Tagged Questions

3
votes
2answers
23 views

How to show that $(\Bbb Q, <)$ contains an order isomorphic copy of $\epsilon_0$ which is the least ordinal satisfies $\omega^\epsilon = \epsilon$?

We define $\epsilon_0 = \sup\{\omega, \omega^\omega,\omega^{\omega^\omega},\omega^{\omega^{\omega^\omega}},\ldots\}$. How to find a subset of $ \Bbb Q$, $(A, <_{\Bbb Q})$ that is isomorphic to ...
4
votes
1answer
46 views

$\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal

I am trying to understand this proof of the existence of an uncountable ordinal. I don't see why $\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal as it is said. For ...
3
votes
2answers
75 views

Why does every countable limit ordinal have cofinality $\omega$?

According to Wikipedia, if $\alpha$ is a countable limit ordinal, then $\mathrm{cf}(\alpha)=\omega$. It is intuitively clear to me that it should be so. Certainly the cofinality of such an ordinal ...
0
votes
2answers
68 views

Closed, unbounded subset of a cardinal.

I missed two lectures in my set theory course, and now I don't understand the homework problems. One is this: let $\kappa$ be a regular uncountable cardinal. Show that the following sets are closed ...
3
votes
1answer
88 views

Diagonal intersection of club sets

Let $C_\alpha\subset\omega_1$ be a club set for every $\alpha<\omega_1$. Show that the diagonal intersection of all the $C_\alpha$'s, that is $\{\alpha<\omega_1:\forall\beta<\alpha,\alpha \in ...
2
votes
2answers
77 views

Another question about Cohen's article

This is another question related to Cohen's Discovery of Forcing: Consider the following passage from p. 1091: Suppose $M$ were a countable model. Up until now we have not discussed the role ...
9
votes
1answer
113 views

What are some good open problems about countable ordinals?

After reading some books about ordinals I had an impressions that area below $\omega_1$ is thoroughly studied and there is not much new research can be done in it. I hope my impression was wrong. ...
13
votes
3answers
251 views

How many positive numbers need to be added together to ensure that the sum is infinite?

The question in the title is naively stated, so let be make it more precise: Let $\sum_{n\in\alpha}a_n$ be an ordinal-indexed sequence of real numbers such that $a_n>0$ for each $n\in\alpha$, where ...
2
votes
2answers
89 views

Ordinals Addition Property

I’m sill trying to figure out how to prove that the following property of ordinals holds Let $x, y, z \in$ On where On is the class of ordinals. If $x < y$ then $\forall z$ we have that ...
15
votes
1answer
130 views

When does ordinal addition/multiplication commute?

I'm looking at basic ordinal arithmetic at the moment, and I am aware that in general, $\alpha+\beta\neq\beta+\alpha$ and $\alpha.\beta\neq\beta.\alpha$ for ordinals $\alpha,\beta$. My question is: ...
3
votes
2answers
54 views

Omega, well ordering and the cumulative hierarchy $V$.

Hi I need help with a problem of set theory. I'm not sure how to prove that the well ordering on $\omega$ isomorphic to $\omega+\omega$ belongs to the level $V_{\omega+\omega}$ in the hierarchy $V$. ...
7
votes
2answers
157 views

Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?

My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
2
votes
2answers
50 views

Is the set of countable successor ordinals countable or uncountable?

I know that $\omega_1$={$\alpha$ : $\alpha$ is a countable ordinal} is uncountable but what about the subset of $\omega_1$ of countable successor ordinals?
2
votes
2answers
106 views

Is the class of all ordinals independent of set theory?

I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
3
votes
3answers
40 views

Limit Ordinal Question

how to prove this result. $\alpha$ is limit ordinal if and only if $\beta<\alpha$ implies $\beta+1<\alpha$ for any $\beta$. I am getting confused because of the definition of limit ...
3
votes
2answers
56 views

Indecomposable limit ordinals

A limit ordinal $\gamma>0$ is said to be indecomposable iff $\nexists\alpha,\beta<\gamma$ such that $\alpha+\beta=\gamma$. In view of this definition, I’m trying to prove the equivalence of the ...
6
votes
1answer
70 views

Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
2
votes
1answer
56 views

Is there a simple characterization of ordinals that cannot be “escaped”?

