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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2answers
40 views

Multiplying two ordinals where one has been raised to power of $\omega$. Term order matters?

When multiplying two ordinals that are both raised to some power, my book says that one adds the exponents. But what happens if one of the exponents is $\omega$ ? Does the order of the terms matter ? ...
0
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1answer
58 views

example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?

I recall reading about a rule for multiplying ordinals where at least one is infinite, and where the cardinality of the multiplier (on right) is larger than the multiplicand (on left). If I recall ...
3
votes
2answers
47 views

How do you multiply this

How can you multiply these ordinal numbers: $(\omega+1)(\omega+1)(\omega2+2)$ I tried and have gotten to this: $(\omega^2+1)(\omega2+2)$ Is that the correct way, or did i made a mistake?
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1answer
48 views

any example of non-zero ordinals $a,b,c$ for which $a < b$, but $a^c \ge b^c$?

I am working from a book that gives the above problem, but no solution ;^(. That is: Show an example of non-zero ordinals $a,b,c$ for which $a < b$, but $a^c \ge b^c$. Exponentiation here ...
12
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10answers
813 views

What are the most prominent uses of transfinite induction outside of set theory?

What are the most prominent uses of transfinite induction in fields of mathematics other than set theory? (Was it used in Cantor's investigations of trigonometric series?)
11
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6answers
800 views

Principle of Transfinite Induction

I am well familiar with the principle of mathematical induction. But while reading a paper by Roggenkamp, I encountered the Principle of Transfinite Induction (PTI). I do not know the theory of ...
1
vote
1answer
51 views

Bringing ordinals to standard polynomial form

By the definition of multiplication and addition of ordinals, the following rules follow- Multiplication- $n\cdot\omega=\omega$ while $\omega\cdot n > \omega$ Addition- $n+\omega=\omega$ while ...
0
votes
3answers
112 views

“Subtracting” Ordinals - possible?

I was wondering whether it is possible to "subtract" ordinals, or in other words - does there exist an ordinal $\gamma$ for every pair of infinite ordinals $\alpha$ and $\beta$ such that ...
0
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1answer
51 views

The order isomorphism theorem without Replacement

In $\sf ZF$, there is the following theorem: Theorem: For any well-ordered set $\langle A,\prec\rangle$, there is a (unique) order isomorphism $F$ from $\langle\alpha,\in\rangle$ to $\langle ...
1
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1answer
43 views

Prove that for $\alpha$ an ordinal, $\alpha \le \omega^\alpha$.

Prove that for $\alpha$ an ordinal, $\alpha \le \omega^\alpha$. I assume the easiest proof is via induction, but I do not know what to do if $\alpha$ is a limit ordinal.
3
votes
1answer
77 views

Cardinals definable using ordinal arithmetics

Let $\kappa$ be an infinite cardinal. $\kappa$ is stable under ordinal addition $+$, ordinal multiplication $.$ and ordinal exponentiation $e: (a,b) \mapsto a^b$, so $\mathcal{K} = ...
4
votes
1answer
95 views

First Uncountable Ordinal Cofinality: Needs AC?

Say $\omega_1$ is the first uncountable ordinal. The reason I care about $\omega_1$ is Any countable subset of $\omega_1$ is bounded (or if you prefer, there is no countable cofinal subset). This ...
0
votes
1answer
60 views

Ordered fields with countable cofinality

Let $(k,+,.,0,1,<_k)$ be an ordered field, $| . |_k = x \mapsto \max(x,-x)$. If $\lambda$ is an ordinal, you can define convergent $\lambda$-sequences: maps $f: \lambda \rightarrow k$ satisfying ...
5
votes
2answers
92 views

Proving that $\alpha\approx|\alpha|$ is not constructive

$\sf ZF$ tells us that for every ordinal $\alpha$ there is a bijection $f:\alpha\to|\alpha|$, more or less by the definition of $|\alpha|$, the cardinal of $\alpha$. What I would like to show is that ...
1
vote
2answers
83 views

What is the cofinality of $2^{\aleph_\omega}$

There is a similar question in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook. The book(Karel Hrbacek&Thomas Jech, ...
3
votes
1answer
142 views

Prob. 6 (c), Sec. 10 in Munkres' TOPOLOGY, 2nd ed: Set of elements having no immediate predecessors in the minimal uncountable well-ordered set

Let $S_{\Omega}$ be the minimal uncountable well-ordered set. Let $X_O$ be the subset of $S_{\Omega}$ consisting of all elements $x$ such that $x$ has no immediate predecessor. Then how to show that ...
5
votes
1answer
83 views

Ordinal arithmetic $(\omega+1) \cdot \omega$ and $\omega \cdot (\omega +1)$

Here is where I am so far: $(\omega+1) \cdot \omega = \sup\{(\omega +1) \cdot n, n \in \omega\} = \omega^2$ and $\omega \cdot (\omega +1) = \omega \cdot \omega + \omega = \omega^2 + \omega$ Hence ...
0
votes
0answers
44 views

ordinal number question

An ordinal i is a natural number if and only if every nonempty subset of i has a greatest element. If a set of ordinals X does not have a greatest element, then sup X is a limit ordinal can someone ...
1
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1answer
48 views

Does $\mathcal{P}(\alpha)$ have maximal well-founded subsets?

