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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Ordinal-indexed exponentation

I have read from somewhere that $2^{\aleph_0}$ is uncountable. However, I have also read from somewhere else that $2^\omega=\omega$. Do the above statements contradict with each other? In ...
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60 views

$\epsilon_0$ is closed under addition, multiplication, exponentiation of ordinals

Question: Let $f: \Bbb{N}\rightarrow \text{Ord}$ (where "Ord" is the set of ordinals) be defined inductively by: $$f(0)=\omega\\ f(n^+)=\omega^{f(n)}$$ Let $\epsilon_0=\{\sup f(i) : i \in \Bbb{N}\}.$ ...
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126 views

Does a $\Pi_2^0$ sentence becomes equivalent to a $\Pi_1^0$ sentence after it has been proven?

I heard that the P vs NP question is equivalent to a $\Pi_2^0$ sentence, and that the Riemann hypothesis is equivalent to a $\Pi_1^0$ sentence. Many known mathematical theorems state that some ...
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3answers
53 views

Ordinals and Countability

I've been watching Prof Su's (Harvey Mudd College) incredible lecture on Ordinals and Transfinite Induction, and I have a few related questions. He starts by drawing and examining three sets of dots, ...
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39 views

Ordinal pair $(α,β)$ such that $α<β$ and $Th(α,<) = Th(β,<)$

A number of weeks ago I was thinking of finding an example of a complete countable theory with only one binary predicate that is not $ω$-categorical. I later realized that $Th(\mathbb{Z},<)$ works, ...
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34 views

Cantor normal form to compute sums and products of ordinals: $\omega^{\beta} c+\omega^{\beta'} c' = \omega^{\beta'}c'$ if $\beta'>\beta$

From Wikipedia on Ordinal arithmetic: The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example, one need merely know that $$\omega^{\...
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7answers
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Why do we classify infinities in so many symbols and ideas?

I recently watched a video about different infinities. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \...
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26 views

How to show that an $\omega$-sequence is club in a limit ordinal $\gamma$.

Suppose that $\gamma$ is a limit ordinal. We say that a subset $X \subseteq \gamma$ is closed if whenever $\delta < \gamma$ is a limit ordinal and $\sup(X \cap \delta) = \delta$, then $\delta \in X$...
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1answer
47 views

Explicit indexing of countable ordinals by sequences of integers?

What I really want is this: A sequence $P_0$, $P_1$,... such that each $P_n$ is a countable partition of $\omega_1$, $P_{n+1}$ is a refinement of $P_n$, and such that if $A_n\in P_n$ for all $n$ and $...
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2answers
44 views

Question regarding countable ordinals

Feel free to suggest a better title. We're going to regard $0$ as a limit ordinal, as people sometimes do. Let $L$ be the set of countable limit ordinals. If $\alpha\in\omega_1$ there exist a unique ...
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4answers
173 views

Is $\omega-1$ finite?

I saw some videos and read some stuff about ordinals, and it came to me that $\omega-1$ should be finite. My logic is that $\omega$ is the smallest transfinite number, so $\omega-1$ should be finite.....
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3answers
93 views

Proving the class of countable ordinals is closed under ordinal exponentiation in ZF

I managed to prove that given the axiom of choice, the class (or is it a set?) of countable ordinals is closed under exponentiation, since the axiom of choice implies that the countable union of ...
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1answer
72 views

Every $\alpha < \kappa^+$ can be embedded in any interval of $\kappa^{<\omega}$.

Let $\kappa$ be a cardinal. I want to show that every ordinal $\alpha < \kappa^+$ can be embedded (as an order) in any interval of $\kappa^{<\omega}$, ordered lexicographically. This is exactly ...
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1answer
16 views

Is the identity a normal function?

I was going to prove the following statement: Let $F$ be a normal function, then $\cup\{0,F(0),F(F(0)),\ldots\}$ is the least ordinal $\alpha$ satisfying $F(\alpha)=\alpha$. In case of $0\neq F(...
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1answer
32 views

Ordinal subtraction as set difference

Consider the following synthetic definitions for ordinal arithmetic: $\alpha + \beta$ is the order type of $\alpha\times\{0\}\cup\beta\times\{1\}$ ordered by reverse lexicographical ordering. $\...
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1answer
31 views

Limit ordinal that cannot be written as $\alpha \cdot \omega$?

I learned from this thread that every limit ordinal can be written as $\omega\cdot\alpha$ for some $\alpha$ but is this also true for $\alpha\cdot\omega$ even though ordinal multiplication is not ...
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1answer
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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2answers
62 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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1answer
48 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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0answers
50 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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80 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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61 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
139 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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2answers
57 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
27 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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1answer
69 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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1answer
134 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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2answers
82 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
56 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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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
36 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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1answer
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
64 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
55 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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96 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 ...
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1answer
50 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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1answer
66 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 ...
5
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1answer
56 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
121 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, ...
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1answer
74 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 $...
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1answer
48 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 \...
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1answer
39 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
81 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 ...
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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 ...
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1answer
94 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 $\...