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
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 ...
0
votes
0answers
36 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
2answers
32 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 ...
1
vote
0answers
40 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 ...
0
votes
2answers
34 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 ...
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 ...
3
votes
0answers
41 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{\...
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 ...
-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 ...
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 ...
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 $\alpha > \beta$?
-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 ...
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 )...
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 ...
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 ...
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
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?
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($\...
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}...
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_{...
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 ...
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(\...
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
40 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
726 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 ...
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 $[P\to P]$ ...
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 ...
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)&=...
0
votes
1answer
39 views

If $ \bigcup N_\alpha $ is stationary, then $ \{ \min(N_\alpha) \}$ is stationary

It's the last set-theoretic question for tonight, I promise. I'm trying to figure out why the following holds true: Suppose we have a regular, uncountable cardinal $ \kappa $ and a disjoint family ...
2
votes
2answers
26 views

Unboundedness of the set of subgroups of a cardinal

I'm trying to figure out why the following is true: Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ \...
2
votes
1answer
50 views

Checking whether some sets are clubs in $ \aleph_2 $ and $ \aleph_1 $

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question: which of the following sets are clubs, contain a club,...
0
votes
0answers
15 views

Order type of a sum ($\bigcup$) of sets

A quick question. Is $$\textrm{ot}(\bigcup\limits_{\gamma <\lambda}\alpha_{\gamma})=\bigcup\limits_{\gamma <\lambda}\textrm{ot}(\alpha_{\gamma})?$$ where $\textrm{ot}$ stands for the order type (...
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
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 $f$ ...
3
votes
2answers
100 views

How well-behaved can well-orders on $\mathbb{R}$ be?

Well-orders on $\mathbb{R}$ are interesting because they witness the "alephness" of the cardinality of the continuum, and they're therefore connected to the continuum problem. It seems reasonable, ...
0
votes
1answer
22 views

Fixed points of normal function form proper class

A normal function is a class function $f$ from the class of ordinals to itself such that $f$ is strictly increasing and continuous. We can easily show that for every ordinal $\alpha$ there is an ...
3
votes
1answer
71 views

Why does $\epsilon_0 = \omega^{\epsilon_0}$

I saw this on wikipedia and wasn't sure why. It seems obvious given the definition but how can we be sure? Also, what would be $\epsilon_0 \bullet \epsilon_0$? Would it be $\epsilon_0$? Thanks
1
vote
0answers
45 views

How many equivalence classes in a set of well-orders of a set

The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on ...
0
votes
2answers
51 views

Can limits to (positive) infinity be replaced by ordinals?

For example, does the following hold? $$\left(1+\frac1\omega\right)^\omega=e$$