Let $\alpha$ denote an ordinal. Then clearly, the powerset $\mathcal{P}(\alpha)$ has well-founded subchains that are maximal with respect to this condition (of being well-founded subchains), since ...
4
votes
1answer
92 views

Ordinal exponentiation, is $3^\mu = \mu$?

I'm revising for my set theory final, and I've been asked to find an ordinal $\mu > \omega$ with $2^\mu = \mu$, then to answer whether $3^\mu = \mu$. The ordinal I picked as $\mu$ was the union ...
1
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0answers
43 views

Existence of increasing pair of labeled trees in an infinite sequence

Assume labeled rooted trees with labels from a fixed set $\{1\ldots m\}$. For a tree $T$, we have: $V(T)$ the set of vertexes, $root(T)$ the root of the tree, $l_T: V(T)\rightarrow \{1\ldots m\}$ ...
2
votes
3answers
70 views

A compact Hausdorff space

It is known that every finite space is compact. Then I am worried whether there exists a compact Hausdorff space $X$ with with ordinal of $X$ is $\omega_0$. Does anyone know about it?
2
votes
1answer
88 views

Ordinal addition is not commutative

The simplest example is $$1+\omega=\omega \neq \omega +1$$ But I think it is also true that $$\omega ^\alpha+\omega^\beta=\omega ^\beta$$ for general ordinals $\alpha < \beta$ but I don't know how ...
0
votes
1answer
59 views

Ordinal arithmetic and limit ordinals

Suppose $1\leq\xi<\omega_1$ is a countable ordinal and that $1\leq\zeta<\omega^\xi$. Is it always true that $\zeta+\omega^\xi=\omega^\xi$? If so, why?
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1answer
37 views

The set of ordinals $< \alpha$ of a given cofinality $\kappa< \text{cf}(\alpha)$ is stationary

I don't understand the last line of the following proof : I understand that we climbed up to $g(\kappa)$ in $\kappa$ steps (i.e. cf$\big(g(\kappa)\big) \leq \kappa$) but can we be sure that we did ...
2
votes
1answer
15 views

Show that a particular set is not a limit ordinal (aim is to define ordinal subtraction)

Let $\alpha$ and $\beta$ be two ordinals with $\beta \leq \alpha$. Define $$ X:= \{\gamma \in \alpha^{+} : \beta + \gamma \leq \alpha\}.$$ I have shown this is an ordinal. Now I need to show it ...
2
votes
2answers
75 views

$|\alpha\times\alpha|=|\alpha|$ for infinite ordinal $\alpha$, definably

I am trying to prove Proposition 1.7 of http://math.bu.edu/people/aki/7.pdf: Proposition 1.7 (Halbeisen–Shelah). If $\aleph_0\le|X|$, then $|{\cal P}(X)|\not\le|{\rm Seq}(X)|$. In words, ...
1
vote
1answer
52 views

Justification for ordinal arithmetic at limit ordinals

Wikipedia tells me that: $$\alpha + \lambda := \bigcup_{\beta < \lambda} \left ( \alpha + \beta \right ) $$ for a limit ordinal $\lambda$. Multiplication and exponentiation, as Wikipedia says, ...
-1
votes
1answer
56 views

Are there ordinals beyond all the $\omega$'s? [closed]

Are there ordinals that are somehow "beyond" all the $\omega$'s?
0
votes
1answer
59 views

Ordinal Exponentiation problem

I've found a problem in Set theory that I can't really get my head around. The problem is: Under ordinal exponentiation find an ordinal $\mu$ such that $\omega < \mu$ and $2^{\mu } = \mu$ (where ...
2
votes
1answer
82 views

How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

I know that with choice, the countable union of countable sets is countable, making $\omega^\omega$ and $\epsilon_0$ both countable. Can we show this without choice? E.g. in the case that $\omega_1$ ...
3
votes
1answer
62 views

Given the axiom of choice, are cardinals ordinals?

