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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Embedding ordinals in $\mathbb{Q}$

All countable ordinals are embeddable in $\mathbb{Q}$. For "small" countable ordinals, it is simple to do this explicitly. $\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in \...
2
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2answers
1k views

Do $\omega^\omega=2^{\aleph_0}=\aleph_1$?

As we know, $2^{\aleph_0}$ is a cardinal number, so it is a limit ordinal number. However, it must not be $2^\omega$, since $2^\omega=\sup\{2^\alpha|\alpha<\omega\}=\omega=\aleph_0<2^{\aleph_0}$,...
4
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1answer
239 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...
7
votes
3answers
1k views

How is $\epsilon_0$ countable?

In Wikipedia, it says that any epsilon number with the index that is countable is countable. How is it? Out of all those numbers, I especially want to know why $\epsilon_0$ is countable. Thanks.
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3answers
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Is the class of cardinals totally ordered?

In a Wikipedia article http://en.wikipedia.org/wiki/Aleph_number#Aleph-one I encountered the following sentence: "If the axiom of choice (AC) is used, it can be proved that the class of cardinal ...
6
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3answers
569 views

Countable ordinals are embeddable in the rationals $\Bbb Q$ — proofs and their use of AC

Yesterday, Asaf Karagila's answer to my question sparked an extensive discussion on ways of proving that all countable ordinals are embeddable in $\Bbb Q$, and whether particular solutions to this use ...
19
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4answers
834 views

Intuition for $\omega^\omega$

I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
10
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1answer
830 views

Uncountable ordinals without power set axiom

Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a ...
5
votes
3answers
529 views

Which set is unwell-orderable?

In textbook it says that every well-orderable set is equipotent to an initial ordinal number. However, of course unwell-orderable cannot equipotent to any ordinal number, but is there any set is ...
15
votes
3answers
2k views

Countable compact spaces as ordinals

I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can ...
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 ...
14
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3answers
460 views

How many positive numbers need to be added together to ensure that the sum is infinite?

The question in the title is naively stated, so let be make it more precise: Let $\sum_{n\in\alpha}a_n$ be an ordinal-indexed sequence of real numbers such that $a_n>0$ for each $n\in\alpha$, where ...
9
votes
3answers
957 views

Uncountability of countable ordinals

According to Wikipedia, there are uncountably many countable ordinals. What is the easiest way to see this? If I construct ordinals in the standard way, $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega +...
6
votes
2answers
534 views

How to think about ordinal exponentiation?

I'm just trying to understand better how to see $\alpha^{\beta}$ for an arbitrary ordinal. I've already know that one can think about $\alpha . \beta$ as $\langle \alpha \times \beta, AntiLex\rangle$ ...
1
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2answers
163 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 ?
14
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1answer
410 views

How do you find the smallest of homeomorphic ordinals?

I am trying to get a better feel for the topology of ordinals and just received a great answer to this question where the Cantor-Bendixson rank and degree turn out to be a complete homeomorphism ...
7
votes
1answer
284 views

what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?

John Conway proved in his book, On Numbers and Games (ch6, theorem 49) that the set of all ordinals smaller than $\omega^{\omega^\omega}$ form a field of characteristic 2 that is isomorphic to $\bar{\...
12
votes
4answers
814 views

I want to know why $\omega \neq \omega+1$.

In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$. I want to know why $\omega \neq \omega+1$.
9
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2answers
290 views

$\varepsilon$-number countability without choice

Let $\alpha\mapsto\varepsilon_\alpha$ be the enumeration of the $\varepsilon$-numbers--that is, those $\alpha$ such that $\omega^\alpha=\alpha$--by the ordinals. If we know that countable unions of ...
3
votes
2answers
335 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
2
votes
3answers
697 views

show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant

show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant.Where $S_{\mathbb{\Omega}}$ denote the first uncountable ordinal. There is a hint that for each ...
2
votes
1answer
100 views

Directed limits of topological spaces and embeddings

Let $(\{X_\alpha\}_\alpha,\{f_{\alpha\beta}:X_\alpha\rightarrow X_\beta\}_{\alpha\preceq\beta})$ be a directed system of topological spaces and $(X,\{g_\alpha:X_\alpha\rightarrow X\}_\alpha)$ its ...
2
votes
2answers
133 views

For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

I don't understand the proof to the a/m claim. How we know that $\eta < \kappa$ and $(\alpha,\beta)<_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone ...
1
vote
1answer
129 views

Prove that $\omega + \omega_1 = \omega \cdot \omega_1 = \omega^{\omega_1} = \omega_1$

I am assuming already that a) the union of countably many countable sets is countable and b) $\omega_1$ is the least uncountable ordinal, so $x < \omega_1$ if and only if $x$ is a countable ...
21
votes
3answers
784 views

Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have $\mathbf{F}(\...
10
votes
2answers
402 views

How many ordinals can we cram into $\mathbb{R}_+$, respecting order?

