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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50 views

Noninital ordinal of a set

If $\mathbb R$ is well ordered by an order $\ll$ such that the ordinal of R and that order $\alpha$ is not initial. How can I show that there is a closed interval $[a,b]$ ($a<b$) such that the ...
3
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1answer
147 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...
3
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1answer
169 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
3
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0answers
141 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
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3answers
93 views

Why $\omega+1$ and $\omega^2$ are not cardinal numbers?

I see the following definition of cardinal number in notes: An ordinal $\alpha$ is a cardinal number if $|\beta|<|\alpha|$ for all $\beta\in\alpha$. Why $\omega+1$ and $\omega^2$ are not cardinal ...
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2answers
92 views

For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

I don't understand the proof to the a/m claim. How we know that $\eta < \kappa$ and $(\alpha,\beta)<_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone ...
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1answer
52 views

Naturally definable order relations

Each and every function $f:X\rightarrow Y$ between two totally ordered sets $(X,\leq_X), (Y,\leq_Y)$ induces a relation $\preceq$ on $X$ by $$x \preceq x' :\equiv \begin{cases} x &\leq_X x' ...
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3answers
203 views

How uncountable is the set of countable ordinals?

From the answers (and comments) to this question on the uncountability of countable ordinals I don't get a lucid picture: How can I see that the uncountability of the set of countable ordinals ...
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1answer
46 views

If $\alpha,\beta$ are at most countable, so are $\alpha+\beta,\alpha\cdot\beta,\alpha^\beta$

If $\alpha$ and $\beta$ are at most countable ordinals, prove the following are at most countable: $\alpha + \beta$ $\alpha \cdot \beta$ $\alpha^{\beta}$
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1answer
41 views

A problem on ordinal arithmetic.

Let $0 < \alpha \leq \beta$ be ordinals. Could anyone advise me on how to prove there exists $\delta $ and $\xi$ such that $\xi < \alpha$ and $\alpha \ . \delta + \xi = \beta?$(ordinal ...
6
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0answers
223 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
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1answer
56 views

Reducing ordinals in the representation of a limit ordinal as $\bigcup_{\beta < \alpha} \beta$

Let $\alpha$ be a limit ordinal such that $\kappa$ < $\alpha$ < $\kappa^{+}$ where $\kappa$ is initial ordinal (so $|\alpha| = \kappa$). I want to know whether or not I can find a sequence ...
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1answer
47 views

Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$

I am getting repeatedly lost trying to approach this question: Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$ I can fairly well grasp the idea that the ...
2
votes
1answer
94 views

Uncountable compact sets in ordinal topology

The question I am interested in is the following: Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied with order topology. Prove ...
2
votes
1answer
65 views

A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
2
votes
1answer
158 views

Let $\gamma $ be an ordinal. Prove there is an ordering preserving $f:\gamma \to \mathbb{R}$ iff $\gamma < \omega_1.$

Let $<$ be the usual ordering on $\mathbb{R}.$ If $\gamma$ is an ordinal, then $f:\gamma \to \mathbb{R}$ is ordering preserving if $\forall \alpha \in\gamma \forall \beta \in \gamma [\alpha \in ...
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2answers
39 views

Existence of certain $\left\langle{\alpha_n | n \in \omega}\right\rangle$

Let $\beta$ be a countable limit ordinal. Prove $\exists$ sequence $\left\langle{\alpha_n | n \in \omega}\right\rangle$ with the following properties: $(1): \alpha_0 = 0\;;$ $ (2): \forall n \in ...
2
votes
1answer
62 views

Functions from ordinals to ordinals

I'm trying to solve this problem, which appears in Schimmerling's "A Course in Set Theory." Problem. Find two functions $$f:\omega\rightarrow\omega\cdot2$$ and ...
2
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2answers
114 views

Proving, for every three ordinals $\alpha,\beta,$ and $\gamma$, that $\alpha^{\beta+\gamma}=\alpha^{\beta}\alpha^{\gamma}$

I am trying to prove, for every three ordinals $\alpha,\beta,$ and $\gamma$, that $\alpha^{\beta+\gamma}=\alpha^{\beta}\alpha^{\gamma}$. Proof is by induction on $\gamma$. Basis: If $\gamma = 0$, ...
2
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1answer
88 views

$\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$

Prove: $\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$. From left to right: if $\omega >\beta: \alpha=\omega^\alpha> ...
3
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1answer
125 views

Ordinals: if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$

I'm trying to proof that if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$, where $\omega$ is the least infinite ordinal. I started with transfinite induction on ...
0
votes
1answer
59 views

How to compute a Von Neumann Successor - ∪∪7?

