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
83 views

Topology on the set of ordinal numbers

This is a problem I encountered while reading Topology : An Outline for a First Course by Lewis E. Ward. Suppose $\Omega$ denotes the smallest ordinal number with uncountably many predecessors. Let ...
2
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1answer
62 views

An ordinal number which satisfies $\omega^{\alpha} = \alpha$

Is there an ordinal such that $\omega^{\alpha} = \alpha$? It seems to me there should be, but I can't explicitly point it out. I know it is possible to prove $\alpha\leq\omega^{\alpha}$, but the ...
4
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1answer
53 views

Order type of the set of all the limit elements in $\omega^{\omega}$

What do you think is the order type of $\{\alpha\in\omega^{\omega}:\alpha\ \text{is a limit ordinal number}\}$? My first thought was $\omega$, but when I visualize the above mentioned set, ...
5
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0answers
88 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
3
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1answer
45 views

Ordinal arithmetic inequality

I have 3 ordinals $\alpha,\beta,\gamma$, $\gamma\neq 0$. Is this implication true? $$\alpha<\beta\implies\gamma\cdot\alpha<\gamma\cdot\beta$$ I have reason to believie it is not, namely ...
4
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1answer
58 views

What's the least class of ordinals closed under successor and the limits of omega-sequences? [duplicate]

What is the smallest class $S$ of ordinals that contains $0$, closed under successor and the limits of omega-sequences, i.e., $$ \forall i \in \omega [\alpha_i \in S] \Rightarrow \sup_{i} \alpha_i \in ...
5
votes
1answer
52 views

Is noetherianity really about cardinals or ordinals?

If $\kappa$ is any cardinal, then one may define a "$\kappa$-Noetherian" ring as a ring such that for any module that has a generating set $S$ satisfying $|S|< \kappa$, then any submodule also has ...
0
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1answer
39 views

Set theory proof for $\min C = \bigcap _{\alpha \in C} \alpha$

Let $C$ be a set of ordinals a) $\sup C$ is the least ordinal $\beta$ such that for all $\alpha\in C$, $a\leq \beta$. b) If $C$ is transitive (and thereby ordinal), then $\sup C \notin C$ if an only ...
2
votes
2answers
107 views

Two ways of defining rank of a set

I am studying set theory, and I have some difficulties in understanding how people define the notion of rank there (I hope, specialists in logic will excuse me for this). As far as I understand, ...
5
votes
1answer
69 views

Does forcing preserve the least undefinable ordinal from a model of ZFC?

Let $M$ be a transitive model of ZFC. For convenience let assume that $M$ is countable. Now let us consider the least undefinable ordinal $\vartheta_M$ which is not definable from elements in $M$ and ...
2
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1answer
47 views

Elements of the power set of $\textit{On}$ are elements of $\textit{On}$?

I don't know the truth value of this statement so this is just my own attempt. I am attempting to prove this in NBG set theory. Let $X \in \textit{P(On)}$ By the definition provided in my notes $X ...
3
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1answer
29 views

Distributivity of ordinal arithmetic

Let greek letters be ordinals. I want to prove $\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma$ by induction on $\gamma$ and I already know it holds true for $\gamma = \emptyset$ and $\gamma$ a ...
0
votes
1answer
27 views

Large ordered family of set inclusions complete?

I'm not an expert on set theory, so this might be trivial: Let $\dots \subseteq X_{\alpha} \subseteq X_{\alpha+1} \subseteq \dots$ be a chain of set inclusions, indexed by ordinals(!), s.t. ...
3
votes
2answers
75 views

$\omega \ \oplus 2 \neq 2\ \oplus \omega $ in NBG set theory

I'm struggling to rigorously prove $\omega \ \oplus 2 \neq 2\ \oplus \omega$ in context of NBG set theory. I haven't really seen a full proof anywhere besides the basic structure. Please direct me ...
5
votes
2answers
120 views

