Questions tagged [ordinals]
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 the positive integers for *transfinite* inductions.
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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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Why do we classify infinities in so many symbols and ideas?
I recently watched a video about different infinities. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \...
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Embedding ordinals in $\mathbb{Q}$
All countable ordinals are embeddable in $\mathbb{Q}$.
For "small" countable ordinals, it is simple to do this explicitly.
$\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in \...
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Conflicting definitions of "continuity" of ordinal-valued functions on the ordinals
I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have $\mathbf{F}(\...
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Principle of Transfinite Induction
I am well familiar with the principle of mathematical induction. But while reading a paper by Roggenkamp, I encountered the Principle of Transfinite Induction (PTI). I do not know the theory of ...
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Examples of transfinite induction
I know what transfinite induction is, but not sure how it is used to prove something. Can anyone show how transfinite induction is used to prove something? A simple case is OK.
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Intuition for $\omega^\omega$
I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
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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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What are some good open problems about countable ordinals?
After reading some books about ordinals I had an impressions that area below $\omega_1$ is thoroughly studied and there is not much new research can be done in it. I hope my impression was wrong. ...
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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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What are the most prominent uses of transfinite induction outside of set theory?
What are the most prominent uses of transfinite induction in fields of mathematics other than set theory?
(Was it used in Cantor's investigations of trigonometric series?)
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When does ordinal addition/multiplication commute?
I'm looking at basic ordinal arithmetic at the moment, and I am aware that in general, $\alpha+\beta\neq\beta+\alpha$ and $\alpha.\beta\neq\beta.\alpha$ for ordinals $\alpha,\beta$.
My question is: ...
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Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$
So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next ...
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Is there an axiomatic approach to ordinal arithmetic?
I've always wondered, is there an axiomatic approach to the arithmetic of ordinal numbers?
If so, I imagine it would be on par with set theory in terms of its proof-theoretic strength.
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Why study cardinals, ordinals and the like?
Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems ...
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