The ordinal $\omega$ is inescapable, in the sense that for any $N \in \omega$ it holds that if $f : N \rightarrow \omega$ is a function, then the supremum (taken in the class of ordinals) of the range ...
15
votes
4answers
236 views

Intuition for $\omega^\omega$

I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
5
votes
1answer
54 views

A question on $\omega_1$

For the liner order space $\omega_1$, does it have countable chain condition? (I kown it is not separable, because for any countable subset $A$ of $\omega_1$, there exists an ordinal $\kappa$ ...
5
votes
1answer
69 views

Induction over all ordinal numbers.

I'm starting to learn set theory. I've already learned transfinite induction and I started to wonder if we could do the same on a well-ordered proper class, like the class of all ordinal numbers. It ...
6
votes
3answers
122 views

Ordinal exponentiation - $2^{\omega}=\omega$

This is my understanding of ordinal arithmetic - two ordinals are the same as one another if there is an order-preserving bijection between them. So for instance $$1+\omega = \omega$$ because if ...
2
votes
2answers
64 views

$\omega_1^{CK} - \omega$ - infinite or finite set? And boundary

I am curious whether $\omega_1^{CK} - \omega$ would result in a finite set or infinite set. Does anyone know what happens? Edit: OK, let me add one more question: Suppose that we take $\omega \cdot ...
1
vote
0answers
81 views

Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
8
votes
1answer
89 views

what does it mean that constructible universe is definable from ordinals?

I know how constructible universe is created, but I also separatedly read that the universe is definable from ordinals - so I am wondering what it really means.
1
vote
1answer
54 views

Disjoint unions of limited cardinality: is the following conjecture/proof correct?

Is the following conjecture about collections of sets correct? And if so, does the proof work? Conjecture. Let $\kappa$ denote an infinite cardinal number, and let $\mathcal{K}$ denote a collection ...
0
votes
1answer
68 views

Does there exist a set theory where the following non well-founded definition of the ordinals is possible?

I was wondering if there is a non well-founded set theory such that the following implementation of the ordinals is possible. $0 = \emptyset$ $1 = \{1\}$ $2 = \{1,2\}$ ... $\omega = ...
1
vote
1answer
32 views

Comparing how “high” the ordinals go in one model of ZFC versus another.

Given two transitive models of ZFC, is there a way of comparing how "high" the ordinals go in one model versus another?
2
votes
0answers
98 views

How to derive Church-Kleene ordinal

Crossing-out: (How does one prove the existence of Church-Kleene ordinal? Also, why is it labeled as $\omega_1^{CK}$? And why is it first ordinal not hyperarithmetical, and is the first admissible ...
3
votes
2answers
93 views

Another basic Ordinal arithmetic question

I am getting confused about ordinal arithmetic I am trying to figure out what $\omega \cdot (\omega+1)$ is. The fact that the distributive law applies from the left side makes me think that the ...
1
vote
1answer
55 views

Help with basic Ordinal arithmetic

I am having trouble with some simple ordinal arithmetic. I am trying to figure out what $(\omega+2)\cdot \omega$ is. If you represent $\omega + 2$ as: $\omega+2 = 0_0 < 1_0 < 2_0 < ... < ...
1
vote
1answer
53 views

A set bijective with an ordinal

Lemma I.10.10 in Kunen's set thoery (2011) claims that a set $A$ is well-orderable iff $A$ is bijective with an ordinal. He does not prove this, and one direction is clear to me. However, I can't ...
3
votes
2answers
96 views

How to think about ordinal exponentiation?

I'm just trying to understand better how to see $\alpha^{\beta}$ for an arbitrary ordinal. I've already know that one can think about $\alpha . \beta$ as $\langle \alpha \times \beta, AntiLex\rangle$ ...
2
votes
2answers
131 views

Does the set of all ordinals strictly dominated by a given set exist in ZF?

