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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7
votes
2answers
229 views

Set of maximal chains in $\langle\mathcal{P}(\mathbb{N}),\subseteq\rangle$

Let $S$ be the set of all maximal chains in the poset $\langle$$\mathcal{P}$$($$\mathbb{N}$$),$$\subseteq$$\rangle$ partitioned into equivalence classes by their order types. How many different order ...
2
votes
2answers
1k views

What does order topology over Ordinal numbers look like, and how does it work?

Space of ordinal numbers are one of the favorite examples of my professor in general topology. I quite understand the idea at the base of ordinal numbers (few things or nothing about the concept of ...
6
votes
1answer
157 views

Admissible ordinals…

a little question about admissible sets: Is every $\mathfrak{M}$-admissible ordinals an admissible ordinal ? where $\mathfrak{M}$ is a $L$-structure over $L=\{R_1,\dots,R_k \}$. Thanks.
7
votes
1answer
272 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
1answer
332 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
117 views

Question on a transfinite construction in algebra

I am studying out of Matsumura's Commutative Ring Theory, and in the first section on modules he proves (following Kaplansky) that every projective module over a local ring is free. My questions have ...
4
votes
0answers
121 views

Transfinite induction and valuation rank

In Engler's valued fields, exercise 3.5.2 goes as follows Construct valuations on $\Bbb{C}$ of rank $\kappa$ for every cardinal $\kappa \leq 2^{\aleph_0}$. The idea behind this (for any ...
2
votes
2answers
176 views

$\wp^{(\omega)}(A)=?$

Let $A$ be a set, $$\wp^{(0)}(A)=A$$ $$\wp^{(n+1)}(A)=\wp(\wp^{(n)}(A))$$ But what sense does $\wp^{(\alpha)}(A)$ make where $\alpha$ is a limit ordinal number? The most natural way is let ...
5
votes
1answer
89 views

Is there a ring with the lattice of ideals isomorphic to $(\omega+1)^{\operatorname{op}}?$

In this question, I gave an example of a ring whose lattice of two-sided ideals is order-isomorphic to $\omega+1$. I've been playing a bit with trying to find rings with a given lattice of ideals ...
1
vote
4answers
280 views

What's the definition of limit of sets(esp. ordinals) in set theory?

By definition of exponent operator on ordinals, we have $$0^\omega=\lim_{\xi\to\omega}0^\xi$$ However, Note that $0^\xi$ is not increasing, so if we still let ...
2
votes
1answer
210 views

Can we get uncountable ordinal numbers through constructive method?

As we know, $2^{\aleph_0}$ is a limit ordinal number, however, it is greater than $\omega$, $\omega+\omega$, $\omega \cdot \omega$, $\omega^\omega$, $\omega\uparrow\uparrow\omega$, and even $\omega ...
2
votes
2answers
963 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 ...
1
vote
1answer
145 views

Cofinal subset of ordinals

Let $\alpha$ be an ordinal and suppose this ordinal has a cofinal subset. Let the cofinality of $\alpha$ be $\beta$. Thus $\beta$ is isomorphic with $B$, cofinal subset of $\alpha$. Then $\beta$ is a ...
5
votes
1answer
119 views

Finite ordinal to the power a = a

Let $\alpha>\omega$ be an ordinal such that $2^\alpha$ = $\alpha$. Then $\alpha$ is an epsilon number? I have tried many different ways, but i can only work with the left side of $\alpha$(e.g, I ...
4
votes
1answer
317 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 ...
2
votes
1answer
70 views

Nonlimit ordinal exponentiation

I know how to compute finite exponentiation limit ordinal expressed in normal form. However, how do i compute for nonlimit ordinal? I know this statement is true. If $r$ is a limit ordinal and $b$ is ...
0
votes
2answers
44 views

Is the class of coefficients of a normal form of an ordinal denumerable?

