\gamma = \alpha$ or $\gamma = \beta$. For, if not, then $\gamma \not= \alpha$ and $\gamma \not= \beta$. Then ...
3
votes
1answer
49 views

What are $\in$-well orderable classes?

$V$ is not $\in$-well ordered but $Ord$ is. Is $Ord$ the unique proper class of the universe which is $\in$-well ordered?
2
votes
1answer
26 views

Finding two functions, $f, g$, such that $\mathrm{sup}(g \circ f[\omega]) < \mathrm{sup} (g[\omega + \omega])$

I've been working for some time on Schimmerling's A course on Set Theory, and, thanks to you guys, I'm now almost finishing chapter 3 on ordinals (hah!). In one of the last exercises, he ask us to ...
3
votes
1answer
85 views

$\aleph_\omega$ and large ordinals?

Assuming the GCH, each time you take the powerset of an aleph number, the subscript increases by one. So there is $\aleph_0, \aleph_1, \aleph_2\ldots \aleph_\omega\ldots \aleph_{\epsilon_0}\ldots$ and ...
0
votes
2answers
58 views

How to define an isomorphism between $^\omega\omega$ and $\omega^\omega$?

Let $^\omega\omega$ be the set of all functions $x: \omega \to \omega$. Define $A = \{x \in ^\omega\omega \; | \; x \text{ has finite support}\}$, where by "finite support" I mean that the set $\{x(n) ...
1
vote
0answers
39 views

ind-completion and ordinals

Given a category $C$, we have its ind-completion $Ind(C)$ whose objects are filtered diagrams in $C$. Assuming the axiom of choice, is any object in $Ind(C)$ isomorphic, in $Ind(C)$, to an ordinal ...
2
votes
3answers
73 views

increasing sequence in $\omega_1$

I want to prove that a (countable) increasing sequence in $\omega_1$ converges to some point. Here is my idea: I first try to find the limit point. We know any point in the increasing sequence (say ...
0
votes
2answers
59 views

Ordinal existence

Is there any ordinal $\alpha$ such that $\omega ^ {\omega ^ \alpha} = \alpha$? Could you please suggest me how to even try to solve this?
0
votes
1answer
48 views

Prove $\omega_1$ is first countable

Given a well order (W,$\le$), where W is uncountable, and $\omega_1$:= {x$\in$W: only countably many y$\in$X s.t. y $\le$ x}, prove $\omega_1$ is first countable. I saw a proof saying that {(a,x]: a ...
1
vote
1answer
56 views

why $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$ doesn't hold?

this rule doesn't hold for all ordinals $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$. I tested many examples but all of them holds for it ! does this hold ? $(B \cup ...
0
votes
1answer
31 views

How to show $A_1 \approx A_2 \iff \mathrm{card}(A_1)=\mathrm{card}(A_2)$

For any set $A_1$ and $A_2$, let us define the relation $\approx$ if there exists a bijection between $A_1$ and $A_2$. Then I want to show that $A_1 \approx A_2 \iff ...
1
vote
1answer
46 views

Insistence on limit ordinal in Kunen's proof of reflection.

I'm reading for an exam on set theory using Kunen's first edition, and we've more or less been told that the reflection theorem will come up. The proof constructively yields an ordinal ...
1
vote
0answers
35 views

$\sigma$-field and Uncountable ordinal

I have been trying to get my head around this question. Any help greatly appreciated. Let $\mathscr{C}$ be any class of subsets $\Omega$ with $\emptyset,\Omega\in \mathscr{C}$. Define ...
3
votes
1answer
72 views

The order-theoretic structure of the ordinal numbers in non-standard models

I have read the following. Proposition. Let $T$ denote the first-order theory of $\mathbb{N}$ in the language of arithmetic. Then any countable non-standard model of $T$ is order isomorphic to a ...
5
votes
0answers
59 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
0
votes
1answer
39 views

