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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How is transifnite recursion applied?
I've been struggling to understand how ordinal addition, multiplication, and exponentiation, along with the Aleph function $\aleph$, are defined using Transfinite Recursion in Jech's Set Theory or ...
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What is cardinality of ordinal exponentiation?
Using von Neumann definition of ordinals, is it true that for all cardinal numbers $a$ and $b$ the following equation holds:
$$
a^b = |a^{(b)}|
$$
where on the left side is the cardinal exponentiation ...
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Find unbounded sequence in an ordinal product of regular uncountable cardinals
I'm a little stuck here (and should mention that I lack experience with unbounded sets in the transfinite):
Say we have two uncountable regular cardinals $\kappa$ and $\lambda$ where $\kappa < \...
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Ordinal number closed for countable sum
I want to find an ordinal number $\beta$ such that
$\sup\{\alpha_1,\alpha_2,\dotsc\}+1<\beta$ for
any (countable) sequences $\alpha_1,\alpha_2,\dotsc< \beta$.
I know that the least uncountable ...
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Hessenberg sum/natural sum of ordinals definition
I was given the following definition of Hessenberg sum:
Definition. Given $\alpha,\beta \in \text{Ord}$ their Hessenberg sum $\alpha \oplus \beta$ is defined as the least ordinal greater than all ...
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Exercise 7.1.6 Introduction to Set Theory by Hrbacek and Jech
This is exercise 7.1.6 of the book Introduction to Set Theory 3rd ed. by Hrbacek and Jech.
Let $h^{*}(A)$ be the least ordinal $\alpha$ such that there exists no function with domain $A$ and range $\...
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If $X_1$,$X_2$ are wosets isomorphic to ordinals $\alpha_1,\alpha_2$ then $X_1\times X_2$ is isomorphic to $\alpha_2\cdot \alpha_1$
I want to prove the following:
Let $X_1$,$X_2$ be wosets, isomorphic to ordinals $\alpha_1,\alpha_2$ respectively. Then $X_1\times X_2$, with the lexicographic order, is isomorphic to $\alpha_2\cdot\...
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How does one construct $\omega + \omega$ in ZFC?
I can't figure out what axioms allow me to construct it. Most of the answers I've encountered say to define $f(x,y) := y = \omega + x$ and use Replacement, but I don't see how that would work. $+$ ...
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What ways are there to define $\aleph$?
I've seen some posts on this website that consist in providing and comparing different proofs of a theorem (e.g. for Taylor's Theorem or trigonometric identities). Currently I'm reading Holz's ...
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Transitive closure of an ordinal is the ordinal itself?
Let $\lambda\in \mathbb{ON}$, then is it true that $\text{trcl}(\lambda)=\lambda$?
I think this is true, since an ordinal is already a transitive set, so it contains all of its subsets, and their ...
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Why does regularity of an ordinal $\gamma$ imply the existence of a sequence $(\delta_n)$ such that $\delta_n<\gamma$ for all $n$?
Source: Set Theory by Kenneth Kunen.
Lemma III.6.2: Let $\gamma$ be any limit ordinal, and assume that $\kappa:=cf(\gamma)>\omega$. Then the intersection of any family of fewer than $\kappa$ club ...
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Proof of the Reflection Theorem in Kunen?
I'm reading Kunen's Set Theory and the last line of the proof of the Reflection theorem (page 131) is a bit puzzling to me. To those not in possession of Kunen at the moment, the book states verbatim:
...
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How can we simplify $\omega^{\omega\times \omega}$?
I am trying to understand the ordinal $\omega^{\omega\times \omega}$, can it be simplified?
I come across this ordinal number as the fixed point of the normal function $\alpha\mapsto \alpha^\omega$ ...
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Being unbounded in a limit ordinal implies order type is also a limit?
Whilst trying to follow a proof from my lecture notes, I stumbled upon the following:
$$\gamma \text{ limit, }A\subseteq \gamma, \sup A=\gamma\implies \text{type}(A)\text{ limit}$$ It sounds true, ...
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Why do we define cardinality only for well-orderable sets?
I'm revising Kunen's Set Theory and he mentions that when we define the cardinality of a set, we should do it with well-orderable sets. Why is this the case? He points to a section which contains many ...
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