# 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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### Question about ordinal space [duplicate]

I have question regarding the ordinal space $[0,\Omega]$ where $\Omega$ the first uncountable number. I know it is not first countable and not separable. i am trying to prove it is compact. it is kind ...
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### Could “ω is an ordinal” be proved without axiom of induction?

Let ω (or all natural numbers) be the set defined by the axiom of infinity in ZF system, ordinal be "transitive and well ordered by ∈", and a set is transitive if all its elements are also its subset. ...
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### Series with terms indexed by transfinite ordinals

I learned this past year how to deal with summation of infinite series $a_0 + a_1 + a_2 + ...$ with indices running over the natural numbers. I've also learned about what comes "after" the natural ...
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### Partial order on the set of ordinal functions

Let $\kappa$ be a regular ordinal. Say that for $f,g: \kappa \rightarrow \kappa$, $f \leq g$ iff $f(\alpha) \leq g(\alpha)$ for sufficiently large $\alpha$. Now define $(X_{\kappa},\preceq)$ as the ...
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### When is the union of a set of ordinals a limit ordinal?

Let $Y$ be a set of ordinals such that $\bigcup Y \not\in Y$. Then $\bigcup Y$ is a limit ordinal. Does this hold? It's taken from Drake's Set Theory, An Introduction to Large Cardinals, page 114, I ...
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### Definition of Ordinals in Set Theory in Layman Terms

I've taken a huge interest in the mathematical concept of infinity and often been contemplating the same over years. But the fundamental concept of set theory ordinals continues to evade my ...
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### Incompressible countable Total-ordering implies well-ordering i

A totally-ordered set (S,<) is incompressible if $(S,<) \cong (T,<)$ and $S \supseteq T$ implies $S = T$. Is it true that if $S$ is incompressible countable and totally-ordered then $S$ is ...
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### pentation of uncountable ordinal

$\omega↑↑↑\omega = \Gamma_0$ Feferman's ordinal collapsing function: $\ θ(Г_Ω) = θ(Ω_2)$ $\ θ(Г_{Ω_2}) = θ(Ω_3)$ ...etc. This means: $θ(Ω_2) = θ(Ω↑↑↑Ω)$ $θ(Ω_3) = θ(Ω_2↑↑↑Ω_2)$ ...etc. ...
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### Ordinal Numbers: is it possible to have an isomorphic copy of an ordinal $\alpha$ inside an ordinal $\beta < \alpha$?

Could there be a function $f:\alpha \to \beta$ order-preserving and injective such that $\alpha$ and $\beta$ are ordinal numbers with $\alpha > \beta$?
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### Is ω really the first ordinal transfinity? [closed]

The Internet claims that ω is the first ordinal transfinity. But what about ω-1? Isn't that a ordinal transfinity, and isn't it before ω? Kind of like ω/2? I guess I lack an understanding of what ω ...
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### Ordinal Arithmetic: $n+\alpha=\alpha$

Let $\alpha$ be an ordinal. Show that $n+\alpha=\alpha$. Proof: Let $W$ be the set with ordinal equal to $\alpha$. Let $A_n=\{a_1,a_2,\cdots, a_n\}$ be the set disjoint from $W$ with its ordinal ...
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### Checking whether some sets are clubs in $\aleph_2$ and $\aleph_1$

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question: which of the following sets are clubs, contain a club,...
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### Order type of a sum ($\bigcup$) of sets

A quick question. Is $$\textrm{ot}(\bigcup\limits_{\gamma <\lambda}\alpha_{\gamma})=\bigcup\limits_{\gamma <\lambda}\textrm{ot}(\alpha_{\gamma})?$$ where $\textrm{ot}$ stands for the order type (...
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### Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
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### Ordinals recursed

Consider any limit ordinal $\alpha$. It seems intuitively obvious to me that for any such ordinal there is $f:ORD \to ORD$ (with $ORD$ being the class of ordinals) with $f(1)=\alpha$ and $f$ ...
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### How well-behaved can well-orders on $\mathbb{R}$ be?

Well-orders on $\mathbb{R}$ are interesting because they witness the "alephness" of the cardinality of the continuum, and they're therefore connected to the continuum problem. It seems reasonable, ...
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### Fixed points of normal function form proper class

A normal function is a class function $f$ from the class of ordinals to itself such that $f$ is strictly increasing and continuous. We can easily show that for every ordinal $\alpha$ there is an ...
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### Why does $\epsilon_0 = \omega^{\epsilon_0}$

I saw this on wikipedia and wasn't sure why. It seems obvious given the definition but how can we be sure? Also, what would be $\epsilon_0 \bullet \epsilon_0$? Would it be $\epsilon_0$? Thanks
The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on ...
For example, does the following hold? $$\left(1+\frac1\omega\right)^\omega=e$$