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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-1
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1answer
24 views

ordinal cardinal task? [on hold]

a) Show that if $\alpha$ is an infinite ordinal then $\alpha+1$ is NOT a cardinal. b) Show that the following statement is false: Every limit ordinal is a cardinal. Help a brother out (please)
-3
votes
0answers
31 views

Strict cardinality, continuum hypothesis, upper bound and more [on hold]

We assume that there is no cardinal strictly between $\omega$ and $2^{\omega}$, in other words we assume the continuum hypothesis. You may use the fact that if $A=_cA^{'}$ and $B=_cB^{'}$, then ...
1
vote
3answers
41 views

Ordinals and Countability

I've been watching Prof Su's (Harvey Mudd College) incredible lecture on Ordinals and Transfinite Induction, and I have a few related questions. He starts by drawing and examining three sets of dots, ...
28
votes
7answers
3k views

Why do we classify infinities in so many symbols and ideas?

I recently watched a video about different infinities. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, ...
4
votes
1answer
34 views

Ordinal pair $(α,β)$ such that $α<β$ and $Th(α,<) = Th(β,<)$

A number of weeks ago I was thinking of finding an example of a complete countable theory with only one binary predicate that is not $ω$-categorical. I later realized that $Th(\mathbb{Z},<)$ works, ...
0
votes
3answers
102 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 ...
0
votes
0answers
30 views

Cantor normal form to compute sums and products of ordinals: $\omega^{\beta} c+\omega^{\beta'} c' = \omega^{\beta'}c'$ if $\beta'>\beta$

From Wikipedia on Ordinal arithmetic: The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example, one need merely know that ...
0
votes
0answers
24 views

How to show that an $\omega$-sequence is club in a limit ordinal $\gamma$.

Suppose that $\gamma$ is a limit ordinal. We say that a subset $X \subseteq \gamma$ is closed if whenever $\delta < \gamma$ is a limit ordinal and $\sup(X \cap \delta) = \delta$, then $\delta \in ...
2
votes
1answer
33 views

Explicit indexing of countable ordinals by sequences of integers?

What I really want is this: A sequence $P_0$, $P_1$,... such that each $P_n$ is a countable partition of $\omega_1$, $P_{n+1}$ is a refinement of $P_n$, and such that if $A_n\in P_n$ for all $n$ and ...
0
votes
2answers
30 views

Question regarding countable ordinals

Feel free to suggest a better title. We're going to regard $0$ as a limit ordinal, as people sometimes do. Let $L$ be the set of countable limit ordinals. If $\alpha\in\omega_1$ there exist a unique ...
3
votes
4answers
148 views

Is $\omega-1$ finite?

I saw some videos and read some stuff about ordinals, and it came to me that $\omega-1$ should be finite. My logic is that $\omega$ is the smallest transfinite number, so $\omega-1$ should be ...
2
votes
3answers
73 views

Proving the class of countable ordinals is closed under ordinal exponentiation in ZF

I managed to prove that given the axiom of choice, the class (or is it a set?) of countable ordinals is closed under exponentiation, since the axiom of choice implies that the countable union of ...
2
votes
1answer
52 views

Every $\alpha < \kappa^+$ can be embedded in any interval of $\kappa^{<\omega}$.

Let $\kappa$ be a cardinal. I want to show that every ordinal $\alpha < \kappa^+$ can be embedded (as an order) in any interval of $\kappa^{<\omega}$, ordered lexicographically. This is exactly ...
0
votes
1answer
12 views

Is the identity a normal function?

I was going to prove the following statement: Let $F$ be a normal function, then $\cup\{0,F(0),F(F(0)),\ldots\}$ is the least ordinal $\alpha$ satisfying $F(\alpha)=\alpha$. In case of $0\neq ...
0
votes
1answer
15 views

Limit ordinal preserved under arithmetic operations

I want to check the following statements: $\alpha+\beta$ is a limit ordinal if and only if $\beta$ is a limit ordinal. $\alpha\cdot\beta$ is a limit ordinal if and only if either $\alpha$ or $\beta$ ...
0
votes
0answers
19 views

Ordinal subtraction as set difference

Consider the following synthetic definitions for ordinal arithmetic: $\alpha + \beta$ is the order type of $\alpha\times\{0\}\cup\beta\times\{1\}$ ordered by reverse lexicographical ordering. ...
2
votes
1answer
22 views

Limit ordinal that cannot be written as $\alpha \cdot \omega$?

I learned from this thread that every limit ordinal can be written as $\omega\cdot\alpha$ for some $\alpha$ but is this also true for $\alpha\cdot\omega$ even though ordinal multiplication is not ...
0
votes
2answers
58 views

A complicated lemma from Munkres's Topology

I don't understand the following lemma of the book Topology by J Munkres : 1- If a set $A$ with the mentioned properties exists there must be an example for it and so, it could help to understand ...
1
vote
1answer
44 views

Necessary and sufficient conditions of a well-ordered set to be of order type $\omega$.

