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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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 ...
0
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0answers
26 views

Commutativity of operations on ordinal numbers? [duplicate]

I read on the Wikipedia for ordinal numbers that the operations of addition and multiplication aren't commutative, more specifically: "addition and multiplication are not commutative: in particular $1 ...
3
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1answer
75 views

Isomorphic ordinals are equal

Let $\alpha$ and $\beta$ be two isomorphic ordinals. Then $\alpha = \beta$. I want to whether the following proof is correct. I already know that there are three prossible cases: $\alpha \in \beta, ...
2
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1answer
27 views

Equal cardinality implies isomorphism of ordering

I think I have proved this (should-be false) lemma. For any set $X$, if $X$ is equipotent to an ordinal $\alpha$, then $X$ can be ordered so that it is isomorphic to $\alpha$. This is the proof ...
4
votes
1answer
71 views

Is it true that $\lvert\alpha+\beta\rvert=\lvert\alpha\rvert+\lvert\beta\rvert$ for ordinals $\alpha$ and $\beta$?

Suppose $\alpha$ and $\beta$ are ordinals. I was asked to prove that $\lvert\alpha+\beta\rvert=\lvert\alpha\rvert+\lvert\beta\rvert$. However, I think to have found a counterexample for this equality, ...
0
votes
1answer
34 views

Ffind an ordinal type of the well-ordering

Let $x_n$ for $n \in N$ are variables of predicate logic. Let $F(L)$ defines set of all formulas of language $L$ and $Σ(L) = L ∪ \{(,), ¬, →, ∀\} ∪ \{x_n | n ∈ N\}$ defines corresponding alphabet. We ...
3
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1answer
45 views

What is $1^\omega$?

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, $\...
2
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0answers
32 views

show that the von Neumann universes $R_{\omega+ \alpha}$ have cardinality $\beth_\alpha$.

Trying to show that the von Neumann universes $R_{\omega+ \alpha}$ have cardinality $\beth_\alpha$. Here is my attempt: Proof by induction on $\alpha$. For $\alpha = 0$, $|R_{\omega+ 0}| = |R_{\...
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1answer
50 views

A question on proof of the Zermelo's theorem “Every set is well-orderable.”

There exists some proof for the theorem. Some of them use Transfinite Recursion. Some of them use same argument with the following. Proof:(copied from proofwiki) "Let $S$ be a set. Let $\mathcal{P}(...
0
votes
2answers
78 views

List of limit ordinals [closed]

I am trying to understand ordinals and cardinals. Seeing a list of limit ordinals would be helpful. A small question outside of this, are the limit ordinals related to cardinals? In what way? ...
0
votes
1answer
47 views

Where is the mistake in this set-theoretic argument?

Let $\omega$ be the first infinite ordinal and for all $n\in \omega $ define $n=\{0,1,2,...,n-1\}$. In particular, $2=\{0,1\}$. Let $f:\omega\rightarrow \omega$, $f(x)=x^2 $. Now $4=f(2)=f(\{0,1\})=...
3
votes
2answers
410 views

Any Set of Ordinals is Well-Ordered

Is this a theorem of ZF or a theorem of ZFC? My suspicion is that it is a theorem of ZFC (the proof I've seen in P. L. Clark's online notes on countable ordinals requires selecting an element from ...
3
votes
1answer
58 views

Is this a way to define an ordinal $\lambda$?

A set $\lambda$ is by definition an ordinal if $\forall x\in\lambda : x\subseteq \lambda$ and $\lambda$ is well-ordered by $\in$ . May I equally define an ordinal as a set $\lambda$ having ...
0
votes
1answer
50 views

Is model theory needed to understand ordinal logic?

By ordinal logic, I mean turing's ordinal logic. I'm going to learn first order logic, elementary set theory, basic computability, and godelian incompleteness as prerequisites for ordinal logic. But, ...
1
vote
1answer
43 views

Comparing Hartogs number to set of all sets of relations on a set

Let $A$ be a set and let $\mathcal{H}(A)$ be the Hartogs number of $A$. Show that $|\mathcal{H}(A)|\neq|\mathcal{P}(\mathcal{P}(A\times A))$. Proof attempt: By contradiction. Suppose that $|\mathcal{...
4
votes
1answer
91 views

Surjective function into Hartogs number of a set

For any set $A$, there is a surjective function $f:\mathcal{P}(A\times A)\longrightarrow\mathcal{H}(A)$. $\mathcal{H}(A)$ is the Hartogs number of $A$. Proof attempt: Suppose $R\subseteq A\times A$. ...
0
votes
3answers
160 views

Best known upper and lower bounds for $\omega_1$

What are the best known lower and upper bounds for $\omega_1$? Are there sharper lower bounds than $\varepsilon_0$? And are there any known upper bounds which can be explicitly constructed like $\...
3
votes
3answers
133 views

Is $\omega_0-1$ infinite?