Given a model of ZFC, is it correct to talk indistinctly about cardinals and initial ordinals, namely, ordinals $\alpha$ such that for every $\beta < \alpha$, there is no bijection between $\alpha$ ...
2
votes
1answer
161 views

Prove the long line is not contractible.

Given the following definition of the long line: Let $\omega_1$ be the first uncountable ordinal and consider $[0,1)$ as an ordinary set. Define the long ray to be the ordered set $\omega_1 \times ...
3
votes
2answers
148 views

Ordinals - motivation and rigor at the same time

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal ...
2
votes
1answer
37 views

Question about $\text{non}(\mathsf{nonstat}_\gamma)$

A lemma in Kunen's (2011) set theory states that if $\gamma$ is a limit ordinal with $\kappa=\text{cf}(\gamma)>\omega$, then $\text{add}(\mathsf{nonstat}_\gamma)$ $=$ ...
0
votes
1answer
72 views

monotonic laws for ordinal subtraction

I have to prove some monotonic laws for ordinals. It's quite comfortable for me to show monotonic laws of ordinal addition (e.g. $\beta\leq\gamma\Rightarrow\alpha+\beta\leq\alpha+\gamma$). But when it ...
1
vote
3answers
66 views

Limit Ordinals as Infinite Ordinals and other questions

I am studying set theory and I am confused in the following: Are limit ordinals the same as infinite ordinals? I would say yes since the least non-zero limit ordinal is $\omega$. Infinite limit ...
2
votes
0answers
27 views

How to prove that every countable compact Hausdorff space is homeomorphic to a well-ordered set with its order topology? [duplicate]

I encountered this problem in a textbook: Let $X$ be a countable compact Hausdorff topological space, I was asked to prove that $X$ is always homeomorphic to a (necessarily countable) topological ...
0
votes
1answer
75 views

Using ordinal arithmetic calculate the following ordinal numbers

(ω + 1) x ω (ω + 1) x 2 For Question #2, I can simplify to the point where I get (ω + (ω + 1)), but I'm not sure how to proceed from there
1
vote
1answer
51 views

Continuous function in order topology

I have to prove that for every $ \alpha$ in $[0, \Omega)$, there exists a continuous function $f$ from $[0, \Omega)$ to $\mathbb R$ such that the pre-image of $0$ is $\{α\}$. I really have no idea ...
30
votes
3answers
3k views

A transfinite epistemic logic puzzle: what numbers did Cheryl give to Albert and Bernard?

I expect that nearly everyone here at stackexchange is by now familiar with Cheryl's birthday problem, which spawned many variant problems, including a transfinite version due to Timothy Gowers. In ...
3
votes
1answer
87 views

A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $\mathbb Q$, I find out that any countable limit ordinal is the union of a sequence of increasing ordinal. Now I'm ...
2
votes
1answer
37 views

Is the $\omega_1$'th element of a well ordered set the first with uncountable predecessors?

As far as I understand the topic, ordinal numbers are defined as sets, but their "purpose" is to generalize the concept of indices beyond infinity. So, can it be said that the element at index ...
1
vote
1answer
187 views

First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. ...
7
votes
0answers
127 views

Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
4
votes
1answer
86 views

Transfinite Induction in Peano Arithmetic

I have heard that Peano Arithmetic (PA) cannot perform transfinite induction up to $\varepsilon_0$. This seems to imply that it can induct up to smaller ordinals, like $\omega$ or $\omega^\omega$ or ...
1
vote
1answer
76 views

Bijection between well-ordered set and ordinal

We are working in ZF. Problem (part 1). Suppose we have well-ordered set $(M, <)$. How to show that there exists a bijection $f$ between this set and some ordinal $x$ that preserves order? I can ...
1
vote
1answer
47 views

Construct an sequence in order set

Assume that we have an uncountable ordered set $\Theta$. Can we construct a sequence $(x_n)$ which $x_1<x_2<...<x_n<...$. In addition, for each $x\in \Theta$, there exists $n$ such that ...
0
votes
1answer
84 views

Help with the definition of ordinal sum

"Given two ordinals $\alpha$ and $\beta$, let $A = (\alpha$ x {0}) $\cup$ $(\beta$ x {1}). Then, define a well ordering on A by: $(v, i) <_{A} (\tau, j) \iff (i \lt j) \lor (i = j$ $\land$ $v ...
1
vote
1answer
30 views

Problem with base case for transfinite induction

I need to prove this using transfinite induction Let $\alpha, \beta , \gamma $ ordinals If $\beta <\gamma$ then $\alpha + \beta < \alpha + \gamma$ I am trying to prove the statement by ...