I wonder if I have taken the correct approach to demonstrating what ∪∪7 is, using Von Neumann's proposal? And if I am indeed correct? My answer appears a little odd to me! Question: What is ∪∪7? By ...
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0answers
48 views

Additively indecomposable ordinals

As a continuation to my question here: Ordinal inequalities. I have tried to write down a proof for this claim: If $\alpha$ is of the form $\omega^\delta$ for some ordinal $\delta$, then, For every ...
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1answer
84 views

Ordinal inequalities

I have a problem with the following exercise: First a definition: $ \mathord{\in} = \{\langle x,y \rangle; x \in y \}$ Exercise: Show that the following conditions are equivalent: a. For every ...
3
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1answer
210 views

Showing ordinal addition and ordinal multiplication are defined

I've been reading absoluteless results in Kunens't latest Set Theory text. After talking about $\Delta_0$ formulas and absoluteness, he mentions that certain concepts are absolute, but not $\Delta_0$. ...
2
votes
1answer
34 views

Can ordinals multiplication be defined by recursion?

I am trying to figure out weather ordinals multiplication be defined by transfinite recursion. It seems a bit problematic to me since multiplication is a function $f: \alpha \times \beta \rightarrow ...
0
votes
2answers
100 views

the Class of Ordinals is a set?

Why the class of ordinals On is not a set? Please dont tell me it's well-ordered till you tell me what you mean by an ordering on something that you dont know in first place if it is a set or not.
3
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2answers
74 views

$f: \omega_1 \to \omega_1$. $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $.

Prove: if $f: \omega_1 \to \omega_1$ is an increasing, continuous, unbounded, function, then $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $. Can anyone give me a tip?
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4answers
42 views

If $A$ is a countable subset of $\omega_1$, then there is an $ \alpha < \omega_1$ such that $A \subseteq \alpha$

Prove : if $A$ is a countable subset of $\omega_1$. Then there exists $ \alpha < \omega_1$ with $ A \subseteq \alpha$. I don't really know where to start, can anyone give a tip first?
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2answers
66 views

An ordinal between an ordinal and its cardinality

Suppose $\alpha, \beta$ are ordinals (the first is not a cardinal) and $|\alpha| \subset \beta \subseteq \alpha$. Then why $|\alpha| \sim \beta?$ I know Cantor–Bernstein theorem does the job ...
2
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3answers
46 views

about definition of a cardinal

Definition: the cardinality of a set $A, |A|$ is the least ordinal s.t. $A \sim \alpha$ Definition: We define a cardinal to be an ordinal $\alpha$ s.t. $\alpha = |\alpha|.$ i.e, an ordinal s.th. ...
2
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1answer
54 views

Distinguishing characteristic and spurious properties

When working in set theory with definitions of somehow primitive notions like ordered pairs (Kuratowski) or ordinal numbers (von Neumann) one is supposed to use only the characteristic properties of ...
2
votes
2answers
46 views

$\bigcup X$ = $\sup(X)$ for a set $X$ of ordinals

Let $X$ be a set of ordinals. Prove that $\bigcup X$ = $\sup(X)$(which is the least ordinal $\ge$ every ordinal in $X$).
1
vote
1answer
60 views

An ordinal is well-ordered by inclusion

Recall that we define $\alpha < \beta$ if $\alpha \in \beta$ and $\alpha \leq \beta$ if $\alpha \in \beta$ or $\alpha = \beta$. Prove that for ordinals $\alpha, \beta$, $\alpha \leq \beta$ iff ...
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vote
1answer
42 views

