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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What's the rank of this well founded relation?

Definition A tree is an ordered list of trees. (N.B these are finite objects and there is a very simple computable bijection of them with $\mathbb N$) Examples [] and [[],[],[]] and [[[]],[[],[],[]],[...
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Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
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what does it mean that constructible universe is definable from ordinals?

I know how constructible universe is created, but I also separatedly read that the universe is definable from ordinals - so I am wondering what it really means.
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How to compute immediate previous ordinal in von Neumann representation?

I don't know how to state the question more clear: why it is so hard to compute immediate previous ordinal in von Neumann representation? This is about Better representaion of natural numbers as sets?...
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Another basic Ordinal arithmetic question: $\omega \cdot (\omega+1)=?$

I am getting confused about ordinal arithmetic I am trying to figure out what $\omega \cdot (\omega+1)$ is. The fact that the distributive law applies from the left side makes me think that the ...
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What is the cardinality of the countable ordinals?

Every countable ordinal $\alpha$ can be written uniquely in Cantor canonical form as a finite arithmetical expression, say $C(\alpha)$. We thus have the 1-1 correspondence between the countable ...
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Proof that for every $X$ $\exists !$ one initial ordinal

The following is an exercise in Just/Weese: Prove that for every set $X$ there exists exactly one initial ordinal $\kappa$ such that $X \approx \kappa$. An ordinal is called initial ordinal if it is ...
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Is $\omega_{\alpha}$ sequentially compact?

For an ordinal $\alpha \geq 2$, let $\omega_{\alpha}$ be as defined here. It is easy to show that $\omega_{\alpha}$ is limit point compact, but is it sequentially compact?
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Lower Bound of countable order-types of linearly ordered sets

On Page 44, Set theory, Jech(2006) (Show that) there are at least $\mathfrak{c}$ countable order-types of linearly ordered sets. [For every sequence $a = \{ a_n : n \in N\}$ of natural numbers ...
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How to show the following equality of ordinal exponentiation: $n^{\omega^\omega}=\omega^{\omega^\omega}$?

Let $n > 1$ be a positive integer. Prove that $n^{\omega^\omega}=\omega^{\omega^\omega}$ I was able to prove that, since $ω$ is a limit ordinal, and $n > 1$, n^\omega= \bigcup_{\beta<\...
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Decreasing sequence of ordinals must stabilize

If $x_1, x_2, x_3, \ldots$ are ordinals such that $x_1\ge x_2\ge x_3\ge \ldots$, then there exists a positive integer $n$ such that $x_n=x_{n+1}=x_{n+2}=\ldots$. I think that this statement is true, ...
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The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals.

I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is: ...