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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2
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2answers
599 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 ...
5
votes
3answers
600 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.
8
votes
3answers
752 views

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 ...
18
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6answers
535 views

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 ...
7
votes
1answer
611 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 ...
8
votes
3answers
582 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 ...
5
votes
2answers
248 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 $\langle \alpha \times \beta, AntiLex\rangle$ ...
17
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4answers
484 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 ...
9
votes
2answers
251 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 ...
4
votes
3answers
221 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 ...
13
votes
3answers
350 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 ...
8
votes
1answer
134 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 ...
2
votes
1answer
101 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 ...
1
vote
1answer
76 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 ...
12
votes
1answer
2k 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 ...
13
votes
1answer
336 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 ...
8
votes
2answers
239 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 ...
6
votes
0answers
215 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& ...
9
votes
1answer
522 views

Surreal numbers without the axiom of infinity

Let $ZF^\times$ denote the set of axioms of Zermelo-Fraenkel set theory without the axiom of infinity. The set $V_\omega$ of all hereditarily finite sets is a model of $ZF^\times$, and ...
7
votes
2answers
303 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" ...
4
votes
1answer
281 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 ...
4
votes
1answer
257 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 ...
2
votes
1answer
173 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?
1
vote
1answer
117 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 ...
6
votes
1answer
75 views

Are there any nontrivial doubly-well-ordered sets?

A set $A$ is well-ordered (by an ordering $R$) if there is an $R$-minimal element in every nonempty subset of $A$. Call $A$ doubly well-ordered (by $R$) if $R$ well-orders $A$ and $R^c$ (converse) ...
6
votes
2answers
313 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, ...
6
votes
1answer
158 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 ...
5
votes
1answer
186 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
2answers
156 views

Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
3
votes
2answers
150 views

Two exercises on set theory about Cantor's equation and the von Neumann hierarchy

Good evening to all. I have two exercises I tried to resolve without a rigorous success: Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where ...
3
votes
3answers
300 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 ...
2
votes
2answers
95 views

What is the cardinality of the open ordinal space $[0,\Omega)$ if we remove it's limit ordinals?

Let $\Omega$ be the first uncountable ordinal. and for any limit ordinal $\lambda < \Omega$ Let $U_\lambda$ an open neighborhood of $\lambda$ with the order topology. what is the cardinality of the ...
1
vote
1answer
202 views

Mechanical definition of ordinals

It seems that one can construct ordinals from bottom up by successively introducing a new symbol each time a limit is taken: $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega +2,\ \ldots,\ \omega\cdot ...
0
votes
2answers
374 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), ...
14
votes
1answer
155 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
10
votes
1answer
347 views

Linearly ordered sets “somewhat similar” to $\mathbb{Q}$

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{\eta} + \pmb{1}$ - order type of $\mathbb{Q}\cap(0, 1]$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. Let's say that a linear ...
19
votes
3answers
613 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 ...
16
votes
2answers
329 views

Is $(\pmb{1} + \pmb{\eta})\cdot\pmb{\omega_1} = \pmb{1} + \pmb{\eta}\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
12
votes
3answers
905 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 ...
7
votes
3answers
229 views

How to prove the Milner-Rado Paradox?

For every ordinal $\alpha<\kappa^+$ there are sets $X_n\subset\alpha$ $(n\in\Bbb{N})$ such that $\alpha=\bigcup_n X_n$ and for each $n$ the order-type of $X_n$ is $\le\kappa^n$. [By ...
6
votes
1answer
221 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 ...
5
votes
2answers
269 views

Can $\omega_1$ be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?

Can $\omega_1$ (the first uncountable ordinal) be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?
10
votes
1answer
167 views

Is $\pmb{\eta}\cdot\pmb{\omega_1} = (\pmb{\eta} + \pmb{1})\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
4
votes
4answers
229 views

Does every ordinal have cardinality no greater than $\aleph_\mathbb{0}$?

My notes say that the ordinals $\omega + 1, \omega + 2, ... , 2 \omega, ... , 3 \omega, ... \omega^2, ... $ are all countable, and hence have cardinality equal to $\omega = \aleph_\mathbb{0}$. So I ...
10
votes
4answers
708 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 $\omega \neq \omega+1$.
6
votes
1answer
199 views

Ordinal arithmetic exercise in Kunen

I'm self studying Kunen (2013 edition) right now, and got stuck on this exercise - I generally lack the tools and/or intuition for these kind of ordinal arithmetic exercises. Exercise. Let ...
3
votes
2answers
129 views

The concept of ordinals

I am trying to understand this concept and have some difficulties. For example, can I say that $\alpha$ is the cardinality of $\{1,2,3,...\}=\Bbb N$? And if so, what is $\alpha +1$? I guess it is ...
3
votes
2answers
196 views

Is there a categorification of (infinite) ordinal arithmetic?

Background for the curious reader: An ordinal $\beta$ is a transitive set in the sense that $\alpha\in\beta$ implies $\alpha\subset\beta$. Any ordinal is naturally well-ordered under $\in$ (so any ...
7
votes
3answers
241 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 ...
6
votes
2answers
295 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 ...