Showing that intersections are not defined

I'm checking to see why intersections are not defined when looking at the class $A$ defined by: $$ A = ON \cup [ON]^2\;,$$ where $ON$ is the class of ordinals and $[ON]^2$ the class of unordered ...
0
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0answers
47 views

Set of ordinals less than a given ordinal

Can anyone enlighten me about how this set can be well-ordered? That is, given any ordinal $\alpha$ the set of all ordinals less than $\alpha$, $W(\alpha)$, is well-ordered. I can show that this set ...
2
votes
2answers
62 views

Euclidean algorithm for ordinals

I am trying to prove the Euclidean algorithm for ordinals. More specifically: For any ordinals $\alpha, \beta$ where $\beta >0$, there are unique $\gamma, \delta$ such that $\alpha = ...
2
votes
1answer
62 views

How would you “count” $\omega^\omega$

$\omega^\omega$ can be seen as the limit of $\omega^n$ which are all countable sets, and is thus countable. For the latter sets, there is an "easy" way list the elements out, but how would you do it ...
2
votes
1answer
101 views

Is the Church-Kleene Ordinal describable with Kleene's $O$?

Kleene's $O$ is an ordinal notation system that uses certain natural numbers to represent transfinite ordinals. It is a recursive notation system (although it's not decidable whether a number ...
0
votes
1answer
45 views

2-well-orderings on the same set

Can you find a set $X$, and two well-orderings $<_{1}, <_{2}$ of $X$ such that $(X, <_{1})$ is not isomorphic to $(X,<_{2})$. I know $X$ cannot be finite because if it were finite then ...
4
votes
1answer
79 views

Is ordinal analysis a non-recursive project?

A recursive ordinal is an ordinal that is the order-type for some recursive relation (i.e. a recursive well-ordering). We can represent recursive ordinals as natural numbers using Kleene's $O$, an ...
2
votes
2answers
71 views

Is the following claim true: “every ordinal has the empty set as one of its elements”

Since we can encode every element as a set, suppose that we consider only sets in which every element is also a set. My question is, is it true that every ordinal has the empty set as one of its ...
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1answer
186 views

Long line is connected and compact

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.
5
votes
1answer
157 views

What is the proof-theoretic ordinal of the first-order theory of real closed fields?

I recently asked a question on MathOverflow, concerning a predicative second-order theory of real numbers. Now the standard way of developing predicativity in the case of second-order arithmetic is ...
3
votes
2answers
161 views

Two exercises on set theory about Cantor's equation and the von Neumann hierarchy

Good evening to all. I have two exercises I tried to resolve without a rigorous success: Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where ...
2
votes
1answer
57 views

A function on ordinals.

Denote by $L$ the set of all non-zero countable limit ordinals. Is there an injection $f\colon \omega_1\times L\to L$ such that $f(\alpha,\beta)>\beta$ for all $(\alpha,\beta)$ in the domain of ...
4
votes
1answer
81 views

Other criteria for a set to be an ordinal.

I'm reading up on ordinals, and I've come across various definitions for what it means to be an ordinal. The main book that I'm following is Kunen's Set Theory book. He defines an ordinal as a set ...
5
votes
3answers
201 views

A few questions about “small” transitive sets.

I'm currently reading about ordinals, working on various exercises since I haven't formally studied the material before. I'm looking at transitive sets. A set $z$ is said to be transitive if and only ...
6
votes
1answer
79 views

Are there any nontrivial doubly-well-ordered sets?

A set $A$ is well-ordered (by an ordering $R$) if there is an $R$-minimal element in every nonempty subset of $A$. Call $A$ doubly well-ordered (by $R$) if $R$ well-orders $A$ and $R^c$ (converse) ...
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1answer
111 views

Construction of an infinite number type and other ideas

Construct an infinite number(?) that has a beginning, an infinite middle, and a end; such as 1000...0001, or 98111...1114 etc. Has this type of number been explored? Under some simple multiplications, ...