# Tagged Questions

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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### Proving that $\alpha\approx|\alpha|$ is not constructive

$\sf ZF$ tells us that for every ordinal $\alpha$ there is a bijection $f:\alpha\to|\alpha|$, more or less by the definition of $|\alpha|$, the cardinal of $\alpha$. What I would like to show is that ...
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### What is the cofinality of $2^{\aleph_\omega}$

There is a similar question in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook. The book(Karel Hrbacek&Thomas Jech, ...
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### Prob. 6 (c), Sec. 10 in Munkres' TOPOLOGY, 2nd ed: Set of elements having no immediate predecessors in the minimal uncountable well-ordered set

Let $S_{\Omega}$ be the minimal uncountable well-ordered set. Let $X_O$ be the subset of $S_{\Omega}$ consisting of all elements $x$ such that $x$ has no immediate predecessor. Then how to show that ...
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### Ordinal arithmetic $(\omega+1) \cdot \omega$ and $\omega \cdot (\omega +1)$

Here is where I am so far: $(\omega+1) \cdot \omega = \sup\{(\omega +1) \cdot n, n \in \omega\} = \omega^2$ and $\omega \cdot (\omega +1) = \omega \cdot \omega + \omega = \omega^2 + \omega$ Hence ...
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### How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

I know that with choice, the countable union of countable sets is countable, making $\omega^\omega$ and $\epsilon_0$ both countable. Can we show this without choice? E.g. in the case that $\omega_1$ ...
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### Given the axiom of choice, are cardinals ordinals?

Given a model of ZFC, is it correct to talk indistinctly about cardinals and initial ordinals, namely, ordinals $\alpha$ such that for every $\beta < \alpha$, there is no bijection between $\alpha$ ...
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### monotonic laws for ordinal subtraction

I have to prove some monotonic laws for ordinals. It's quite comfortable for me to show monotonic laws of ordinal addition (e.g. $\beta\leq\gamma\Rightarrow\alpha+\beta\leq\alpha+\gamma$). But when it ...
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### Limit Ordinals as Infinite Ordinals and other questions

I am studying set theory and I am confused in the following: Are limit ordinals the same as infinite ordinals? I would say yes since the least non-zero limit ordinal is $\omega$. Infinite limit ...
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### How to prove that every countable compact Hausdorff space is homeomorphic to a well-ordered set with its order topology? [duplicate]

I encountered this problem in a textbook: Let $X$ be a countable compact Hausdorff topological space, I was asked to prove that $X$ is always homeomorphic to a (necessarily countable) topological ...
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### Using ordinal arithmetic calculate the following ordinal numbers

(ω + 1) x ω (ω + 1) x 2 For Question #2, I can simplify to the point where I get (ω + (ω + 1)), but I'm not sure how to proceed from there
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### Continuous function in order topology

I have to prove that for every $\alpha$ in $[0, \Omega)$, there exists a continuous function $f$ from $[0, \Omega)$ to $\mathbb R$ such that the pre-image of $0$ is $\{α\}$. I really have no idea ...
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### A transfinite epistemic logic puzzle: what numbers did Cheryl give to Albert and Bernard?

I expect that nearly everyone here at stackexchange is by now familiar with Cheryl's birthday problem, which spawned many variant problems, including a transfinite version due to Timothy Gowers. In ...
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### A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $\mathbb Q$, I find out that any countable limit ordinal is the union of a sequence of increasing ordinal. Now I'm ...
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### Bijection between well-ordered set and ordinal

We are working in ZF. Problem (part 1). Suppose we have well-ordered set $(M, <)$. How to show that there exists a bijection $f$ between this set and some ordinal $x$ that preserves order? I can ...
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### Problem with base case for transfinite induction

I need to prove this using transfinite induction Let $\alpha, \beta , \gamma$ ordinals If $\beta <\gamma$ then $\alpha + \beta < \alpha + \gamma$ I am trying to prove the statement by ...
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### Uncountable subset of first uncountable ordinal set

Given $g: \omega_1\rightarrow \omega_1$ is a function such that if $x\neq 0$, then $g(x)<x$ ($g$ is not necessarily continuous). Prove that there exists $t\in \omega_1$ such that $\ f^{-1}(t)$ is ...
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### If $\alpha$ and $\beta$ are ordinals then $\alpha \in \beta \Leftrightarrow \alpha \subsetneq \beta$

Def 1. $x$ is $\underline{transitive}$ if $\forall y \forall z (z \in y \in x \Rightarrow z \in x)$. Def 2. $x$ is $\underline{ordinal}$ if $x$ is transitive and all elements of $x$ are transitive. ...
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### Why are the countable ordinals a set?

The countable ordinals are themselves either countable or uncountable. They cannot be countable since that would involve a set with itself as an element, so they are uncountable. If they are ...
I have been (recreationally) trying to expand the notion of ordinal numbers in the same way that the natural numbers $\mathbb N$ are extended to the integers $\mathbb Z$. My objective is to be able ...