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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14
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2answers
488 views

A transfinite epistemic logic puzzle: what numbers did Cheryl give to Albert and Bernard?

I expect that nearly everyone here at stackexchange is by now familiar with Cheryl's birthday problem, which spawned many variant problems, including a transfinite version due to Timothy Gowers. In ...
3
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1answer
36 views

A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $\mathbb Q$, I find out that any countable limit ordinal is the union of a sequence of increasing ordinal. Now I'm ...
2
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1answer
23 views

Is the $\omega_1$'th element of a well ordered set the first with uncountable predecessors?

As far as I understand the topic, ordinal numbers are defined as sets, but their "purpose" is to generalize the concept of indices beyond infinity. So, can it be said that the element at index ...
0
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1answer
57 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. ...
7
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0answers
90 views

Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
4
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1answer
58 views

Transfinite Induction in Peano Arithmetic

I have heard that Peano Arithmetic (PA) cannot perform transfinite induction up to $\varepsilon_0$. This seems to imply that it can induct up to smaller ordinals, like $\omega$ or $\omega^\omega$ or ...
1
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1answer
27 views

Bijection between well-ordered set and ordinal

We are working in ZF. Problem (part 1). Suppose we have well-ordered set $(M, <)$. How to show that there exists a bijection $f$ between this set and some ordinal $x$ that preserves order? I can ...
1
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1answer
34 views

Construct an sequence in order set

Assume that we have an uncountable ordered set $\Theta$. Can we construct a sequence $(x_n)$ which $x_1<x_2<...<x_n<...$. In addition, for each $x\in \Theta$, there exists $n$ such that ...
0
votes
1answer
38 views

Help with the definition of ordinal sum

"Given two ordinals $\alpha$ and $\beta$, let $A = (\alpha$ x {0}) $\cup$ $(\beta$ x {1}). Then, define a well ordering on A by: $(v, i) <_{A} (\tau, j) \iff (i \lt j) \lor (i = j$ $\land$ $v ...
1
vote
1answer
20 views

Problem with base case for transfinite induction

I need to prove this using transfinite induction Let $\alpha, \beta , \gamma $ ordinals If $\beta <\gamma$ then $\alpha + \beta < \alpha + \gamma$ I am trying to prove the statement by ...
1
vote
1answer
39 views

Uncountable subset of first uncountable ordinal set

Given $g: \omega_1\rightarrow \omega_1$ is a function such that if $x\neq 0$, then $g(x)<x$ ($g$ is not necessarily continuous). Prove that there exists $t\in \omega_1$ such that $\ f^{-1}(t)$ is ...
2
votes
2answers
45 views

If $\alpha$ and $\beta$ are ordinals then $\alpha \in \beta \Leftrightarrow \alpha \subsetneq \beta$

Def 1. $x$ is $\underline{transitive}$ if $\forall y \forall z (z \in y \in x \Rightarrow z \in x)$. Def 2. $x$ is $\underline{ordinal}$ if $x$ is transitive and all elements of $x$ are transitive. ...
2
votes
1answer
75 views

Why are the countable ordinals a set?

The countable ordinals are themselves either countable or uncountable. They cannot be countable since that would involve a set with itself as an element, so they are uncountable. If they are ...
2
votes
1answer
32 views

Extending the Ordinals analogously to the Integers

I have been (recreationally) trying to expand the notion of ordinal numbers in the same way that the natural numbers $\mathbb N$ are extended to the integers $\mathbb Z$. My objective is to be able ...
3
votes
1answer
39 views

What is the least ordinal $\beta$ for which the function $f_\beta(n)$ in fast-growing hierarchy is incomputable?

Fast-growing hierarchy consists of a transfinite succession of faster growing functions $f_\alpha$: $f_0(n) := n+1$, $f_{\alpha+1}(n) := f^n_\alpha(n)$, $f_{\alpha}(n) := f_{\alpha[n]}(n)$ if ...
0
votes
1answer
16 views

An upper bound of a countable set of countable ordinals

Let $X$ be a countable set of countable ordinals. Is there any upper bound of this set under $\omega_1$? In other words, can we find an ordinal $\alpha<\omega_1$ which is larger than any elements ...
1
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0answers
37 views

A question regarding Silver's Theorem

I am reading Jech's book Introduction to Set Theory. As an introduction to Silver's theorem, it is stated that: If $\aleph_\alpha$ is a strong limit cardinal and if $k < \aleph_\alpha$ and ...
4
votes
1answer
37 views

How to prove $cf(\aleph_{\omega_1})=\omega_1$?

