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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Peano axiomatics holds on the least limit ordinal

I want to show, that the Peano axiomatics holds on the set $\omega$, where $\omega$ is a least limit ordinal. To be more specific, how to get formalization within the first order logic theory? If $\...
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When a transfinite sequence converges with respect the order topology? Is the limit of a transfinite sequence necessarly its supremum?

Let be $s_\theta$ a transfinite sequence $(\alpha_\nu)_{\nu\in\theta}$ of length an ordinal theta $\theta$. So any ordinal is totally ordered with respect the usual order relation so that any ordinal ...
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Definitions of nice recursive analogues of cofinality?

The cofinality of an ordinal is the minimal order type of a cofinal subset of that ordinal. In the study of admissible ordinals, some concepts that mimic the idea of cofinality appear. One common ...
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Reference for Analysis book in which natural numbers constructed from sets

Could anyone suggest books on Mathematical/Real Analysis that construct natural numbers through sets not Peano axioms? I find construction of natural numbers through sets more convenient. So I ...
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Can Hilbert's Hotel be explained by a difference between ordinal numbers and cardinal numbers

In taking a philosophy of maths course I have been very curious about the notion of infinity, and whether or not it is paradoxical. One thing I have frequently thought is that "infinity" as ...
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$ \mathbb{X}=\left(\left[0,\omega_{1}\right]\times\left[0,\omega\right]\right)\setminus\left\{ \left(\omega_{1},\omega\right)\right\} $ is not T4

I need to show that the subspace $\mathbb{X}=\left(\left[0,\omega_{1}\right]\times\left[0,\omega\right]\right)\setminus\left\{ \left(\omega_{1},\omega\right)\right\} $ of $\left[0,\omega_{1}\right]\...
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Show that omega_1 is normal?

I've already proven the following statement. If $A$ and $B$ are disjoint closed subsets of $\omega_1$, then at least one of them must be bounded. This proof is supposed to be used to prove that $\...
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Obscure question in Smullyan and Fitting: "strengthening of definition by finite recurrence"

Context: self-study from Smullyan and Fitting's "Set Theory and the Continuum Problem" (revised 2010 edition), chapter 3, section 8, Definition by Finite Recursion. They give Theorem 8.1 ...
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Unexplained notation in Devlin's Joy of Sets: theorem 1.7.11, Condition for Woset to be Isomorphic to Ordinal

Context: self-study from Devlin's "The Joy of Sets" -- theorem 1.7.11. Let $(X, \le)$ be a woset such that for each $a \in X$, $X_a$ is isomorphic to an ordinal. Then $X$ is isomorphic to ...
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Are there ordinals $\alpha,\beta$ such that $\alpha+\beta \neq \beta+\alpha$ but $\alpha2+\beta2=\beta2+\alpha2$?

I know that if $\alpha+\beta=\beta+\alpha$, then by associativity $\alpha2+\beta2=\beta2+\alpha2$. Was wondering if the reverse holds? Edit: It can't be possible. $\alpha+\beta < \beta+\alpha$. $\...
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closed set and the supremum in order topology

Suppose $\alpha$ is an ordinal endowed with the order topology, i.e. its basic open sets are generalized open intervals. Given $C \subseteq \alpha$, I want to show (1) implies (2) : (1) For all ...
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Is it true that $\aleph(A_n)<c\implies \sum_{n\in\mathbb{N}}\aleph(A_n)<c$? [duplicate]

Let $\{A_n\mid n\in \mathbb{N}\}$ be disjoint family of sets such that $\forall n\in\mathbb{N},\; \aleph(A_n)<c$. Is it true that $\aleph(\bigcup_{n\in\mathbb{N}}A_n)=\sum_{n\in\mathbb{N}}\aleph(...
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Prove $cf(\alpha) \leq cf(F(\alpha))$ [closed]

I'm trying to prove $cf(\alpha) \leq cf(F(\alpha))$. The other main assumptions are that $\alpha$ is a limit ordinal and F is continuously increasing. I know I'm supposed to use the squeeze tactic, ...
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Transfinite Construction, an intuitive interpretation.

Theorem (Transfinite Construction). Let $W$ be a well-ordered set, and $E$ an arbitrary class. Assume: For each $x\in W$, there is a given rule $R_x$ that associates with each $\varphi\colon W(x)\to E$...
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Is the union of a set of ordinals a nonlimit ordinal?