Proving the Powerset Axiom for hereditarily finite sets

Consider $\mathsf{ZF}$, and relace the Axiom of Infinity with its negation. This gives us the theory of hereditarily finite sets. Its universe is $V_\omega$. Intuitively, I feel that I can construct ...
7
votes
1answer
87 views

Fixed points in the enumeration of inaccessible cardinals

Let inaccessible cardinal mean uncountable regular strong limit cardinal. Consider $\mathsf{ZFC}$ with an additional axiom: For every set $x$ there is an inaccessible cardinal $\kappa$ such that ...
0
votes
1answer
30 views

Definition of a recursive ordinal

I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows: "...an ordinal $\alpha$ is said to be recursive if there is a recursive ...
1
vote
2answers
45 views

Is $\varnothing$ a limit ordinal?

This question is from Goldrei's Classic Set Theory: Let $\lambda$ be an ordinal. If $\cup\lambda=\lambda$, then $\lambda$ is a limit ordinal. But what if $\lambda=\varnothing$? I think I am ...
2
votes
2answers
84 views

Why is every limit ordinal a multiple of $\omega$?

An ordinal $\alpha$ is a limit ordinal if and only if $\alpha = \omega\cdot\beta$ for some $\beta$. I have checked through transfinite induction that $\omega\cdot\beta$ is always a limit ordinal, but ...
1
vote
0answers
60 views

Subset of rational numbers isomorphic to $\omega^{\omega+1}$

Does anyone know how to find/prove a rational subset isomorphic to $\omega^{\omega+1}$? I have couple of ideas but I find it hard to prove.
5
votes
1answer
77 views

ordinal isomorphism theorem

The only proof of the theorem stating that every well-order is order-isomorphic to a unique ordinal of which I am aware relies on the Axiom of Replacement. Is this necessary?
3
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0answers
50 views

The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
2
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0answers
45 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say ...
1
vote
0answers
28 views

Cantor-Bendixson rank of a first countable space

This question has been bothering me for quite a while, so let me ask it here. Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index? By a Cantor-Bendixson index I ...
3
votes
2answers
98 views

Countable ordinal

I haven't done ordinal algebra for a long time and I can't remember how to prove that $\omega + \omega$ is a countable ordinal. Precisely what is the bijection between $\omega$ and $\omega \cdot 2$ ? ...
0
votes
1answer
23 views

Union of a chain of cardinalities?

I was trying to understand the union of a chain of cardinalities and I found this equation $$\kappa=\bigcup_{\alpha<\kappa} \alpha$$ for any cardinal $\kappa$ in the answers to this question. Can ...
3
votes
1answer
71 views

Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms

I need to show that for any countable ordinal $\alpha$ there is a set A $\subseteq \mathbb{Q}$ such that (A, <) is isomorphic to ($\alpha, \in$). To do it I am supposed to show the following ...
2
votes
2answers
113 views

Can all ordinals be obtained by taking successors and countable limits?

Define a class $K$ of ordinals inductively as follows: $0=\emptyset\in K$. For all $\alpha\in K$, the succesor of $\alpha$ is also an element of $K$. For every function $f\colon \mathbb N\to K$, the ...
2
votes
0answers
23 views

Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
0
votes
1answer
30 views

Can't the first (/second) transinfinite ordinal replace the first (/second) uncountable ordinal in several counterexamples?

An example of a sequentially compact but not compact space is $\omega_1$. Indeed, any sequence of countable ordinals either has infinite elements below some countable ordinal, or has an ascending ...
4
votes
2answers
195 views

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
votes
1answer
56 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
votes
0answers
25 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
votes
1answer
69 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 ...
2
votes
1answer
26 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
votes
1answer
44 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
votes
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}| = ...
1
vote
1answer
49 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 ...
0
votes
2answers
76 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 ...
3
votes
2answers
370 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
56 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
47 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
42 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 ...
4
votes
1answer
90 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
154 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 ...
2
votes
3answers
92 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
49 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 ...