I am reading Jech's & Hrbacek's book "Introduction to set theory" and this is question 2.2 from chaper 9: Prove $cf(\aleph_{\omega_1})=\omega_1$ Isn't it clear from the fact that ...
0
votes
2answers
38 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) ...
1
vote
1answer
47 views

Do ordinal sets exist?

An ordinal is woset, where for all members $a$ in the set, $a$'s segment is $a$ itself. But, the definition of $a$'s segment is: $X_a=\{x\in X:x \subsetneq a\}$ But, given that definition, how can ...
0
votes
1answer
14 views

Given stationary set $A$ of $k$, there exists an ordinals $\alpha$ such that $A$ contains all limit ordinals greater then $\alpha$

Is it true that given stationary set $A$ of $k$, there exists an ordinal $\alpha < k$ such that $A$ contains all limit ordinals of $k$ greater then $\alpha$? I think it is true because the set of ...
1
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1answer
38 views

What is wrong with the following function $f:ON \rightarrow ON$?

Define the following function on ON (The class of ordinals): $f(0)=0$ and for every $\alpha >0$, if $\alpha$ is a successor then $f(\alpha) = \alpha -1$ and if $\alpha$ is limit then, $f(\alpha) = ...
2
votes
1answer
24 views

If $f\colon\kappa\rightarrow\kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded.

If $f\colon\kappa\rightarrow \kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded. This is from Jech's book (page 103) so ...
1
vote
0answers
28 views

Ordinals that are not cardinals [duplicate]

I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are ...
0
votes
1answer
32 views

A question about well-ordered subsets of partially ordered sets

Let $S$ be an infinite partially ordered set. Is there a name for the smallest ordinal number which is not ordinally similar to any subset of any chain of $S$? Assume that we are working in ZF. If the ...
3
votes
1answer
42 views

Does every countable sequence of ordinals converge?

Let $A=\{ \alpha_n | n \in \omega \}$ a monotonic increasing sequence. Can we say that there exists an $\alpha \in ON$ such that $\alpha = \lim_{n \in \omega} \alpha_n$? i am asking this because I am ...
4
votes
1answer
40 views

How do we know that $\omega_1$ exists in ZF? [duplicate]

In ZF, not all "collections of objects" are sets. For example, there is no set of all sets, and there is no set of all ordinals. So, how do we know that there is a set of all countable ordinals? In ...
1
vote
1answer
16 views

Lexicographic order on $\alpha^\beta$ is well-ordered

Suppose that $\alpha$ and $\beta$ are ordinals, and define the following order on $\alpha^\beta$, the set of all functions $\beta\to\alpha$ with finite support: $$f\,R\,g\iff\exists ...
4
votes
2answers
35 views

Is $f: \mathcal{P}(\omega) \to \omega \cup \lbrace \omega \rbrace, \ x \mapsto \bigcup \lbrace n \in \omega: n \subset x \rbrace $ surjective?

Intro: I am currently preparing for an exam and I have found this particular question in a previous exam of the same class. It is a 'simple' question where you only have to write the answer without ...
1
vote
1answer
53 views

finite sequences,linear order, lexicographic order, well order, cardinal

Let $\lambda$ be an infinite cardinal.Consider the lexicographic order on $\lambda^{< \omega}$. Why this order is not a well-order: how may an infinite long descending chain look like? Why every ...
0
votes
2answers
38 views

Is there a strict superset of the reals with total order?

The ordinals immediately come to mind, but I am mainly interested if there are new elements bounded between two real numbers $[a,b]\subseteq\mathbb{R}$ that can be added to the reals while preserving ...
2
votes
1answer
77 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
0answers
56 views

Compact, sequential spaces

A compact, Hausdorff space $X$ is sequential if each for each $A\subset X$ and $x\in \overline{A}$, there exists a countable set $A_0\subset A$ such that $x\in \overline{A}_0$. I am asked to show ...
1
vote
0answers
13 views

Exponentiation on order types

How is exponentiation defined on order types? We know that $2^\omega=\omega$. What is $2^{\omega^*}$? Is it $\omega^*$? $\eta$? $\lambda$? I'm guessing $\eta$, but I'm not sure. $\omega$ is the ...
0
votes
0answers
14 views

Can the direction of failing distributivity on ordinals be turned into an inequality?

