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
1answer
47 views

Could “ω is an ordinal” be proved without axiom of induction?

Let ω (or all natural numbers) be the set defined by the axiom of infinity in ZF system, ordinal be "transitive and well ordered by ∈", and a set is transitive if all its elements are also its subset. ...
2
votes
1answer
328 views

First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. ...
0
votes
0answers
14 views

Question about ordinal space [duplicate]

I have question regarding the ordinal space $[0,\Omega]$ where $\Omega$ the first uncountable number. I know it is not first countable and not separable. i am trying to prove it is compact. it is kind ...
2
votes
2answers
33 views

Series with terms indexed by transfinite ordinals

I learned this past year how to deal with summation of infinite series $a_0 + a_1 + a_2 + ...$ with indices running over the natural numbers. I've also learned about what comes "after" the natural ...
3
votes
0answers
43 views

$\varepsilon_\omega$ reachable using a Corless-Veblen $\varphi_1$?

$\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ve}{\varepsilon} \newcommand{\om}{\omega} \newcommand{\al}{\alpha} \newcommand{\ph}{\varphi} \DeclareMathOperator{\W}{W} \DeclareMathOperator{\...
1
vote
0answers
43 views

Partial order on the set of ordinal functions

Let $\kappa$ be a regular ordinal. Say that for $f,g: \kappa \rightarrow \kappa$, $f \leq g$ iff $f(\alpha) \leq g(\alpha)$ for sufficiently large $\alpha$. Now define $(X_{\kappa},\preceq)$ as the ...
1
vote
2answers
72 views

Theorem $($Transfinite Induction and Construction$)$ (James Dugundji - 5.1, Page - 40)

Let $W$ be a well ordered set, and let $Q \subset W$. If $[ W(x) \subset Q] \Rightarrow [ x \in Q]$ for each $x \in W$, then $Q = W$. For each $a \in W$, the set $W(a) = \{x \in W; (x\prec a) \...
0
votes
2answers
35 views

When is the union of a set of ordinals a limit ordinal?

Let $Y$ be a set of ordinals such that $\bigcup Y \not\in Y$. Then $\bigcup Y$ is a limit ordinal. Does this hold? It's taken from Drake's Set Theory, An Introduction to Large Cardinals, page 114, I ...
5
votes
3answers
212 views

An easy to understand definition of $\omega_1$?

I have two things I'm not sure in 100% about them. The first, is $\omega_1$. I have a little "feeling" of it, but if I'll be asked to define it - I don't know where to begin from. Perhaps it is ...
1
vote
4answers
147 views

Definition of Ordinals in Set Theory in Layman Terms

I've taken a huge interest in the mathematical concept of infinity and often been contemplating the same over years. But the fundamental concept of set theory ordinals continues to evade my ...
0
votes
1answer
33 views

Does this certain modular property hold for functions on recursive ordinals?

Let $f$ be a function with the following properties. 1) The domain and codomain of $f$ are the recursive ordinals. 2) $f$ is nondecreasing. 3) The set of fixed points of $f$ is unbound. ...
0
votes
1answer
43 views

Do all computable functions on ordinals satisfy this certain modular property?

Consider the following property of a non decreasing function $f$ whose domain and codomain are each the set of ordinals less than the Church-Kleene ordinal, and whose set of fixed points is unbound. ...
5
votes
2answers
50 views

Infinite Ordinal Sum

When working with ordinal numbers, would it be correct to say that: $$ \sum_{i=0}^{\infty}1 = \omega$$ Or does this simply not make sense? In the ordinals, does the notation $\sum^\infty_{i=0}$ even ...
0
votes
1answer
57 views

Need help in understanding some basic ordinal concepts

I'm trying to construct a proof that there is no homeomorphism from a subspace of $\omega_1$ to $\Bbb{Q}$ with the usual topology, but first I need to clear up some concepts. This is the definition ...
-1
votes
1answer
27 views

Closed disjoint sets in $\omega_1$ implies one of them is countable

Let $A, B \subseteq \omega_1$ be disjoint, closed sets. Show that one of A or B must be countable. Since $\omega_1$ is uncountable, at least one of A or B must be countable, I want to show that ...
3
votes
1answer
42 views

What are the clopen sets in $\omega_1$ as a topological space?