I have read in another answer Is infinity an odd or even number? that the $\omega_0$ is the "smallest infinity", but is $\omega_0-1$ not also infinite?
0
votes
1answer
51 views

Vanishing Cantor-Bendixson derivative if and only if scattered

In the question asked here, it is claimed in a comment and answer of Brian M. Scott that a subset of $\mathbb{R}$ is scattered if and only if it has vanishing Cantor-Bendixson derivative. I thought ...
3
votes
1answer
115 views

What are all the finite transitive sets?

In set theory, we come across the definition of an ordinal as a well-ordered transitive set. And then we show that $\omega$ and all its elements are ordinals. My question whether there are finite ...
1
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1answer
89 views

Countable Ordinals

In an intro topology class we briefly brought up ordinal numbers during a conversation of transfinite induction. I believe I understand how the ordinal numbers work, at least up to $\omega^\omega$. ...
-1
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2answers
51 views

What would be an example of a well-ordering of the natural numbers of type $\omega^3$? [closed]

What would be an example of a well-ordering of the natural numbers of type $\omega^3$ ?
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1answer
43 views

Is it the case that for all ordinal numbers $a$ and $b$, $|a^b|=|a|^{|b|}$? [closed]

Is it the case that for all ordinal numbers $a$ and $b$, $|a^b|=|a|^{|b|}$?
1
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0answers
26 views

(Ordinal number)How to prove that epsilon-naught is countable without using Axiom of Choice? [duplicate]

How to prove that epsilon-naught is countable without using Axiom of Choice? or, Can we explicitly show that there is isomorpism between epsilon naught and subset of rational number?
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1answer
130 views

There is no infinite descending chain of ordinals

I'm currently reading a set theory book and it uses the fact that There is not infinite descending chain of ordinals. But I can't find the proof. (And also I remember Topology by Munkres uses ...
3
votes
1answer
31 views

Can ordinal numbers be generalized to arbitrary classes?

The following is an obvious potential, albeit incomplete, method of generalization. One class paired with an ordering $s$ would be of the same order type as another $t$ iff there exists an order-...
1
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0answers
62 views

Ordinal Numbers within Simple Type Theory

A little preamble: I recently came across the system of higher-order logic known as simple type theory, and I was immediately intrigued by it, as it seemed to be exactly what I was looking for as an ...
0
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0answers
43 views

What axioms satisfy a limit ordinal?

Let us consider the pair $(\alpha,\in)$ where $\alpha$ is a limit ordinal. The question is what axioms of ZFC satisfies that pair. I think it satistifies all axioms but I'm not sure...
0
votes
1answer
54 views

Is there some result that says a theory cannot prove the consistency of any of its extensions?

Is there some result that says a (sufficiently strong) theory cannot prove the consistency of any of its extensions? Or something along these lines?? More generally, is there a result that says a (...
7
votes
1answer
73 views

Inconsistencies in ordinal arithmetic (normal form)

I'm trying to understand how to bring ordinals of the form $(\omega +n)\omega$ to normal form. Unfortunately, I am confused by what seem to be inconsistent answers to very similar questions: In a ...
1
vote
1answer
24 views

Proving basic fact about ordinals

In a set of online lecture notes, I saw the following proposition. Let $C$ be a set of ordinals. Then $\sup \left\{ \alpha +\beta:\beta\in C \right\} =\alpha +\sup C$. How can I prove this?
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3answers
83 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 $\omega+\omega+\cdots+\...
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0answers
36 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 ...
9
votes
0answers
143 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 ...
5
votes
2answers
52 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
votes
2answers
39 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?
0
votes
1answer
101 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 ...
3
votes
0answers
86 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 ...
3
votes
2answers
84 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 $\omega^*...
2
votes
1answer
84 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 ...
7
votes
0answers
147 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 < \...
1
vote
2answers
226 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 ...
5
votes
1answer
117 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 ...
1
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1answer
82 views

formal definition of ordinal addition by recursion

I'm reading Kunen's Set Theory, An Introduction to Independence Proofs (1980). On page 26 he explains how to introduce ordinal addition through recursion. For the sake of convenience i'll give the ...
2
votes
1answer
42 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
votes
2answers
48 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
votes
1answer
67 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
51 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?
0
votes
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 ...