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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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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First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. ...
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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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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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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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An easy to understand definition of $\omega_1$?

I have two things I'm not sure in 100% about them. The first, is $\omega_1$. I have a little "feeling" of it, but if I'll be asked to define it - I don't know where to begin from. Perhaps it is ...
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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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Does this certain modular property hold for functions on recursive ordinals?

Let $f$ be a function with the following properties. 1) The domain and codomain of $f$ are the recursive ordinals. 2) $f$ is nondecreasing. 3) The set of fixed points of $f$ is unbound. ...
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Do all computable functions on ordinals satisfy this certain modular property?

Consider the following property of a non decreasing function $f$ whose domain and codomain are each the set of ordinals less than the Church-Kleene ordinal, and whose set of fixed points is unbound. ...
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Infinite Ordinal Sum

When working with ordinal numbers, would it be correct to say that: $$\sum_{i=0}^{\infty}1 = \omega$$ Or does this simply not make sense? In the ordinals, does the notation $\sum^\infty_{i=0}$ even ...
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Need help in understanding some basic ordinal concepts

I'm trying to construct a proof that there is no homeomorphism from a subspace of $\omega_1$ to $\Bbb{Q}$ with the usual topology, but first I need to clear up some concepts. This is the definition ...
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Closed disjoint sets in $\omega_1$ implies one of them is countable

Let $A, B \subseteq \omega_1$ be disjoint, closed sets. Show that one of A or B must be countable. Since $\omega_1$ is uncountable, at least one of A or B must be countable, I want to show that ...
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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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Epsilon numbers

Let $\alpha$ be an ordinal number and define $f_\alpha$ as: $f_\alpha(0) = \alpha + 1$ $f_\alpha(n+1) = \omega^{f_a(n)}$ Let $S(\alpha) = \sup\{f_a(n)\ |\ n \in \omega\}$ Then $S(\alpha)$ is an ...
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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 ...
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Existence and Uniqueness of an ordinal

$\underline{\mathbf{Problem:}}$ If $\alpha < \beta$ then $\exists$ a unique $\gamma$ st. $\alpha+\gamma=\beta$, where '$+$' denotes ordinal addition. If someone would be so kind to check the ...
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Is the limit of a recursive sequence of recursive ordinals itself a recursive ordinal?

Is the limit of a recursive sequence of recursive ordinals itself a recursive ordinal? If so, is there a nice proof of this?