How do I prove that $$ \{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V,$$ assuming $A\in V$? I know that if AC is assumed, this set is equal to $\mbox{card}(A):=\mu_\alpha(\alpha\approx A)$, and ...
5
votes
1answer
93 views

$\varepsilon$-numbers

An ordinal $\xi$ is an $\varepsilon$-number (where does this name come from?) if $\omega^\xi = \xi$. Is the set of all countable $\varepsilon$-numbers closed in $\omega_1$? In other words, does an ...
4
votes
2answers
59 views

Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
4
votes
2answers
98 views

Two questions regarding Ordinal Numbers.

I'm trying to prove that there is an uncountable ordinal all which members are countable ordinal. This is fairly easy if I can state that the class of all countable ordinals is a set and then take the ...
7
votes
2answers
112 views

Non-measurable subset of $\omega_1$

Consider $\omega_1$ equipped with the order topology. Then Borel subsets of $\omega_1$ are precisely those which contain a closed and unbounded set or the complement contains such a set. There must be ...
4
votes
2answers
121 views

Ordinals definable over $L_\kappa$

Suppose $\kappa$ is an uncountable cardinal, with $L_\kappa$ an admissible set (i.e. a model of Kripke–Platek set theory). Let $<_\gamma \subseteq \kappa \times \kappa$ denote a wellordering of ...
1
vote
2answers
150 views

What is the cardinality of the countable ordinals?

Every countable ordinal $\alpha$ can be written uniquely in Cantor canonical form as a finite arithmetical expression, say $C(\alpha)$. We thus have the 1-1 correspondence between the countable ...
2
votes
1answer
94 views

Proving properties of enumeration of infinite cardinals

I am doing the following exercise from Just/Weese: where $F$ is defined as follows: (a) I'm not sure whether the following passes as "convince myself": Apparently $F$ is defined for all ...
5
votes
1answer
135 views

If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$?

Greets This is from exercise 3.4 of Thomas Jech's "Set Theory", stated: "Show that $\Gamma(\alpha\times{\alpha})\leq{\omega^{\alpha}}$. Thus $\Gamma(\alpha\times{\alpha})=\alpha$ for all ...
0
votes
1answer
128 views

Does every infinite well-ordered set have an initial segment isomorphic to $\mathbb N$?

This is an exercise question from Chapter 2 of A Course in Galois Theory by D.J.H. Garling: Suppose that $(A,\leq)$ is an infinite well-ordered set. Show that there is a unique element $a$ such that ...
0
votes
0answers
62 views

Problem 1.45 (Boolean-valued ordinals) (J.Bell, boolean-valued models and indipendence proof, 3rd edition)

Problem 1.45 (Boolean-valued ordinals) (J.Bell, boolean-valued models and indipendence proof, 3rd edition) (i) Show that, for any formula φ(x), $ [[ ∃α .φ(α)]] = \bigvee_α [[φ( \hat{α})]]$ and ...
3
votes
1answer
31 views

Proof of every cofinal subclass of $\mathbf{ON}$ is proper

Can you please tell me if my proof of the following claim is correct? Thank you! Claim: Every cofinal subclass of $\mathbf{ON}$ is proper. Proof: Let $A \subset \mathbf{ON}$ be cofinal. Assume $A$ ...
1
vote
1answer
119 views

How to show the following equality of ordinal exponentiation: $n^{\omega^\omega}=\omega^{\omega^\omega}$?

Let $n > 1$ be a positive integer. Prove that $n^{\omega^\omega}=\omega^{\omega^\omega}$ I was able to prove that, since $ω$ is a limit ordinal, and $n > 1$, $$n^\omega= ...
1
vote
1answer
71 views

Decreasing sequence of ordinals must stabilize

If $x_1, x_2, x_3, \ldots$ are ordinals such that $x_1\ge x_2\ge x_3\ge \ldots$, then there exists a positive integer $n$ such that $x_n=x_{n+1}=x_{n+2}=\ldots$. I think that this statement is true, ...
2
votes
1answer
59 views

The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals.

I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is: ...
9
votes
3answers
630 views

Curious facts about ordinal numbers

I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule ...
1
vote
1answer
113 views

Ordinal non-commutative addition example

I know this is a newbie question, so please bare with me :) I'd like to prove within ZF axiomatic set theory that the addition of two ordinals is not commutative. In particular, I'd like to prove ...

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