I want to find some necessary and sufficient conditions of a well-ordered set to be of order type $\omega$. To be specific, let $A$ be a well-ordered set, if each element of $A$ except the smallest ...
0
votes
0answers
54 views

Faster growing function than the fast growing heiarchy under Church-Kleene?

Is there a computable function that grows faster than any function in a fast growing hierarchy with index less than the Church–Kleene ordinal, where computable fundamental sequences are used? If the ...
2
votes
0answers
42 views

A question on ordinal arithmetic.

I have to order these two ordinals and I was just wondering if I have done it correctly. $\omega^\omega + \omega^3$ and $\omega + \omega^3 +\omega^\omega$ I have worked out that $\omega + \omega^3 ...
2
votes
1answer
31 views

Does the cofinality always induce a continuous function?

Let $\alpha$ be a limit ordinal and let $\lambda = cf(\alpha)$ be its cofinality. This means that there exists a strictly increasing $\lambda$-sequence $\langle \alpha_\xi \mid \xi < \lambda ...
1
vote
2answers
47 views

Order types of subsets of ordinals and transfinite induction

I am currently trying to practice the technique of transfinite induction with the following problem: Suppose that $X$ is a non-empty subset of an ordinal $\alpha$, so that $X$ is well-ordered by ...
2
votes
0answers
51 views

Trichotomy of Ordinals. Is $K$ a set?

We want to prove the "Trichotomy of Ordinals": Definiton: An ordinal is a transitive set with elements that are all transitive. Definiton: $\alpha$ and $\beta$ ordinals are comparable if one of the ...
2
votes
2answers
134 views

Power two of ordinal

if $\omega$ is ordinal number of set $\mathbb{N}$ can I compute $(\omega + 1)^2$ like below $\begin{align*} (\omega + 1)^2 & = \ (\omega + 1).(\omega + 1)\\ & = \ ((\omega + ...
1
vote
1answer
22 views

Cardinal exponentiation question of infinite cardinals.

I got a little confused with a question about cardinal exponentiation: Let $\beta$ be an ordinal, and let $K_\alpha$ , $\alpha < \beta$, be infinite cardinals with $K= ...
5
votes
1answer
65 views

Does forcing preserve the least undefinable ordinal from a model of ZFC?

Let $M$ be a transitive model of ZFC. For convenience let assume that $M$ is countable. Now let us consider the least undefinable ordinal $\vartheta_M$ which is not definable from elements in $M$ and ...
1
vote
2answers
79 views

Elementary Set Theory, cofinal subset, cofinality, ordinal, totally ordered set problem

Definition 1. Let $\langle u,<\rangle$ be totally ordered set, $v\subset u$. $v$ is cofinal subset of $u$ means that for all $a\in u$, there exist $b\in v$ ($a\le b$). Definition 2. Let ...
0
votes
1answer
60 views

Arithmetic of uncountable ordinal

Assume $\alpha$ is an ordinal such that $\alpha \geq \omega_1$. Is it true then that $\alpha = \omega + \alpha$ with respect to ordinal arithmetic?
9
votes
1answer
127 views

Commutative addition on the ordinals

It is well known that ordinal addition is not commutative (for example $\omega+1\neq 1+\omega$), but it is associative. My question regards a new kind of addition defined as: $$a\oplus b = ...
1
vote
1answer
50 views

Ordinal Fractions

Is any fraction $\,{x}\big/ {y}\,$ an ordinal number and if so, does ordinal $\,1 = \big\lbrace0,\dots,y - 1\big/y\big\rbrace\,$ instead of $\,\left\lbrace0\right\rbrace\,$? "If (X, <=) is a well ...
1
vote
1answer
31 views

Strictly increasing function from $\alpha$ to $\aleph_0$

Given that $\alpha < \aleph_1$, does there exists a function $f: \alpha \rightarrow \aleph_0$ that is a strictly increasing function? This seems like it would be true since $|\alpha| = \aleph_0$, ...
1
vote
0answers
36 views

Strictly increasing function from $\alpha< \aleph_1$ to $\mathbb{R}$ [duplicate]

I know that there is no increasing function $f: \aleph_1 \to \mathbb{R}$, so it seems like for $\alpha < \aleph_1$, there should exists a function $f: \alpha \rightarrow \mathbb{R}$ that is ...
0
votes
1answer
30 views

Limit ordinal in the exponent [duplicate]