Let $A$ be an arbitrary set of nonlimit ordinals less than a given $\kappa$, and let the union $A$ be also less than $\kappa$. Does this union have to be a nonlimit ordinal if $\kappa=\omega$; $\...
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Order Isomorphism between $\omega_1$ and $\omega_1 \times \omega_1$

It is well known that $\omega_1$ and $\omega_1 \times \omega_1$ are order isomorphic when the latter is equipped with canonical well ordering. In fact, it can be explicitly given by $$\phi: \omega_1 \...
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$\gamma<\theta$ iff $\gamma\in\theta$

I'm struggling with my homework: Let $\gamma,\theta$ be order types. Then, $\gamma<\theta$ iff $\gamma\in\theta$. I already proved that if $\gamma\in\theta$, then $\gamma<\theta$. To do the ...
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What is the minimal theory for subset relation to be connective over those kinds of ordinals?

If we define ordinal as being a set of all transitive proper subsets of it. Formally $ord(X) \iff X=\{Y \subsetneq X \mid trs(Y) \}$ Where: $trs(X) \iff \forall x \in X \,(x \subseteq X)$ What is the ...
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Axiomatization of $(\mathbb Z, <)$

I'm interested in the axiomatization of the total order $(\mathbb Z, <)$. My idea is to have first the axioms for a total order: $\exists x : x = x$ $\forall x : \lnot(x < x)$ $\forall x : \...
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Questions about the induction on cardinals

From Hereditary Cardinality and Rank : For an infinite cardinal $\kappa$, $$\forall x,\ \textrm{hcard }x<\kappa\rightarrow\textrm{rank }x<\kappa$$ We can show this by induction on $\kappa$. ...
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What does the Kirby-Paris Theorem mean?

I was recently reading about the Hydra game and how it always terminates. What amazed me was how this is not provable in Peano arithmetic. Now as far as I understand PA, it constructs the natural ...
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Relating the cofinality and regular cardinals

A couple of questions about the first part of this answer. $\kappa$ is a regular if and only if whenever $\lambda<\kappa$, and $\{A_i\mid i<\lambda\}$ is such that $|A_i|<\kappa$, then $|\...
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3 votes
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Ordinals without set theory?

I'm interested in whether ordinal numbers can be described by a first-order theory without presupposing ZFC or any particular set theory. Such a theory might look like Peano arithmetic, but ...
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The relation between rank and von Neumann's Hierarchy

$\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\ord}{\text{Ord}}$ (This question, in a sense, is a follow up to ℋolo's answer.) Let $V_\alpha$ be the members of the von Neumann Hierarchy: $$...
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Explain why transfinite induction does not assume that a property must be true for zero.

THIS QUESTION IS NOT A DUPLICATE OF THIS ONE!!! So I would like to discuss the following proof of transfinite induction which is taken by the text Introduction to Set Theory wroten by Karell Hrbacek ...
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Find the smallest cardinal $\kappa$ such that $\omega + \omega$ is the supremum of $\kappa$ smaller ordinals

I'm asked to find the smallest cardinal $\kappa$ such that $\omega + \omega$ is the supremum of $\kappa$ smaller ordinals. By definition, $\omega + \omega = \sup \{\omega + n : n < \omega\}$. I ...
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About the definition of hereditary cardinality

I've seen these two definitions of sets that are hereditarily of cardinality $< \kappa$. $x$ is hereditarily of cardinality $< \kappa$ iff $|trcl(x)| < \kappa$ $x$ is hereditarily of ...
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The axiom of regularity in a fact about $V_\omega$

I'm working out the details in this proof (from here), and I have some questions about the second part ("For the other direction, ..."). First, why is it possible to assume WLOG that $A$ is ...
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Is true that $\bigcup α=α$ if $\alpha$ is a limit ordinal?

Definition Given a set $A$ the membership relation on $A$ is the relation defined by the identity $$ \in_A:=\{a\in A\times A:a_1\in a_2\} $$ Definition A set $A$ is said transitive if ech its element $...
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9 votes
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The definition of cofinality of an ordinal

For an ordinal $\alpha$, define $cf(\alpha)$ to be the smallest ordinal $\beta$ such that there is $f:\beta\to\alpha$ such that $\bigcup f(\beta)=\alpha$. Why is this condition equivalent to $(\forall ...
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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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2 votes
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Minimal models in strong set theories

Given some theory $T$, let $M(T)$ denote the height of the minimal model of $T$, i.e. $\min\{\eta: L_\eta \vDash T\}$. Obviously there are some famous examples, e.g. $M(\textsf{KP}) = \omega$, and $M(\...
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Is every limit ordinal an infinite sum of smaller ordinals?

If $\left\langle \alpha_\gamma : \gamma < \beta\right\rangle$ is a sequence of ordinals of length $\beta$, we define the sum $\sum_{\gamma < \beta} \alpha_\gamma$ inductively as follows: If $\...
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Transfinite limit of an orbit of a dynamical system and rationalizability of symmetric games.

