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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15 views

Limit ordinal preserved under arithmetic operations

I want to check the following statements: $\alpha+\beta$ is a limit ordinal if and only if $\beta$ is a limit ordinal. $\alpha\cdot\beta$ is a limit ordinal if and only if either $\alpha$ or $\beta$ ...
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60 views

A complicated lemma from Munkres's Topology

I don't understand the following lemma of the book Topology by J Munkres : 1- If a set $A$ with the mentioned properties exists there must be an example for it and so, it could help to understand ...
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47 views

Necessary and sufficient conditions of a well-ordered set to be of order type $\omega$.

I want to find some necessary and sufficient conditions of a well-ordered set to be of order type $\omega$. To be specific, let $A$ be a well-ordered set, if each element of $A$ except the smallest ...
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48 views

A question on ordinal arithmetic.

I have to order these two ordinals and I was just wondering if I have done it correctly. $\omega^\omega + \omega^3$ and $\omega + \omega^3 +\omega^\omega$ I have worked out that $\omega + \omega^3 +...
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1answer
32 views

Does the cofinality always induce a continuous function?

Let $\alpha$ be a limit ordinal and let $\lambda = cf(\alpha)$ be its cofinality. This means that there exists a strictly increasing $\lambda$-sequence $\langle \alpha_\xi \mid \xi < \lambda \...
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59 views

Faster growing function than the fast growing heiarchy under Church-Kleene?

Is there a computable function that grows faster than any function in a fast growing hierarchy with index less than the Church–Kleene ordinal, where computable fundamental sequences are used? If the ...
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60 views

Trichotomy of Ordinals. Is $K$ a set?

We want to prove the "Trichotomy of Ordinals": Definiton: An ordinal is a transitive set with elements that are all transitive. Definiton: $\alpha$ and $\beta$ ordinals are comparable if one of the ...
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2answers
137 views

Power two of ordinal

if $\omega$ is ordinal number of set $\mathbb{N}$ can I compute $(\omega + 1)^2$ like below $\begin{align*} (\omega + 1)^2 & = \ (\omega + 1).(\omega + 1)\\ & = \ ((\omega + 1).\omega)+(\...
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56 views

Order types of subsets of ordinals and transfinite induction

I am currently trying to practice the technique of transfinite induction with the following problem: Suppose that $X$ is a non-empty subset of an ordinal $\alpha$, so that $X$ is well-ordered by $\...
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1answer
25 views

Cardinal exponentiation question of infinite cardinals.

I got a little confused with a question about cardinal exponentiation: Let $\beta$ be an ordinal, and let $K_\alpha$ , $\alpha < \beta$, be infinite cardinals with $K= \sum_{\alpha<\beta}K_\...
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65 views

Arithmetic of uncountable ordinal

Assume $\alpha$ is an ordinal such that $\alpha \geq \omega_1$. Is it true then that $\alpha = \omega + \alpha$ with respect to ordinal arithmetic?
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132 views

Commutative addition on the ordinals

It is well known that ordinal addition is not commutative (for example $\omega+1\neq 1+\omega$), but it is associative. My question regards a new kind of addition defined as: $$a\oplus b = \text{max}\{...
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80 views

Elementary Set Theory, cofinal subset, cofinality, ordinal, totally ordered set problem

Definition 1. Let $\langle u,<\rangle$ be totally ordered set, $v\subset u$. $v$ is cofinal subset of $u$ means that for all $a\in u$, there exist $b\in v$ ($a\le b$). Definition 2. Let $\...
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1answer
53 views

Ordinal Fractions

Is any fraction $\,{x}\big/ {y}\,$ an ordinal number and if so, does ordinal $\,1 = \big\lbrace0,\dots,y - 1\big/y\big\rbrace\,$ instead of $\,\left\lbrace0\right\rbrace\,$? "If (X, <=) is a well ...
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0answers
38 views

Strictly increasing function from $\alpha< \aleph_1$ to $\mathbb{R}$ [duplicate]

I know that there is no increasing function $f: \aleph_1 \to \mathbb{R}$, so it seems like for $\alpha < \aleph_1$, there should exists a function $f: \alpha \rightarrow \mathbb{R}$ that is ...
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1answer
34 views