How would you prove that $$n^{\gamma} = m^{\gamma}$$ for every limit ordinal $\gamma$ and $n,m$ finite ordinals? It's a rather short solution problem, but I can't construct any slick answer for it. ...
2
votes
1answer
49 views

Prove that $1^\alpha + 2^\alpha = 3^\alpha $ if $ \alpha $ is a limit ordinal

I am trying to prove the following statement: Suppose $ \alpha $ is a limit ordinal. Then $ 1 ^\alpha + 2^\alpha = 3^\alpha$. I'm not sure how to grasp this. Obviously induction won't work, since ...
6
votes
1answer
77 views

Topology on the set of ordinal numbers

This is a problem I encountered while reading Topology : An Outline for a First Course by Lewis E. Ward. Suppose $\Omega$ denotes the smallest ordinal number with uncountably many predecessors. Let ...
5
votes
4answers
359 views

How do I show there isn't an order isomorphism b/w the two sets $\{1, 2, 3,…\}$ and $\{1, 2, 3, …, \omega \}$

That is, how can I prove there isn't a bijection $f$ from one set to the other such that $f(x) < f(y)$ iff $x < y$?
2
votes
1answer
59 views

An ordinal number which satisfies $\omega^{\alpha} = \alpha$

Is there an ordinal such that $\omega^{\alpha} = \alpha$? It seems to me there should be, but I can't explicitly point it out. I know it is possible to prove $\alpha\leq\omega^{\alpha}$, but the ...
4
votes
1answer
53 views

Order type of the set of all the limit elements in $\omega^{\omega}$

What do you think is the order type of $\{\alpha\in\omega^{\omega}:\alpha\ \text{is a limit ordinal number}\}$? My first thought was $\omega$, but when I visualize the above mentioned set, ...
5
votes
0answers
74 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
2
votes
2answers
110 views

Can all ordinals be obtained by taking successors and countable limits?

Define a class $K$ of ordinals inductively as follows: $0=\emptyset\in K$. For all $\alpha\in K$, the succesor of $\alpha$ is also an element of $K$. For every function $f\colon \mathbb N\to K$, the ...
3
votes
1answer
40 views

Ordinal arithmetic inequality

I have 3 ordinals $\alpha,\beta,\gamma$, $\gamma\neq 0$. Is this implication true? $$\alpha<\beta\implies\gamma\cdot\alpha<\gamma\cdot\beta$$ I have reason to believie it is not, namely ...
4
votes
1answer
56 views

What's the least class of ordinals closed under successor and the limits of omega-sequences? [duplicate]

What is the smallest class $S$ of ordinals that contains $0$, closed under successor and the limits of omega-sequences, i.e., $$ \forall i \in \omega [\alpha_i \in S] \Rightarrow \sup_{i} \alpha_i \in ...
5
votes
1answer
52 views

Is noetherianity really about cardinals or ordinals?

If $\kappa$ is any cardinal, then one may define a "$\kappa$-Noetherian" ring as a ring such that for any module that has a generating set $S$ satisfying $|S|< \kappa$, then any submodule also has ...
4
votes
3answers
298 views

Indecomposable limit ordinals

A limit ordinal $\gamma>0$ is said to be indecomposable iff $\nexists\alpha,\beta<\gamma$ such that $\alpha+\beta=\gamma$. In view of this definition, I’m trying to prove the equivalence of the ...
2
votes
2answers
106 views

Two ways of defining rank of a set

I am studying set theory, and I have some difficulties in understanding how people define the notion of rank there (I hope, specialists in logic will excuse me for this). As far as I understand, ...
0
votes
1answer
39 views

Set theory proof for $\min C = \bigcap _{\alpha \in C} \alpha$

Let $C$ be a set of ordinals a) $\sup C$ is the least ordinal $\beta$ such that for all $\alpha\in C$, $a\leq \beta$. b) If $C$ is transitive (and thereby ordinal), then $\sup C \notin C$ if an only ...
2
votes
1answer
46 views

Elements of the power set of $\textit{On}$ are elements of $\textit{On}$?

I don't know the truth value of this statement so this is just my own attempt. I am attempting to prove this in NBG set theory. Let $X \in \textit{P(On)}$ By the definition provided in my notes $X ...
0
votes
1answer
27 views

Large ordered family of set inclusions complete?

I'm not an expert on set theory, so this might be trivial: Let $\dots \subseteq X_{\alpha} \subseteq X_{\alpha+1} \subseteq \dots$ be a chain of set inclusions, indexed by ordinals(!), s.t. ...
3
votes
1answer
29 views

Distributivity of ordinal arithmetic

Let greek letters be ordinals. I want to prove $\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma$ by induction on $\gamma$ and I already know it holds true for $\gamma = \emptyset$ and $\gamma$ a ...