By Cantor's normal form theorem, any ordinal $r$ can be expressed as: $r=\omega^{k_1}a_1 + \omega^{k_2}a_2 + \ldots$. ($k_1>k_2>\ldots$) I want to know whether the class of all $a_i$'s is ...
0
votes
1answer
42 views

Relation between two distinct class of ordinals

Class of ordinals is defined as follows: If $A$ is 'any' well-ordered set, there exists an element $a$ in the class such that $A$ is similar to $a$. If $A$ is a well-ordered set and $a$,$b$ are ...
0
votes
2answers
200 views

Membership-well-ordering and class of ordinal numbers

I learnt how to construct a well ordered class of oridinals, ORD which satisfies 2 conditions. 1. If A is any well ordered set, there exists $a$$\in$ORD such that A is similar to $a$. 2. If A is a ...
2
votes
1answer
162 views

Ordinal exponentiation of limit ordinal

Prove that If $a$ is a limit ordinal and $p$ and $q$ are finite ordinals, then $(ap)^q$ = $a^q$$p^q$. In my book, the right side of the equality is written as $a^qp$, but i think its typo since ...
1
vote
1answer
231 views

Is there a general explicit way to construct bijections between countable ordinals?

Is there a general - and possibly easy! - explicit way to "create" bijective functions between countable ordinals? I mean for example a bijection between $\omega$ and $\omega+\omega$, or a bijection ...
0
votes
2answers
233 views

Ordinal Exponentiation and transfinite induction

Let $a\ne0$ be any ordinal By transfinite recursion theorem, we can define a function f:OR→OR such that i) $f(0)=1$ ii) $f(b^+)=f(b)a$ iii) $f(b)=\sup\{f(r)|r<b\}$ if $b$ is a limit ordinal. Then ...
0
votes
2answers
126 views

Real line, field of real numbers and $\omega_1$ topological space difference

Real line is separable. Then, why is $\omega_1$ topological space not separable? IF this is true, doesn't this settle continuum hypothesis? Also, does the field of real numbers have anything to do ...
5
votes
2answers
77 views

Mappings from $\omega_1$

We've been studying various properties of $\omega_1$ (equipped with the order topology), and I recently came across these questions. Can anyone help? If $f$ maps $\omega_1$ onto a metric space $X$ ...
5
votes
2answers
286 views

Constructing the reals from fractions of ordinals

We can construct the positive rationals from ratios of positive integers (and thus from pairs of finite ordinals). Can we analogously construct the reals from pairs of countable ordinals?
2
votes
1answer
81 views

Can we prove that some non-cardinal ordinals are regular?

Usually the notion of cofinality is used for cardinals, though there seems to be no problem defining it for non-cardinal limit ordinals also: the cofinality of a limit $\alpha$ is the least $\gamma$ ...
6
votes
2answers
377 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, ...
0
votes
1answer
361 views

the sup of a set of ordinals

Let $A$ a set of ordinals. We know that $\sup A:=\bigcup A$ is an ordinal. Frequently, in proofs, one use that it is a limit ordinal. I would want to know when it is. To show that it is limit : let ...
8
votes
1answer
273 views

Definable order types without infinity axiom.

Denote by $ZF^\times$ the theory of $ZF$ without the axiom of infinity. We know that $V_\omega$, the set of all hereditarily finite sets in a model of $ZF$, is a model of $ZF^\times$. We further know ...
4
votes
2answers
89 views

Well-ordered set, $\langle\mathbb{N},<^{(n)}\rangle$.

I want to prove the following statement: In well-ordered set $\langle\mathbb{N},<\rangle$, moving $0,1,2,3,...,n-1$ to the end, retaining that order, results in a well-ordered set $\langle \Bbb ...
3
votes
1answer
140 views

What is this function on ordinals called?