Strictly increasing function from $\alpha$ to $\aleph_0$

Given that $\alpha < \aleph_1$, does there exists a function $f: \alpha \rightarrow \aleph_0$ that is a strictly increasing function? This seems like it would be true since $|\alpha| = \aleph_0$, ...
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1answer
35 views

Limit ordinal in the exponent [duplicate]

How would you prove that $$n^{\gamma} = m^{\gamma}$$ for every limit ordinal $\gamma$ and $n,m$ finite ordinals? It's a rather short solution problem, but I can't construct any slick answer for it. ...
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1answer
53 views

Prove that $1^\alpha + 2^\alpha = 3^\alpha $ if $ \alpha $ is a limit ordinal

I am trying to prove the following statement: Suppose $ \alpha $ is a limit ordinal. Then $ 1 ^\alpha + 2^\alpha = 3^\alpha$. I'm not sure how to grasp this. Obviously induction won't work, since ...
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85 views

Topology on the set of ordinal numbers

This is a problem I encountered while reading Topology : An Outline for a First Course by Lewis E. Ward. Suppose $\Omega$ denotes the smallest ordinal number with uncountably many predecessors. Let $\...
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1answer
63 views

An ordinal number which satisfies $\omega^{\alpha} = \alpha$

Is there an ordinal such that $\omega^{\alpha} = \alpha$? It seems to me there should be, but I can't explicitly point it out. I know it is possible to prove $\alpha\leq\omega^{\alpha}$, but the ...
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1answer
54 views

Order type of the set of all the limit elements in $\omega^{\omega}$

What do you think is the order type of $\{\alpha\in\omega^{\omega}:\alpha\ \text{is a limit ordinal number}\}$? My first thought was $\omega$, but when I visualize the above mentioned set, $\omega^2$...
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91 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
3
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1answer
48 views

Ordinal arithmetic inequality

I have 3 ordinals $\alpha,\beta,\gamma$, $\gamma\neq 0$. Is this implication true? $$\alpha<\beta\implies\gamma\cdot\alpha<\gamma\cdot\beta$$ I have reason to believie it is not, namely ...
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62 views

What's the least class of ordinals closed under successor and the limits of omega-sequences? [duplicate]

What is the smallest class $S$ of ordinals that contains $0$, closed under successor and the limits of omega-sequences, i.e., $$ \forall i \in \omega [\alpha_i \in S] \Rightarrow \sup_{i} \alpha_i \in ...
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1answer
55 views

Is noetherianity really about cardinals or ordinals?

If $\kappa$ is any cardinal, then one may define a "$\kappa$-Noetherian" ring as a ring such that for any module that has a generating set $S$ satisfying $|S|< \kappa$, then any submodule also has ...
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1answer
40 views

Set theory proof for $\min C = \bigcap _{\alpha \in C} \alpha$

Let $C$ be a set of ordinals a) $\sup C$ is the least ordinal $\beta$ such that for all $\alpha\in C$, $a\leq \beta$. b) If $C$ is transitive (and thereby ordinal), then $\sup C \notin C$ if an only ...
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2answers
119 views

Two ways of defining rank of a set

I am studying set theory, and I have some difficulties in understanding how people define the notion of rank there (I hope, specialists in logic will excuse me for this). As far as I understand, ...
5
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1answer
73 views

Does forcing preserve the least undefinable ordinal from a model of ZFC?

Let $M$ be a transitive model of ZFC. For convenience let assume that $M$ is countable. Now let us consider the least undefinable ordinal $\vartheta_M$ which is not definable from elements in $M$ and $...
2
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1answer
47 views

Elements of the power set of $\textit{On}$ are elements of $\textit{On}$?

I don't know the truth value of this statement so this is just my own attempt. I am attempting to prove this in NBG set theory. Let $X \in \textit{P(On)}$ By the definition provided in my notes $X \...
4
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1answer
38 views

Distributivity of ordinal arithmetic

Let greek letters be ordinals. I want to prove $\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma$ by induction on $\gamma$ and I already know it holds true for $\gamma = \emptyset$ and $\gamma$ a ...
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1answer
28 views

Large ordered family of set inclusions complete?

