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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1answer
681 views

Is it really impossible to lose the Hydra game?

In a number of blog posts I found the claim that the game described below cannot be lost, which is to say, every possible strategy is a winning strategy. In each case, a sketch proof is given that ...
0
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1answer
47 views

Does the MK axiom of infinity and the Power set axiom hold in the cumulative hierarchy?

Question: Define the cumulative hierarchy inductively as: $$V_0 = \emptyset$$ For each ordinal $\alpha$: $$V_{\alpha+1} = \mathcal{P}(V_\alpha)$$ ie the power set of $\alpha$ and for any nonzero ...
1
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1answer
27 views

Order-preserving maps on a directed-complete poset

Definitons. A poset $P$ is directed if every finite subset has an upper bound in $P$. It is directed-complete if every directed subset has a least upper bound in $P$. For any poset $P$ let $[P\to P]$ ...
0
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0answers
17 views

Ordinals defined by successor and union

Let $M$ be the smallest set of ordinals satisfying $0\in M$ $x\in M\implies x+1\in M$ $S\subseteq M\implies \bigcup S\in M$ What does $M$ look like? Is it all ordinals? Is it all countable ...
5
votes
4answers
507 views

Trying to understand the sum of ordinals

I'm having a hard time with the functions defined by transfinite induction, in particular I have the sum of two ordinals defined as follows: \begin{align}\alpha+0&=\alpha \\ ...
0
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1answer
37 views

If $ \bigcup N_\alpha $ is stationary, then $ \{ \min(N_\alpha) \}$ is stationary

It's the last set-theoretic question for tonight, I promise. I'm trying to figure out why the following holds true: Suppose we have a regular, uncountable cardinal $ \kappa $ and a disjoint family ...
2
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2answers
25 views

Unboundedness of the set of subgroups of a cardinal

I'm trying to figure out why the following is true: Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ ...
2
votes
1answer
44 views

Checking whether some sets are clubs in $ \aleph_2 $ and $ \aleph_1 $

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question: which of the following sets are clubs, contain a ...
0
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0answers
13 views

Order type of a sum ($\bigcup$) of sets

A quick question. Is $$\textrm{ot}(\bigcup\limits_{\gamma <\lambda}\alpha_{\gamma})=\bigcup\limits_{\gamma <\lambda}\textrm{ot}(\alpha_{\gamma})?$$ where $\textrm{ot}$ stands for the order type ...
3
votes
1answer
106 views

Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
0
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0answers
20 views

Ordinals recursed

Consider any limit ordinal $\alpha$. It seems intuitively obvious to me that for any such ordinal there is $f:ORD \to ORD$ (with $ORD$ being the class of ordinals) with $f(1)=\alpha$ and $f$ ...
3
votes
2answers
94 views

How well-behaved can well-orders on $\mathbb{R}$ be?

Well-orders on $\mathbb{R}$ are interesting because they witness the "alephness" of the cardinality of the continuum, and they're therefore connected to the continuum problem. It seems reasonable, ...
0
votes
1answer
18 views

Fixed points of normal function form proper class

A normal function is a class function $f$ from the class of ordinals to itself such that $f$ is strictly increasing and continuous. We can easily show that for every ordinal $\alpha$ there is an ...
3
votes
1answer
66 views

Why does $\epsilon_0 = \omega^{\epsilon_0}$

I saw this on wikipedia and wasn't sure why. It seems obvious given the definition but how can we be sure? Also, what would be $\epsilon_0 \bullet \epsilon_0$? Would it be $\epsilon_0$? Thanks
1
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0answers
39 views

How many equivalence classes in a set of well-orders of a set

The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on ...
0
votes
2answers
48 views

Can limits to (positive) infinity be replaced by ordinals?

For example, does the following hold? $$\left(1+\frac1\omega\right)^\omega=e$$
1
vote
0answers
15 views

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 ...
1
vote
0answers
53 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 $\epsilon_0=\{\sup f(i) : i \in \Bbb{N}\}.$ ...
1
vote
1answer
106 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 ...
1
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3answers
46 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, ...
4
votes
1answer
37 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, ...
0
votes
0answers
33 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 ...
28
votes
7answers
3k views

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, ...
0
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0answers
25 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 ...
2
votes
1answer
37 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 ...
0
votes
2answers
35 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 ...
3
votes
4answers
159 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 ...
2
votes
3answers
82 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 ...
2
votes
1answer
62 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 ...
0
votes
1answer
15 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 ...
0
votes
1answer
28 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. ...
1
vote
1answer
27 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 ...
0
votes
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 $\beta$ ...
0
votes
2answers
59 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 ...
1
vote
1answer
45 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 ...
2
votes
0answers
46 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 ...
2
votes
1answer
31 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 ...
0
votes
0answers
58 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 ...
2
votes
0answers
56 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 ...
2
votes
2answers
136 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
vote
2answers
50 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 ...
1
vote
1answer
23 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= ...
0
votes
1answer
62 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?
9
votes
1answer
130 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 = ...
1
vote
2answers
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 ...
1
vote
1answer
52 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 ...
1
vote
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 ...
1
vote
1answer
33 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 $|\alpha| = \aleph_0$, ...
0
votes
1answer
32 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. ...
2
votes
1answer
50 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 ...