A set which is $\in$-transitive and well ordered by $\in$ is an ordinal

A set $x$ is said to be $\in$-transitive if $\forall y$ $\forall z$($ y \in x$ and $ z \in y \Rightarrow z \in x$). A set $x$ is said to be an ordinal if $x$ and every member of $x$ is ...
0
votes
1answer
47 views

Ordinal arithmetic question

The question is: find $\delta \le \omega_1$ and $\rho \lt \omega$ such as $\omega_1 = \omega \cdot \delta + \rho$. My guess is that $\delta = \omega_1$ and $\rho = 0$, abut I don't know how to prove ...
2
votes
1answer
51 views

Ordinal $10^\omega$

$10^\omega$ = $10 \cdot 10 \cdot 10 \cdot ...= \lim_{\alpha \lt \omega} (10^\alpha) = \omega$. Are my thoughts correct? Is this sufficient explanation, given the ordinal arithemtic proved from ZFC?
2
votes
1answer
431 views

Biggest countable ordinal number

I need to find biggest countable ordinal number. But I am sure there is no one, if I understand proofs and definitions correctly. So here are my idea: Suppose there is biggest countable ordinal number ...
1
vote
1answer
54 views

Proving a relation between Ordinal Exponentiation and its corresponding set

(Characterization of Ordinal Exponentiation) Let $\alpha$ and $\beta$ be ordinals. For f:$\alpha \rightarrow \beta$, let s(f)={$\epsilon < \alpha|f(\epsilon) \not= 0$}. Let S($\alpha, ...
1
vote
2answers
94 views

What do you need to know about ordinals to understand Zorn's Lemma's proof?

I'm trying to understand the proof of Zorn's Lemma but the one which does not use ordinals (Halmos' proof) is extremely long and I really feel I get lost somewhere along the way. On the other hand, ...
-1
votes
1answer
18 views

Finite set of rank $\alpha$

Defining the rank function $\in$-recursively as $\text{rank}(x)=\bigcup\{\text{rank}(y)^{+}\mid y\in x\}$ for a set $x$, for which ordinals $\alpha$ does there exist a finite set of rank $\alpha$? And ...
-1
votes
1answer
30 views

Different types of well-ordering of $\mathbb{N}$

The question is: "Find 3 non-isomorpic with each other well-orderings of $\mathbb{N}$. Define their ordinals and arrange them in magnitude." The question in ZFC, but I only know that such ...
0
votes
1answer
40 views

Are these ordinals cardinals?

Consider $\omega_2 \times \omega$ and $\omega \times \omega_2$ with ordinal arithmetic. Then $| \omega_2 \times \omega | =\omega_2$ and $| \omega \times \omega_2 | =\omega_2$ Does this imply ...
15
votes
5answers
1k views

Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems ...
1
vote
1answer
21 views

Countable limit ordinal as limit of $\omega$ ordinals

I've got a most probably silly question, but I can't find the answer to it: If $\alpha$ is a countable limit ordinal, how can we be sure that there exist ordinals $\alpha_n$ such that $\alpha = ...
4
votes
2answers
100 views

Proof that $\omega_1$ is countable (where is the flaw?)

I have discovered a "proof" of the fact, that $\omega_1$ can be countable, contradicting its definition. It mustn't assume axiom of choice, so I guess there are some parts of this construction which ...
0
votes
2answers
38 views

Element of ordinal a subset of the same ordinal

I've got a very short question in set theory. I am currently reading P. T. Johnstone's book Notes on Logic and set theory, and the proof of the fact that Every element of an ordinal is a subset of ...
1
vote
2answers
48 views

Is a limit ordinal necessarily a cardinal?

Maybe this is a trivial question. I see that every infinite cardinal is necessarily a limit ordinal, but is the converse true ?
1
vote
1answer
43 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
26 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 ...
0
votes
1answer
32 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
72 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
31 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
70 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 ...