I'm not an expert on set theory, so this might be trivial: Let $\dots \subseteq X_{\alpha} \subseteq X_{\alpha+1} \subseteq \dots$ be a chain of set inclusions, indexed by ordinals(!), s.t. $X_{\alpha}...
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2answers
76 views

$\omega \ \oplus 2 \neq 2\ \oplus \omega $ in NBG set theory

I'm struggling to rigorously prove $\omega \ \oplus 2 \neq 2\ \oplus \omega$ in context of NBG set theory. I haven't really seen a full proof anywhere besides the basic structure. Please direct me ...
5
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2answers
122 views

Proving the Powerset Axiom for hereditarily finite sets

Consider $\mathsf{ZF}$, and relace the Axiom of Infinity with its negation. This gives us the theory of hereditarily finite sets. Its universe is $V_\omega$. Intuitively, I feel that I can construct ...
7
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1answer
93 views

Fixed points in the enumeration of inaccessible cardinals

Let inaccessible cardinal mean uncountable regular strong limit cardinal. Consider $\mathsf{ZFC}$ with an additional axiom: For every set $x$ there is an inaccessible cardinal $\kappa$ such that $\...
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1answer
31 views

Definition of a recursive ordinal

I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows: "...an ordinal $\alpha$ is said to be recursive if there is a recursive well-...
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2answers
45 views

Is $\varnothing$ a limit ordinal?

This question is from Goldrei's Classic Set Theory: Let $\lambda$ be an ordinal. If $\cup\lambda=\lambda$, then $\lambda$ is a limit ordinal. But what if $\lambda=\varnothing$? I think I am ...
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2answers
91 views

Why is every limit ordinal a multiple of $\omega$?

An ordinal $\alpha$ is a limit ordinal if and only if $\alpha = \omega\cdot\beta$ for some $\beta$. I have checked through transfinite induction that $\omega\cdot\beta$ is always a limit ordinal, but ...
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0answers
62 views

Subset of rational numbers isomorphic to $\omega^{\omega+1}$

Does anyone know how to find/prove a rational subset isomorphic to $\omega^{\omega+1}$? I have couple of ideas but I find it hard to prove.
5
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1answer
78 views

ordinal isomorphism theorem

The only proof of the theorem stating that every well-order is order-isomorphic to a unique ordinal of which I am aware relies on the Axiom of Replacement. Is this necessary?
3
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54 views

The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
2
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0answers
45 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say $(\omega^...
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0answers
29 views

Cantor-Bendixson rank of a first countable space

This question has been bothering me for quite a while, so let me ask it here. Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index? By a Cantor-Bendixson index I ...
3
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2answers
103 views

Countable ordinal

I haven't done ordinal algebra for a long time and I can't remember how to prove that $\omega + \omega$ is a countable ordinal. Precisely what is the bijection between $\omega$ and $\omega \cdot 2$ ? ...
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1answer
23 views

Union of a chain of cardinalities?

I was trying to understand the union of a chain of cardinalities and I found this equation $$\kappa=\bigcup_{\alpha<\kappa} \alpha$$ for any cardinal $\kappa$ in the answers to this question. Can ...
3
votes
1answer
72 views

Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms

I need to show that for any countable ordinal $\alpha$ there is a set A $\subseteq \mathbb{Q}$ such that (A, <) is isomorphic to ($\alpha, \in$). To do it I am supposed to show the following ...
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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 ...
2
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0answers
25 views

Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
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1answer
30 views

Can't the first (/second) transinfinite ordinal replace the first (/second) uncountable ordinal in several counterexamples?

An example of a sequentially compact but not compact space is $\omega_1$. Indeed, any sequence of countable ordinals either has infinite elements below some countable ordinal, or has an ascending ...
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2answers
197 views

Can every set be expressed as the union of a chain of sets of lesser cardinality?

If a set $S$ has countably many elements $\{x_n\}$, it can be expressed as a union of a chain of finite sets $$ \{x_1\} \subset \{x_1,x_2\} \subset $$ But what about a set of arbitrary cardinality $...
2
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1answer
57 views

Order type of final segment of countable limit ordinal

Suppose $\gamma$ is a countable limit ordinal, with only finitely many limit ordinals less than it. If $\zeta$ is the greatest limit ordinal less than $\gamma$, does this necessarily imply that the ...