Define a function $f:\mathbf{ON}\to\mathbf{ON}$ as follows: for each ordinal number $\alpha$ let $$f(\alpha)=\operatorname{ord(}\lbrace \beta<\alpha|\text{ }\beta\textrm{ is a limit ...
4
votes
1answer
263 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 ...
5
votes
1answer
222 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 ...
0
votes
2answers
230 views

Transitive subset of set of natural numbers

Let $A$ be a nonempty subset of $\omega$, the set of natural numbers. I want to prove this statement: If $\bigcup A=A$ then $n\in A \implies n^+\in A$. Help...
5
votes
1answer
900 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 ...
20
votes
3answers
742 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 ...
22
votes
6answers
826 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 ...
2
votes
3answers
300 views

A sum of ordinals and a bijection

Let $a=a_{i\in n}$ is an ordinal-indexed family of ordinals. I call a structured sum of $a$ an order isomorphism from the lexicographically ordered set $\{ \left( i ; x \right) \; | \; i \in n, x\in ...
3
votes
1answer
157 views

countable product of $\omega_1 + 1$

How can I prove that the topological countable product of $\omega_1 + 1$ (with the order topology) is the union of $\omega_1$ closed nowhere dense sets?
-3
votes
1answer
136 views

Projections of ordinal numbers [closed]

Let $p$ is a product of two non-zero ordinal numbers. Knowing $p$ we can restore each of these two ordinals, right? How can we denote each of these two ordinals (supposing we know the value of $p$)? ...
1
vote
1answer
82 views

Product of ordinals notation

How to denote product of ordinals: $\cdot$ or $\times$? I'm not sure which of these two multiplication symbols to use.
-1
votes
2answers
160 views

Order of ordinals

Let $R_i$ is an ordinal for every $i\in n$ (where $n$ is also an ordinal) and $S_{i,j}$ is an ordinal for every $i\in n$ and $j\in m_i$ (where $m_i$ is also an ordinal for every $i\in n$). What is a ...
0
votes
1answer
97 views

An invertible function on ordinals

Let $R=R_{i\in n}$ is a family of ordinals where $n$ is also an ordinal. Let $q_i(x) = \sum_{j \in i} R_j + x$ for every $x\in R_i$. Is $q_i$ an invertible function?
4
votes
3answers
587 views

Definition of cofinality

Let $\alpha$ be a limit ordinal. We define $\operatorname{cf}\alpha$ to be the least limit ordinal $\beta$ such that there is an increasing $\beta$-sequence $\langle \alpha_{\xi} \mid \xi < ...
2
votes
1answer
129 views

Definition of a $\gamma$-filtered ordinal

This definition is part of Hovey's book on model categories: Let $\gamma$ be a cardinal. An ordinal $\alpha$ is $\gamma$-filtered if it is a limit ordinal and, if $A\subseteq \alpha$ and $|A|\leq ...
2
votes
1answer
178 views

Well ordering and maximal principle

I have to prove that given any poset $(P,\preceq)$ there exists a chain $S$ such that it is maximal (meaning that if $S\subseteq S'$ then $S=S'$). The book contains a proof using the axiom of ...
5
votes
2answers
172 views

The set of fixed points is closed (ordinals)

I am given that $F$ is a definite unary operation on the ordinals which is normal. That is, $$x<y\Rightarrow F(x)<F(y)$$ and furthermore $$F(y)=\sup\{F(x): x<y\}$$ if $y$ is a limit ordinal. ...
7
votes
1answer
217 views

If $\alpha\leq \beta,\gamma\leq \delta$ then $\alpha+\gamma\leq \beta+\delta$

The title is one of my homework assigments, (and the above greek letters are ordinal numbers) but this is the first one containing inequalities, and I dont really know how to do it. My guess is ...
6
votes
1answer
165 views

What are these operations on ordinals, analogous to multichoose, called?

Given well-ordered sets $\alpha$ and $\beta$, define $\left(\!\!{\alpha\choose \beta}\!\!\right)$ to be the set of weakly decreasing functions from $\beta$ to $\alpha$, ordered lexicographically; this ...
3
votes
2answers
418 views

Is the supremum of an ordinal the next ordinal?

I apologize for this naive question. Let $\eta$ be an ordinal. Isn't the supremum of $\eta$ just $\eta+1$? If this is true, the supremum is only necessary if you conisder sets of ordinals.