1
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0answers
18 views

Subsets of $ \mathbb Q $ of order type $ \omega^{\alpha}$ for each countable ordinal $\alpha $.

My introductory text in Set Theory (Stillwell) includes an exercise (6.3.1) asking for an explicit example of a subset of $ \mathbb Q $ or order type $ \omega^2 $. This seems straight forward enough. ...
1
vote
0answers
25 views

Comprehending ordinals: from $\omega^\omega$ through $\omega^{\omega^2}$ to $\varepsilon_0$

I am currently trying to comprehend ordinal numbers by finding an order on some countable set (like natural numbers or tuples of natural numbers) that is isomorphic to some ordinal. For an instance, ...
1
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2answers
43 views

What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?

I've been working on a particular set theory problem for a while and essentially I've hit a roadblock because of this question. I just need to know what this ordinal is, because I have a sneaking ...
2
votes
1answer
40 views

Cardinality: is it true that $|X^\mathcal{inj}| \leq 2^{|X|}$?

Definition. Whenever $X$ is a set, write $X^\mathcal{inj}$ for the collection of all injections $f$ such that: $f$ has codomain $X$ There exists an ordinal $\alpha$ such that $f$ has domain ...
1
vote
1answer
18 views

In what sense $\alpha \times \alpha$ is the initial segment $(0,\alpha)$ in $Ord \times Ord$?

This is from Jech's book on set theory: "We define a well ordering of the class $Ord \times Ord$ of ordinal pairs. Under this well ordering, each $\alpha \times \alpha$ is an initial segment of ...
0
votes
1answer
19 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
17 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 $\gamma = \omega^{\alpha}$ ...
1
vote
1answer
37 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
42 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
127 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
31 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 ...
1
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3answers
54 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
54 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
31 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 ...
1
vote
1answer
32 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 ...
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 ...
0
votes
2answers
61 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) ...
0
votes
1answer
33 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 ...
2
votes
1answer
53 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
449 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
57 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
votes
1answer
20 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
31 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
42 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
25 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 = ...
0
votes
2answers
44 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
53 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
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1answer
44 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
27 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
48 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 ...
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 $n\in\mathbb{N}$?
2
votes
1answer
89 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 ...
0
votes
1answer
24 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
2answers
32 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 ...
1
vote
1answer
36 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
31 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 ...
2
votes
2answers
90 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
61 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
64 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
56 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
24 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 $k+\eta=l+\eta$ ...
1
vote
1answer
107 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
49 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
54 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\} \\ & ...
1
vote
1answer
24 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 $\alpha$ ...
1
vote
1answer
43 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
49 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?
1
vote
1answer
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
1
vote
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 $a\ge 0, a<b$, ...