Consider a dynamical system and let $v(t)=(v^1(t),\dots,v^n(t))$ be an orbit. Suppose we are able to prove, for some $k=1,\dots,n$: $$\lim_{t\to\infty} v^k(t)=0$$ Of course, we cannot be sure that $\...
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Correct cardinals vs Stable ordinals

I recently learnt about correct and stable cardinals and ordinals. A cardinal $\kappa$ is called $\Sigma_n$-correct iff $V_\kappa \prec_n V$. An ordinal $\alpha$ is called $\Sigma_n$-stable iff $L_\...
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Is there a measurable function from $[0,1]$ to $ω_1$?

Does there exist a measurable function from $[0,1]$ (with the Lebesgue measure) to $ω_1$ that induces the Dieudonné measure? Definitions: $ω_1$ is the set of all countable ordinals, equipped with its ...
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2 votes
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Iterated $\Pi^1_1$-reflection and non-Gandiness underrepresented in ordinal analyses?

The admissible ordinals $\alpha$ s.t. the supremum of $\alpha$-recursive well-orderings is less than the next admissible $>\alpha$ are called non-Gandy, and according to Madore's "A Zoo of ...
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Definitions of ordinal addition and multiplication

For any two sets $\alpha$ and $\beta$, let $\alpha+\beta = \{\alpha', \alpha+\beta'\}$ and $\alpha\cdot\beta = {\alpha\cdot\beta'+\alpha'}$, where $\alpha'$ and $\beta'$ are variables taking as values ...
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Questions about $\omega_1$ as a space (The Set of All Countable Ordinals)

The Set of All Countable Ordinals, $\omega_1$ I'm trying to understand several things regarding $\omega_1$ and trying to get a better intuition. I have four questions regarding this space (in bold), ...
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The cardinality of the preimage of an ordinal

Define a cardinal as an ordinal $\kappa$ such that for all ordinals $\alpha < \kappa$, $\alpha$ injects into $\kappa$ but is not bijective to $\kappa$. Let $\kappa$ be an infinite cardinal. I'm ...
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How do you proove that TREE(3) is finite in layman terms?

Apologies in advance. I am just a layman who knows about ordinals and know about TREE(n) but I don't know how to prove it is finite. Is there a simple transfinite ordinal proof? I don't know anything ...
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Can we conclude $rng(f) \subseteq \beta$ for some $\beta \in B \cap \omega_1$?

Consider the language $\mathcal{L}=\{ \in\}$. Let $\mathcal{A}$ be the $\mathcal{L}$-structure $(V_\theta, \in)$ for some $\theta> \omega$. Let $\mathcal{B}$ be an elementary substructure of $\...
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Exercises on transfinite induction

Define by transfinite recursion $V(0)=\emptyset, V(\alpha+1)=\mathcal P(V(\alpha)), V(\alpha)=\cup_{\beta < \alpha}V(\beta)$ for $\beta$ a limit ordinal. I'm trying to show the following properties:...
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Is this set of ordinals an ordinal?

Consider the language $\mathcal{L}=\{ \in\}$. Let $\mathcal{A}$ be an $\mathcal{L}$-structure whose domain is some "sufficiently large" Von-Neumann universe and which interprets $\in$ in the ...
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Exponentiating limit ordinals

I have been reading about ordinal arithmetic and I came across this definition for ordinal exponentiation, when the exponent is a limit ordinal and the base isn't. Here is the definition: $$\begin{...
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What is the intuitive meaning of a transitive set?

I'm now working my way through some basic set theory, and I came across the concept of an ordinal. In the book (and many other books), a transitive set is first defined as a set $x$ that satisfies $\...
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Isomorphism between the first the uncountable ordinal $\omega_{1}$ and real numbers

Recently, i have studied the proof of why the set of all countable ordinals $\Gamma$ is an ordinal and why it is an uncountable. (Basically, since it contains all countable ordinals as predecessors, ...
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Supremum in well orderings

Problem *x7.27 from Moschovakis' textbook is asking to define a definite operation $\sup \mathscr E $, such that for every family $\mathscr E$ of well-ordered sets, $\sup \mathscr E$ has the following ...
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Can we define the predicate "ordinal" in ZF-Reg. by recursion?

Working in $\sf ZF-Reg.$ can we define the unary predicate "is an ordinal", denoted by "$\operatorname {od}$", meaning is a von Neumann ordinal, in a recursive manner? The usual ...
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Are the ordinals $\alpha$ so that the stable ordinals $< \alpha$ are unbounded in $\alpha$ equivalent to the nonprojectible ordinals $\alpha$

An ordinal $\alpha$ is called nonprojectible if and only if $\forall \beta (\beta \in X \cap \alpha \implies \min\{\gamma \in X: \beta \in \gamma\} \in \alpha)$, where $\delta \in X \iff L_\delta \...
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