I am reading about $\omega_1$ https://dantopology.wordpress.com/2009/10/11/the-first-uncountable-ordinal/ I know that certain subsets in $\omega_1$ is both open and closed. For example, let $a = \...
0
votes
1answer
57 views

Incompressible countable Total-ordering implies well-ordering i

A totally-ordered set (S,<) is incompressible if $(S,<) \cong (T,<)$ and $S \supseteq T$ implies $S = T$. Is it true that if $S$ is incompressible countable and totally-ordered then $S$ is ...
-2
votes
1answer
46 views

pentation of uncountable ordinal

$\omega↑↑↑\omega = \Gamma_0$ Feferman's ordinal collapsing function: $\ θ(Г_Ω) = θ(Ω_2)$ $\ θ(Г_{Ω_2}) = θ(Ω_3)$ ...etc. This means: $ θ(Ω_2) = θ(Ω↑↑↑Ω) $ $ θ(Ω_3) = θ(Ω_2↑↑↑Ω_2) $ ...etc. ...
1
vote
1answer
33 views

Ordinal Numbers: is it possible to have an isomorphic copy of an ordinal $\alpha$ inside an ordinal $\beta < \alpha$?

Could there be a function $f:\alpha \to \beta$ order-preserving and injective such that $\alpha$ and $\beta$ are ordinal numbers with $\alpha > \beta$?
1
vote
3answers
105 views

Sum of finite ordinals: $\lambda_1+\lambda_2+\dots+\lambda_n+\dots=\omega$

Prove: $\lambda_1+\lambda_2+\dots+\lambda_n+\dots=\omega$, where $\lambda_i$ are finite ordinal nonzero numbers. I tried like this. $A_i$ set and ord($A_i$)$=\lambda_i, i=\{1,2,...\}$ and ord($\...
-4
votes
1answer
75 views

Is ω really the first ordinal transfinity? [closed]

The Internet claims that ω is the first ordinal transfinity. But what about ω-1? Isn't that a ordinal transfinity, and isn't it before ω? Kind of like ω/2? I guess I lack an understanding of what ω ...
4
votes
1answer
343 views

Epsilon numbers

Let $\alpha$ be an ordinal number and define $f_\alpha$ as: $f_\alpha(0) = \alpha + 1$ $f_\alpha(n+1) = \omega^{f_a(n)}$ Let $S(\alpha) = \sup\{f_a(n)\ |\ n \in \omega\}$ Then $S(\alpha)$ is an ...
18
votes
1answer
3k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
0
votes
1answer
41 views

Existence and Uniqueness of an ordinal

$\underline{\mathbf{Problem:}}$ If $\alpha < \beta $ then $\exists$ a unique $\gamma$ st. $\alpha+\gamma=\beta$, where '$+$' denotes ordinal addition. If someone would be so kind to check the ...
2
votes
2answers
55 views

A 'bad' definition for the cardinality of a set

My set theory notes state that the following is a 'bad' definition for the cardinality of a set $x:$ $|x|=\{y:y\approx x\}$ $(y\approx x\ \text{iff} \ \exists\ \text{a bijection}\ f:x\rightarrow y )...
2
votes
1answer
207 views

Uncountable compact sets in ordinal topology

The question I am interested in is the following: Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied with order topology. Prove ...
1
vote
2answers
73 views

$\aleph_1$ and $\omega_1$, what are they?

Sorry for my ignorant question but.. I understand that some sources says that $\aleph_1$ is the cardinality of the real numbers (ℝ) because In set theory $$\mathfrak{c} = 2^{\aleph_0} $$ and the ...
1
vote
1answer
68 views

Every transitive $\in$-linearly ordered set is $\in$-well ordered without axiom of foundation

I try to prove that every ordinal number $(\alpha,\in)$ is well-ordered, where an ordinal number is defined as a transitive $\in$-linearly ordered set. So all I have to show is, that every non-empty ...
0
votes
1answer
42 views

Ordinals recursed

Consider any limit ordinal $\alpha$. It seems intuitively obvious to me that for any such ordinal there is $f:ORD \to ORD$ (with $ORD$ being the class of ordinals) with $f(1)=\alpha$ and $f$ ...
1
vote
2answers
68 views

A specific example of transfinite induction

I am trying to understand better how transfinite induction can be applied in different concrete problems. Here is an example that seems relevant, but I am stuck on it. Consider a couple of points $p, ...
2
votes
1answer
82 views

Well-orderings of $\mathbb R$ without Choice

The question is about well-ordering $\mathbb R$ in ZF. Without the Axiom of Choice (AC) there exists a set that is not well-ordered. This could occur two ways: a) there are models of ZF in which $\...
0
votes
1answer
66 views

Ordinal Arithmetic: $n+\alpha=\alpha$

Let $\alpha$ be an ordinal. Show that $n+\alpha=\alpha$. Proof: Let $W$ be the set with ordinal equal to $\alpha$. Let $A_n=\{a_1,a_2,\cdots, a_n\}$ be the set disjoint from $W$ with its ordinal ...
0
votes
3answers
135 views

$(\omega +3)\cdot\omega=\omega\cdot\omega$ [duplicate]

