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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22 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
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1answer
27 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. ...
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1answer
30 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 $\alpha > \beta$?
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3answers
99 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($\...
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1answer
65 views

Is ω really the first ordinal transfinity? [on hold]

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
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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
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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
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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
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2answers
54 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
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1answer
200 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 ...
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2answers
71 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
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1answer
67 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
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1answer
41 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 $f$ ...
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2answers
63 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
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1answer
81 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
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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 ...
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3answers
129 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?
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0answers
37 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
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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
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2answers
29 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
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2answers
127 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?
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1answer
57 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 ...
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1answer
35 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(\...
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1answer
252 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 ...
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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
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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
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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?
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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
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1answer
715 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 ...
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1answer
52 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 ...
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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
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1answer
28 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
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1answer
117 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 ...
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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 ...
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1answer
82 views

formal definition of ordinal addition by recursion

I'm reading Kunen's Set Theory, An Introduction to Independence Proofs (1980). On page 26 he explains how to introduce ordinal addition through recursion. For the sake of convenience i'll give the ...
0
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1answer
37 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
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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
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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,...
1
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1answer
120 views

Does a $\Pi_2^0$ sentence becomes equivalent to a $\Pi_1^0$ sentence after it has been proven?

I heard that the P vs NP question is equivalent to a $\Pi_2^0$ sentence, and that the Riemann hypothesis is equivalent to a $\Pi_1^0$ sentence. Many known mathematical theorems state that some ...
2
votes
1answer
112 views

Prove $\bigcup \omega_1 = \omega_1$

This is a question regarding ordinals and probably requires some background knowledge of ordinal arithmetic. I will list what I hope is the relevant information for this question: $\omega_1$ is ...
0
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0answers
14 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 (...
0
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1answer
31 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. $\...
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2answers
98 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, ...
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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
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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
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0answers
40 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 ...
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2answers
51 views

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

For example, does the following hold? $$\left(1+\frac1\omega\right)^\omega=e$$
1
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0answers
15 views

Ordinal-indexed exponentation

I have read from somewhere that $2^{\aleph_0}$ is uncountable. However, I have also read from somewhere else that $2^\omega=\omega$. Do the above statements contradict with each other? In ...
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0answers
60 views

$\epsilon_0$ is closed under addition, multiplication, exponentiation of ordinals

Question: Let $f: \Bbb{N}\rightarrow \text{Ord}$ (where "Ord" is the set of ordinals) be defined inductively by: $$f(0)=\omega\\ f(n^+)=\omega^{f(n)}$$ Let $\epsilon_0=\{\sup f(i) : i \in \Bbb{N}\}.$ ...
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3answers
53 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, ...