I've been pondering the following question. How can we measure the amount of "space" above an element $p$ in a partially ordered set $P$? One way would be to try to cram the elements of ...
9
votes
0answers
310 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& \boldsymbol{+}&...
11
votes
4answers
3k views

Examples of transfinite induction

I know what transfinite induction is, but not sure how it is used to prove something. Can anyone show how transfinite induction is used to prove something? A simple case is OK.
9
votes
0answers
141 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
9
votes
2answers
1k views

Cardinal Arithmetic versus Ordinal Arithmetic

I am reading Philosophy, not Set Theory, so please excuse the naivety of my question. My question concerns the wildly different character of ordinal arithmetic versus cardinal arithmetic. The "...
9
votes
0answers
145 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
4
votes
2answers
623 views

Why does every countable limit ordinal have cofinality $\omega$?

According to Wikipedia, if $\alpha$ is a countable limit ordinal, then $\mathrm{cf}(\alpha)=\omega$. It is intuitively clear to me that it should be so. Certainly the cofinality of such an ordinal ...
2
votes
1answer
221 views

A number system

Can we have a number system $S$ of cardinality continuum such that for every $x \in S$, there is a unique $y \in S$, such that for all $z>x$ in S, $x<y\le z$ holds?
8
votes
3answers
633 views

Ordinal exponentiation - $2^{\omega}=\omega$

This is my understanding of ordinal arithmetic - two ordinals are the same as one another if there is an order-preserving bijection between them. So for instance $$1+\omega = \omega$$ because if $...
7
votes
2answers
360 views

Is $V$ under ZFC really a proper class?

Is $V$, the union of the von Neumann hierarchy, necessarily a proper class? Or is it only a proper class after you assume that it contains every set? (In that case, $V$ can't be a "set of all sets" ...
6
votes
2answers
614 views

How to define countability of $\omega^{\omega}$ and $\omega_1$? in set theory?

How is the ordinal $\omega_1$ defined? I know that it is a supremum of all smaller ordinals, but then $\omega^\omega$ is also a supremum of all smaller ordinals. How can we distinguish these two ...
5
votes
1answer
355 views

Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals.

I don't know if this question is appropriated for this site. Anyway, I'm searching for an isomorphism of order $f:K \longrightarrow \epsilon_o $, such that $(K, \leq)$ is a subset(proper or not) of $(\...
5
votes
1answer
1k views

Union of ordinals

From Jech (pg. 20): If $X$ is a nonempty set of Ordinals, than $\bigcup X$ is an ordinal. This should be easy enough to prove but I don't see how; also, I guess this is not the case if $X$ is a ...
4
votes
2answers
141 views

What is the cardinality of the open ordinal space $[0,\Omega)$ if we remove neighborhoods of each limit ordinal?

Let $\Omega$ be the first uncountable ordinal with the order topology. For each limit ordinal $\lambda < \Omega$ let $U_\lambda$ be an open neighborhood of $\lambda$. What is the cardinality of $[0,...
4
votes
3answers
323 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 ...
4
votes
1answer
271 views

What is the smallest possible value of $\omega_1$ in $\mathrm{ZF}$?

It is consistent with $\mathrm{ZF}$ that a countable union of countable sets may be uncountable. As far as I understand it, this is because in absence of $\mathrm{AC}$ we cannot necessarily choose a ...
1
vote
1answer
138 views

Cardinal numbers

Suppose $m, n$ are infinite ordinal numbers. $$a) m=n → |m|=|n|$$ $$b)|m|=|n| →m=n$$ $$c)m<n→ |m|<|n|$$ $$d)|\max{(m,n)}|< |m|+|n|$$ $$e)|m|<|n| →|m|^{|n|}<|n|^m$$ Which of the above ...
0
votes
2answers
555 views

How far do known ordinal notations span?

What is the largest known ordinal number $\alpha$ such that a uniform notation scheme has been developed for all ordinals up to $\alpha$ (there should be no "gaps" in what ordinals are representable), ...
8
votes
1answer
182 views

Do there exist totally ordered sets with the 'distinct order type' property that are not well-ordered?

Define that the order type of an element $x$ in a totally ordered set $X$ is the order type of $\{w \in X\mid w < x\}$. Under this definition, distinct elements of a well-ordered set have distinct ...
7
votes
1answer
180 views

Sequence of surjections imply choice

I am reading a paper where a side remark said that if a sequence $\langle g_\beta\colon\omega\to\beta\mid\beta<\omega_1\rangle$ is a sequence of surjections then $\omega_1$ is regular. I have ...
6
votes
2answers
385 views

Operations on ordinal numbers

Let $w$ be an ordinal for a denumerable set. Prove that$(w+w)w=ww$ Let A and B be sets. Let A be ordered by G and B by H. Let f be an isomorphism such that x≦y in G implies f(x)≦f(y) in H. Now, ...
5
votes
1answer
230 views

How do we justify functions on the Ordinals

In ZFC there seem to be two ways to define a function, as an ordered triple of domain, codomain and the set of ordered pairs that make its graph, or a relation over the two sets. Whichever we use ...
4
votes
1answer
101 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...
4
votes
2answers
497 views

Order-isomorphic with a subset iff order-isomorphic with an initial segment

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of ...
2
votes
2answers
113 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 ...