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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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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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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Can we implement $\omega^{CK}_1$ using $\omega^{CK}_2$ as an oracle?

Let $\omega^{CK}_1$, $\omega^{CK}_2$ denote the first two admissible ordinals greater than $\omega$. Suppose we have an unknown well-ordering of $\mathbb{N}$ of the order type $\omega^{CK}_2$ as an ...
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Colourful class function

Background. We're in $\mathsf{ZFC}$, and I can use the principle of $\epsilon$-induction, but not (directly) the $\epsilon$-recursion. Problem. Let $F : V \to V$, where $V$ is the class of all the ...
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How Should I Do the Inductive Case of this Proof on Ordinals?

Question Prove that for all ordinals $\alpha$, $V_\alpha=\{x:\rho (x)<\alpha\}$. Note: The function $\rho$ is a rank function. Attempt I did the proof by induction on ordinals. I started with the ...
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Strictly decreasing function from an ordinal to $\mathbb{R}$

Context: We are working in $\mathsf{ZFC}$. Problem: Given a poset $(P,<)$, let $\text{Dec}(P)$ be the set of all strictly decreasing function $f : \alpha \to P$, where $\alpha$ is an ordinal number....
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Finding the first term of a Goodstein sequence whose expression has maximum exponent

Let $n$ be a natural number. When constructing the Goodstein sequence $(n)_{k}$, we start with $(n)_{1}=n$ written in complete base $2$, and we proceed recursively. That is, if $(n)_{k}$ has been ...
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Proving $\alpha\subset\beta\implies\alpha\in\beta$ for ordinals $\alpha$ and $\beta$

From Jech's Set Theory: Lemma 2.11. (iii) If $α\ne β$ are ordinals and $α ⊂ β$, then $α ∈ β$. Proof: If $α ⊂ β$, let $γ$ be the least element of the set $β − α$. Since $α$ is transitive, it follows ...
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Which are the countable ordinals $\lambda$ such that, for every sequence of ordinals $\alpha_i\ (i\in\mathbb{N})$ such that $\$it is strictly increasing for all sufficiently large $i$ $\ $$\alpha_i&... 3 votes 1 answer 146 views Is the empty set always an 'implicit member' of all sets under a pure set theory? Pure set theory, wherein all objects considered are sets —whose elements are themselves sets, and so forth— is usually thought of as building itself up in an ex nihilo fashion off the empty set \... • 532 1 vote 0 answers 68 views How is transifnite recursion applied? I've been struggling to understand how ordinal addition, multiplication, and exponentiation, along with the Aleph function \aleph, are defined using Transfinite Recursion in Jech's Set Theory or ... • 5,166 2 votes 1 answer 90 views What is cardinality of ordinal exponentiation? Using von Neumann definition of ordinals, is it true that for all cardinal numbers a and b the following equation holds:$$ a^b = |a^{(b)}|$$where on the left side is the cardinal exponentiation ... • 59 0 votes 1 answer 27 views 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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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 ...
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 ...