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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How do you find the smallest of homeomorphic ordinals?

I am trying to get a better feel for the topology of ordinals and just received a great answer to this question where the Cantor-Bendixson rank and degree turn out to be a complete homeomorphism ...
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3answers
1k views

Countable compact spaces as ordinals

I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can ...
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203 views

Computable increasing function from $\omega_1^{CK} \to \mathbb{R}$

If $f:\omega_1 \to [0,1]$ such that $f(0)=0$ and such that $f(\alpha)<1$ and $\alpha<\beta$ imply $f(\alpha)<f(\beta)$ then there is a $\gamma$ such that $f(\gamma)=1$. Non-constructive ...
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329 views

Equivalent definitions of ordinals?

The first definition of an ordinal number I found was that an ordinal number is the $\in$-image of a well-ordered set $(A,\lt)$. From this definition it was derived that an ordinal is just the set of ...
3
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1answer
191 views

If $\kappa<\aleph_\alpha$, then $\kappa\leq\alpha$?

I'm having a hard time following this proof. Here $\aleph(\alpha)$ is the cardinality of $Z(\alpha)$, the set of all ordinals $\gamma$ such that $|\gamma|\leq\alpha$. Also, $\aleph_1=\aleph(\aleph_0)$ ...
3
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110 views

Small question on proof that any ordinal is the index of some initial ordinal

I'm reading a proof of a theorem, but an apparently trivial case has tripped me up. The theorem goes as For any ordinal $\alpha$, there is a initial ordinal $\phi$ such that $i(\phi)=\alpha$. Here ...
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2answers
327 views

Is $V$ under ZFC really a proper class?

Is $V$, the union of the von Neumann hierarchy, necessarily a proper class? Or is it only a proper class after you assume that it contains every set? (In that case, $V$ can't be a "set of all sets" ...
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427 views

Can any infinite ordinal be expressed as the sum of a limit ordinal and a finite ordinal?

I've been browsing through Jech's and Levy's texts on set theory, and the ideas of ordinals come up fairly quickly. The idea of a limit ordinal is introduced, which is an ordinal with no maximum ...
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845 views

A transitive set of ordinals is an ordinal

This is Exercise III.2.20 of Bourbaki's Set Theory. (Von Neumann ordinals are actually called "pseudo-ordinals" by Bourbaki, but I simply call them ordinals here) Let $X$ be a transitive set, and ...
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2answers
115 views

Transfinite sequences of topologies

I am not that familiar with arguments using ordinals so this may be pretty simple. Let $T$ be a topology on a set $X$. For each ordinal $\alpha$ let $T_{\alpha}$ be a topology on $X$ satisfying ...
6
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Proving $0$ is an ordinal

In Introduction to Set Theory by J. Donald Monk, he defines ordinal as follows. Definition (1): $A$ is an ordinal iff $A$ is $\in$-transitive and each member of $A$ is $\in$-transitive. $A$ is ...
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The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
7
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2answers
830 views

Non-existence of a surjection $\aleph_n \to \aleph_{n+1}$, without the axiom of choice

Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ ...
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4answers
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Simple example of uncountable ordinal

Can you make a simple example of an uncountable ordinal? With simple I mean that it is easy to prove that the ordinal is uncountable. I know that the set of all the countable ordinals is an ...
4
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2answers
368 views

Order-isomorphic with a subset iff order-isomorphic with an initial segment

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of ...
7
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1answer
710 views

Uncountable ordinals without power set axiom

Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a ...