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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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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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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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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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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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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Constructible subsets of an ordinal - an alternate definition?
Work over ZFC. Recall the standard definition of $\mathrm{Def}(X)$, the set of constructible subsets of the set $X$. This can be considered as taking all those subsets of $X$ that satisfy some first-...
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Does every ordinal have a well-defined "next" limit ordinal?
I would like to know whether every ordinal $\alpha$ has a well-defined "next limit ordinal", i.e. a least limit ordinal $\beta$ such that $\beta > \alpha$.
I understand from this ...
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Ordinals multiplication: Does $a^2 b^2=b^2a^2$ imply $ab=ba$? [duplicate]
I found this question in one of set theory past exam: If $\alpha$ and $\beta$ are two ordinals such that $\alpha^2\beta^2=\beta^2\alpha^2$, does it necessarily imply $\alpha\beta=\beta\alpha$?
Clearly ...
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Intuition of Ordinal Number
I am learning the concept of ordinal numbers.
In the book of Set Theory and Metric Spaces by I. Kaplansky (Sec. 3.2, pg. 55), the author states
We attach to every well-ordered set an ordinal number; ...
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References that give the cofinality of ordinal addition, multiplication and exponentiation
$\newcommand{\cf}{\operatorname{cf}}$
Let $\alpha$, $\beta$ be ordinals. I believe that we have
\begin{align*}
\cf(\alpha+\beta)=\cf(\beta),\quad\beta\neq 0;\quad\cf(\alpha\cdot\beta)=\begin{cases}\cf(...
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