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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65 views

What is ordinal expression of $\infty$? [closed]

$\infty$ - is cardinal expression? ?Origin of a line ray is ordinal expression of $\infty$, if distance from the $\infty$ to the $0$ origin of line ray?
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52 views

Let $\omega$ be the ordinal of the well ordered set $\mathbb{N}$, what does means $\omega^\omega$?

Let $\omega$ be the ordinal of the well ordered set $\mathbb{N}$, then we can define the ordered sum and product, and so the sum and product of ordinal. Is easy to see that ...
1
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21 views

Derivatives of ordinal order

This question actually arises from this answer to another question, which contains the sentence A function is smooth is it has derivatives of infinite order. While the author surely didn't ...
7
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67 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
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1answer
437 views

Is there an axiomatic approach to ordinal arithmetic?

I've always wondered, is there an axiomatic approach to the arithmetic of ordinal numbers? If so, I imagine it would be on par with set theory in terms of its proof-theoretic strength.
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2answers
40 views

How to prove the not-so-long rays are homeomorphic to the reals?

The long ray (half of the long line) is an interesting topological space. It is defined as the order topology on $\omega_1$$\times [0, 1)$ with lexicographic order. Basically, it is an uncountable ...
5
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2answers
32 views

How many limits of limits are there in $\omega_1$?

We know there are $\omega_1$-many limit ordinals less than $\omega_1$. What about ordinals that are limits of limit ordinals? Are there also $\omega_1$-many of these?
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1answer
43 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
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75 views

Height of an ordered field

I'm studying ordered fields, and a specific notion regarding ordered fields that I will denote here by their "height". If $k$ is an ordered field, and $\alpha$ is a non-empty ordinal, a ruler of ...
0
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1answer
54 views

How to view beth number as ordinal?

The following is the snippet of the paragraph given in my lecture note: (Note: $\beth$ is the beth-number.) (When I introduced $\beth_\alpha$ I told you that it was to be a particular size of ...
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2answers
57 views

Cantor-Bendixson rank of a closed countable subset of the reals, and scattered sets

I am reading the notes in the following link, and I am a bit unclear about the connection between scattered sets, countable sets, and sets for which the Cantor-Bendixson derivative is eventually the ...
3
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2answers
68 views

Proving that $\langle \omega+1, \leq \rangle$ and $\langle \omega + \omega^*, \leq \rangle$ are not elementarily equivalent

This is an exercise from Kees Doet's Basic Model Theory. The basic idea is to show that an ordering of type $\omega + 1$ is not isomorphic to an ordering of type of $\omega + \omega^*$, where ...
3
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1answer
71 views

Extending the Ordinals analogously to the Integers

I have been (recreationally) trying to expand the notion of ordinal numbers in the same way that the natural numbers $\mathbb N$ are extended to the integers $\mathbb Z$. My objective is to be able ...
2
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1answer
73 views

order of infinite countable ordinal numbers

I'm trying to understand ordinal arithmetic. If one had an ordered list of the some subset of countable ordinal numbers, what order would the following 6 countably infinite ordinals be in? If the ...
5
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1answer
106 views

The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
7
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114 views

What well-orders are definable over $V$?

Let $V$ be a (the) universe of sets, and $On^V$ denote the ordinals of $V$. It is well known that there are formulae that seem to define orderings `longer than' $On^V$. For example: $\alpha < ...
11
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10answers
704 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?)
3
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1answer
67 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} = ...
1
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1answer
89 views

Well ordering of type epsilon one

I have been very interested in the countable ordinals for awhile now, but one thing has eluded me despite my research into the subject. What is a well-ordering of the natural numbers corresponding to ...
9
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3answers
790 views

Uncountability of countable ordinals

According to Wikipedia, there are uncountably many countable ordinals. What is the easiest way to see this? If I construct ordinals in the standard way, $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega ...
2
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1answer
34 views

limit and infinite ordinals: same thing? [duplicate]

I'd like to clarify my understanding re: limit and infinite ordinals. This post says: An initial infinite ordinal is a limit ordinal. Is it true the other way around? That is if I have a limit ...
2
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2answers
33 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 ? ...
3
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2answers
41 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
46 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 ...
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1answer
44 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 ...
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3answers
80 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 ...
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1answer
26 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
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1answer
41 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 ...
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1answer
38 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.
4
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1answer
75 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
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1answer
40 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
87 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 ...
0
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2answers
54 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, ...
2
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1answer
63 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
60 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 ...
4
votes
1answer
85 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 ...
0
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0answers
26 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 ...
2
votes
3answers
62 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?
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0answers
39 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
2answers
66 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, ...
2
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1answer
51 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
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1answer
39 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?
4
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1answer
80 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...
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2answers
30 views

Lexicographic order on $\alpha^\beta$ is well-ordered

Suppose that $\alpha$ and $\beta$ are ordinals, and define the following order on $\alpha^\beta$, the set of all functions $\beta\to\alpha$ with finite support: $$f\,R\,g\iff\exists ...
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1answer
35 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
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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 ...
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1answer
33 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, ...
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1answer
52 views

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

Are there ordinals that are somehow "beyond" all the $\omega$'s?
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1answer
46 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 ...
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1answer
49 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 $\omega_1$ ...