Show that $(\omega +3)\cdot\omega=\omega\cdot\omega$. Is this just $(\omega +3)\cdot\omega=(\omega +\omega)\cdot\omega=\omega\cdot\omega$? Also, could someone suggest a good book for set theory?
0
votes
0answers
40 views

Any initial infinite ordinal is of the form $\omega_{\alpha}$

Let $\gamma$ be an initial infinite ordinal. Show that $\gamma=\omega_{\alpha}$ for some ordinal $\alpha$ The definition I use for $\omega_{\alpha}$ as follows: Suppose $\omega_{\beta},\aleph_{\beta}...
3
votes
1answer
33 views

In what sense $\alpha \times \alpha$ is the initial segment generated by $(0,\alpha)$ 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 $...
2
votes
2answers
30 views

Does there exist an increasing unbounded sequence in $\omega_{1}$?

I ask for some forgiveness, I am quite unfamiliar with actually working with ordinals. My question is: Does there exist a sequence $(\alpha_{k})_{k \in \mathbb{N}}$ such that $\alpha_{k} < \alpha_{...
7
votes
2answers
129 views

Is the limit of a recursive sequence of recursive ordinals itself a recursive ordinal?

Is the limit of a recursive sequence of recursive ordinals itself a recursive ordinal? If so, is there a nice proof of this?
0
votes
1answer
59 views

Is there a key difference between the principle of transfinite induction and the principle of transfinite induction for ordinals?

In a recent question I was asked to prove the principle of transfinite induction for ordinals but I mistakenly proved the principle of transfinite induction, since I have a only a vague understanding ...
1
vote
1answer
37 views

Show that there exists a fixed point for this (set theoretic) class function

I see that this question might be trivial but I can't seem to figure it out myself: Suppose that $F:ON\to ON$ is a class function: that is, for every ordinal $\alpha$ there is unique ordinal $F(\...
2
votes
1answer
255 views

Prove that ordinal multiplication is left distributive

Suppose $\alpha$, $\beta$ and $\gamma$ are ordinals. Prove the distributive law $\alpha \cdot ( \beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$. The following is my proof: Proof: We use ...
1
vote
1answer
41 views

Why can we choose a greatest ordinal $\beta$, such that $\omega^\beta\leq \alpha$?

I am reading a proof of Cantor's normal form theorem. In it, I read: for arbitrary $\alpha>0$ let $\beta $ be the greatest ordinal such that $\omega^\beta \leq \alpha$. Why should such an ...
0
votes
1answer
37 views

Some ordinal arithmetic exercises

I reckon it's easier if I don't open an extra thread for each of these small exercises. I'm trying to get a better grasp of ordinal arithmetic. If someone could please give me feedback on my solutions....
2
votes
1answer
19 views

Ordinal Arithmetic Identity

Let $\alpha$ be an Ordinal and $S$ a set of Ordinals. Is it the case that $$\alpha\bigcup\limits_{x\in S} x=\bigcup\limits_{x\in S} \alpha x$$ and if so how could one prove this?
0
votes
2answers
74 views

Does $1-\frac{1}{2^\omega}$ equal 1? What about $1-\frac{1}{2^{\aleph_0}}$?

(Correct if I'm wrong on any of this.) Recently, I've been learning about transfinite ordinals and cardinals. For some cases, I understand the difference between ordinals and cardinals, for instance ...
14
votes
1answer
727 views

Is it really impossible to lose the Hydra game?

In a number of blog posts I found the claim that the game described below cannot be lost, which is to say, every possible strategy is a winning strategy. In each case, a sketch proof is given that ...
0
votes
1answer
53 views

Does the MK axiom of infinity and the Power set axiom hold in the cumulative hierarchy?

Question: Define the cumulative hierarchy inductively as: $$V_0 = \emptyset$$ For each ordinal $\alpha$: $$V_{\alpha+1} = \mathcal{P}(V_\alpha)$$ ie the power set of $\alpha$ and for any nonzero ...
5
votes
4answers
525 views

Trying to understand the sum of ordinals

I'm having a hard time with the functions defined by transfinite induction, in particular I have the sum of two ordinals defined as follows: \begin{align}\alpha+0&=\alpha \\ \alpha+s(\beta)&=...
1
vote
1answer
30 views

Order-preserving maps on a directed-complete poset

Definitons. A poset $P$ is directed if every finite subset has an upper bound in $P$. It is directed-complete if every directed subset has a least upper bound in $P$. For any poset $P$ let $[P\to P]$ ...
3
votes
1answer
119 views

Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
0
votes
0answers
17 views

Ordinals defined by successor and union

Let $M$ be the smallest set of ordinals satisfying $0\in M$ $x\in M\implies x+1\in M$ $S\subseteq M\implies \bigcup S\in M$ What does $M$ look like? Is it all ordinals? Is it all countable ...