Is it true that for all ordinals $\alpha_1, \alpha_2, \beta$: $$(\alpha_1 + \alpha_2)\cdot\beta \leq \alpha_1\cdot\beta+\alpha_2\cdot\beta\;?$$
0
votes
0answers
20 views

If $x,y$ are ordinals and $x \cong y$ then $x = y$

If $x,y$ are ordinals and $x \cong y$ then $x = y$ The proof goes as follows; Suppose $f$ is an isomorphism between $x$ and $y$ then show $f$ is the identity map. Assume that $f$ is not the ...
1
vote
1answer
70 views

Is this the real definition of an ordinal number?

I have to prove that if $\alpha$ is ordinal and $\beta \in \alpha$ then $\beta$ is oridinal. To do this I wanted to prove that for every $x\in\beta$, $x\in\alpha$. This seems like it should be true, ...
1
vote
1answer
21 views

Show $\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$

$\alpha$ is a limit ordinal $\leftrightarrow \alpha \neq 0$ and $\cup \alpha = \alpha$ Sorry if this question has been asked already but I couldn't find it on this site. I assume by definition ...
1
vote
1answer
34 views

Prove $\alpha+\beta=\beta$ if $\alpha < \beta$

$\alpha$ and $\beta$ are ordinal numbers and $\beta$ is limit ordinal. I assume the proof is relatively simple because it's been omitted in my textbook. Could someone also provide an intuitive way to ...
1
vote
1answer
26 views

Difference between a criteria of well-ordered and transitive in terms of an ordinal?

From what I gather a set $x$ is $transitive$ if whenever $y \in z , z \in x \rightarrow y \in x$ And one of the properties of a well-ordered set is that $x \in y , y \in z \rightarrow x \in z$ Now ...
5
votes
4answers
277 views

Uses of ordinals

Are there any interesting theorems outside of set theory that use ordinals in their proofs? The only example I know of is Goodstein's theorem, and I haven't been able to find anything else. In other ...
2
votes
1answer
51 views

Order preserving injection from first uncountable ordinal to the real numbers

Can there be an ordering preserving (1-1) map $f \colon \langle\aleph_1,\in\rangle \rightarrow \langle\mathbb R, \lt\rangle$? I remember that if $f$ is order preserving then consider the following ...
3
votes
1answer
52 views

if $\alpha\leq\beta$ then there is $\gamma$ such as $\alpha+\gamma=\beta$

I'm trying to prove the statement in the title, a.e existence of ordinal substraction. I think it can be done with transfinite induction, but do any of you have a better/easier way? thanks
1
vote
1answer
45 views

Epsilon number cannot be a successor?

Let $\omega$ be the canonical representation of the natural numbers. A epsilon number $\alpha$ is a ordinal number such that $\alpha=\omega^\alpha$ holds. I suspect that a epsilon number cannot be a ...
4
votes
1answer
69 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 ...
2
votes
2answers
67 views

Subsets of the rationals order isomorphic to transfinite ordinals

Let $\langle\mathbb Q,\lt\rangle$ be the set of rational numbers with the usual $\lt$ ordering. Find subsets $A$ and $B$ of $\mathbb Q$ which, with the usual ordering, are order isomorphic to the ...
3
votes
0answers
28 views

For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
0
votes
1answer
32 views

Proof of the Principle of Transfinite Induction for On

Principle of Transfinite Induction for On: If $C\neq\varnothing, C\subseteq$ On then $\exists\alpha\in C\forall\beta\in C[\alpha\in\beta\vee\alpha=\beta]$ Proof: (1) As $C\neq\varnothing$, let ...
0
votes
1answer
57 views

Take $S$ to be a set of ordinals, show $\cup S=\sup(S)$

Ok so firstly I can see how this makes sense. If you have say: $S=\{0,1,2\}$ with $0 = \emptyset, 1 = \{0\} = \{\emptyset\}, 2 = \{0,1\} = \{\emptyset,\{\emptyset\}\}$. $$\cup S = ...
0
votes
1answer
44 views

Can set containing not only all ordinal numbers exist?

I've always seen the same proof of non-existence of set of all ordianal numbers. It was based on a fact that such a set would fit into definition of ordinal number, and the set itself would be an ...