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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Is $\operatorname{rank}(A)\subseteq\operatorname{TC}(A)$?
This question came to me when I was thinking about rank and transitive closures. Let $A$ be a set of rank $\alpha$ and let $\operatorname{TC}(A)$ denote the transitive closure of $A$. Is it true then ...
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Can $\sf{ST}$ construct an infinite class wellordered ordered by $\in$?
Assume the axioms of Extensionality, Empty Set, and Adjunction (meaning that $S\cup\{x\}$ forms a set for any $S,x$). Notice that we do not have Specification as an axiom, which makes this theory very ...
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If I have a sequence $a_0, a_1, a_2, \cdots$ , then is expressing the limit of this sequence as $a_\omega$ sensible?
If I have a sequence created by some rule which comes to a limit , then I can express it as $a_0, a_1,a_2,\cdots$.
If I said $\lim_{n \to \infty} a_n = a_{\omega} $ , is that a sensible thing to do ?
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Link between a theory’s proof-theoretic ordinal and the fastest-growing function it can prove total
When I’ve tried to read up on proof-theory I’ve come across this point multiple times - that given a well-founded fast-growing hierarchy, the index of the fastest-growing function f that T can prove ...
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Is $\aleph_0$ "bigger", or "smaller" than $\omega$? What sets are of cardinality $\omega$ or $\aleph_0$?
I am new to set theory and need a little clarification.
Cardinals are generalization of natural number, so they can tell us how big a set is. After we run out of natural numbers 1,2,3,... 10000000,.......
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Order types of modified gap-embedding relations on finite trees
Kruskal's tree theorem states that finite trees are well-quasi-ordered under homeomorphic embedding (i.e. inf-preserving injections). In order to prove the Robertson-Seymour theorem, the "...
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Showing that $\bigcap A$ is the least element for the set $A$ where $A$ is a set of ordinals.
The notes I am reading define a set $x$ to be an ordinal provided $x$ is transitive and every element in $x$ is transitive.
Let $A$ be a set of ordinals. I have shown that $\bigcap A$ is an ordinal. I ...
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What is the largest known "computational" ordinal
I am interested in the computational implementation of ordinals. What I mean by that, is a data structure T and a function/algorithm "compare" that takes two arguments of type "T" ...
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A sequence of continuum hypotheses
The continuum hypothesis asserts that $\aleph_{1}=\beth_{1}$. Both it and its negation can be consistent with ZFC, if ZFC is consistent itself.
The generalised continuum hypothesis asserts that $\...
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Can cardinality $\kappa$ exist where $\forall n\in\mathbb{N} \beth_n<\kappa$,$\kappa<|\bigcup_{n\in\mathbb{N}}\mathbb{S}_n|$,$|\mathbb{S}_n|=\beth_n$
The Wikipedia article on Beth numbers defines $\beth_\alpha$ such that $\beth_{\alpha} =\begin{cases}
|\mathbb{N}| & \text{if } \alpha=0 \\
2^{\beth_{\alpha-1}} & \text{if } \alpha \text{ is a ...
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Can the class of ordinals be extended even further? [duplicate]
Is it possible for anything to come after all ordinals? I don't see why not. For example, one can take a non-ordinal set $S$, and then add in all the ordered pairs $(\alpha, S)$ to $ON$, where $\alpha$...
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Stuck on Jech's Set Theory Exercise 2.3
From Jech's Set Theory:
Exercise (2.3). If $X$ is inductive, then $X\cap\text{Ord}$ is inductive. $\textbf{N}$ is the least nonzero limit ordinal, where $\textbf{N} = \bigcap\{X:X\text{ is inductive}\...
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Ordinal addition with limit ordinals, as in Kunen.
This definition of ordinal adddition is taken from Kenneth Kunens "Set Theory: An Introduction to Independence Proofs":
$\alpha + \beta = \text{type}(\alpha \times \{0\} \cup \beta \times \{...
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Is the first-order theory of the class of well-ordered sets the same as the first-order theory of the class of ordinals?
Consider the class $K$ of well-ordered sets $(W,\leq)$. Although that class is not first-order axiomatizable, it has an associated first-order theory $Th(K)$. Now consider the class of ordinals $On$, ...
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Set of ordinals isomorphic to subsets of total orders
Background. Given a poset $(S,<)$ we'll indicate with $\tau(S,<)$ the set of all the ordinals which are isomorphic to a well ordered subset of $(S,<)$. We're in $\mathsf{ZFC}$.
Questions.
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