0
votes
2answers
57 views

How to define an isomorphism between $^\omega\omega$ and $\omega^\omega$?

Let $^\omega\omega$ be the set of all functions $x: \omega \to \omega$. Define $A = \{x \in ^\omega\omega \; | \; x \text{ has finite support}\}$, where by "finite support" I mean that the set $\{x(n) ...
1
vote
1answer
43 views

Cardinality of Orderings of $\mathbb{R}$

For a finite set $S$ there are $\vert S\vert!$ orderings of its elements. What is the cardinality of all orderings of $\mathbb{N}$? What would $$\vert \mathbb{N}\vert!$$ mean? Is it ...
3
votes
2answers
54 views

Embedding $\omega_1$ into the direct sum/product of $\Bbb R$'s and $\Bbb N$'s

I'm trying to find a way to visualize the first uncountable ordinal $\omega_1$. This is rather difficult, as the visualization tactic that I often use for countable ordinals - namely, the "matchstick" ...
7
votes
1answer
80 views

Order-preserving injections of ordinals into $[0,1]$

The usual "matchstick" representation of an ordinal number can be thought of as an order-preserving injection of that ordinal into the interval [0,1]. For example, here's a representation of ...
2
votes
1answer
64 views

A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
0
votes
0answers
21 views

Under what circumstances does the least $x \in X$ such that [whatever] still have this property when mapped into $Y$?

Let $X \lesssim Y$ denote well-ordered sets and $f : X \rightarrow Y$ denote the unique such order embedding whose image is downward-closed. Then its not necessarily the case that for all ...
6
votes
1answer
76 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) ...
2
votes
2answers
119 views

detecting cofinality in ordinals

Suppose K is an unbounded well ordered set, and suppose K is minimal in the sense that each proper initial segment of K has strictly smaller cardinality. Suppose J is an unbounded well ordered set of ...
1
vote
1answer
94 views

The classification of order types

In response to a previous question which I asked concerning the order type of the Rationals, a reference was made (by MDJ) to a theorem of Cantor stating that any two countable, dense, linearly ...
5
votes
2answers
255 views

The order type of the rationals.

Herewith another mind-numbingly naive question from a reader of philosophy. My question concerns the order type of the rational numbers. Omega squared seems a natural first choice, but obviously ...
14
votes
2answers
218 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 ...
4
votes
1answer
82 views

If $(W,<)$ is a well-ordered set and $f : W \rightarrow W$ is an increasing function, then $f(x) ≥ x$ for each $x \in W$

I could use a hand understanding a proof from Jech's Set Theory. Firstly, note that Jech defines that an increasing function $f : P \rightarrow Q$ is a function that preserves strict inequalities ...
2
votes
0answers
66 views

Is there a nice characterization of posets induced by trees?

Define that a tree in $X$ is a set of ordinal-indexed sequences with codomain $X$ that is closed under the operations of restricting to an ordinal. (I do not know if this definition is standard.) ...
0
votes
1answer
116 views

The limit elements of well-ordered proper classes

For any well-ordered class $W$ with no maximum element, we can define a successor function $f_W : W \rightarrow W$ by asserting that $f_W(w) = \mathrm{min}\{x > w \mid x \in W\}.$ This allows us to ...
2
votes
1answer
82 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
3
votes
3answers
118 views

Set $A$ has the cardinality of $\aleph_0$. Prove some properties of partial order $ \langle \mathcal P (A), \subseteq \rangle$.

I try to solve the following task: Set $A$ has the cardinality of $\aleph_0$. The truth is that in the partial order $ \langle \mathcal P (A), \subseteq \rangle$: (answer true or false) a) ...
3
votes
3answers
121 views

Which ordinals embed in $2^\omega$ ordered by inclusion?

Which ordinals can be embedded in the power set of $\omega$ ordered by inclusion? I see that $\omega\cdot\omega$ can (and therefore anything less than that): we can partition $\omega$ into $\omega$ ...
4
votes
3answers
226 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 ...
8
votes
1answer
137 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 ...
5
votes
3answers
143 views

How to show that $(\Bbb Q, <)$ contains an order isomorphic copy of $\epsilon_0$ which is the least ordinal satisfies $\omega^\epsilon = \epsilon$?

We define $\epsilon_0 = \sup\{\omega, \omega^\omega,\omega^{\omega^\omega},\omega^{\omega^{\omega^\omega}},\ldots\}$. How to find a subset of $ \Bbb Q$, $(A, <_{\Bbb Q})$ that is isomorphic to ...
3
votes
2answers
255 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 ...
8
votes
2answers
243 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 ...
7
votes
1answer
115 views

Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
8
votes
2answers
214 views

Ordinal interpretation of Friedman's $n$?

I heard that Kruskal's tree theorem can be turned into a finite form that creates an extremely fast growing function because ordinals could be encoded into trees. On this wiki page it mentions that ...
7
votes
3answers
244 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 ...
4
votes
2answers
159 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 ...
0
votes
1answer
139 views

Proving of existence of limit ordinal

How to prove that for every ordinal $\alpha$ there exist limit ordinal $\beta$ ,such that$\alpha\in\beta$ ? Thank you.
5
votes
1answer
93 views

Lower Bound of countable order-types of linearly ordered sets

On Page 44, Set theory, Jech(2006) (Show that) there are at least $\mathfrak{c}$ countable order-types of linearly ordered sets. [For every sequence $a = \{ a_n : n \in N\}$ of natural numbers ...
4
votes
1answer
101 views

Generalization of ordinals to well-founded sets?

In set theory, the ordinal numbers are objects that represent the order types of well-ordered sets. Is there a similar class of objects that represent the order types of well-founded sets? If so, ...
10
votes
3answers
463 views

An order type $\tau$ equal to its power $\tau^n, n>2$

In this question we are concerned only with linear (aka total) order types. By a cardinality of an order type we understand a cardinality of an instance of this type, which obviously does not depend ...
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 ...
16
votes
2answers
331 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. ...
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. ...
7
votes
2answers
206 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 ...
5
votes
1answer
258 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 ...
19
votes
6answers
543 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 ...
8
votes
4answers
259 views

Is there an element with no fixed point and of infinite order in $\operatorname{Sym}(X)$ for $X$ infinite?

Let $X$ be an infinite set. Let $\operatorname {Sym}(X)$ denote the group of all bijections from $X$ onto itself. I have been thinking about the existence of elements of infinite order in this group. ...
4
votes
1answer
275 views

The “canonical” representative of an order type

An ordinal (in von Neumann sense) can be thought as the "canonical" representative of the order type (i.e. of the class of all order-isomorphic sets) of a well-ordered set. This is very convenient, ...
7
votes
4answers
289 views

Supremum and ordinals

Question: Show that sup{$ \xi \dotplus 1 : \xi \in A $} is the least ordinal that is greater than each element of $A$. I tried to get a better feel for this by letting $A=3=${$0,1,2$}. Then I know ...