Questions tagged [transfinite-recursion]
Questions dealing with set-theoretic functions defined by transfinite recursion.
169 questions
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Cartesian product interpretation of transfinite numbers up to $\epsilon_0$ as sets of tuples; difficult to compose $-$, $/$
Interpreting expressions of $k\in\mathbb{N}, \omega, $ and $\text{+ - */^}$ as types of tuples
Section 1 - Preamble
The ordinal-like expressions tend to cause confusion, so in this question I define ...
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242
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Constructing the set of all finite tuples using $F$, showing that it has size $\epsilon_0$, and that it must contain all finite tuples?
Ok, this construction is still a bit wrong and still a bit vague, but I have a better one. I can't delete the question as too many people have commented, but I've learned enough more that I think a ...
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Recursive definition of the atomic formulas in Boolean Valued Models of ZFC
I've been reading J. L. Bell's book "Set Theory: Boolean-Valued Models and Independence Proofs" and when the book gives this definition of the Boolean truth for atomic formulas, $u\in v$, $...
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Reverse mathematics of Spector-Gandy theorem
E community, I am new to reverse maths. I would like to know how I can prove the equivalence between the proof of Spector-Gandy theorem (in Sack's book)
and some second order arithmetic subsystem. ...
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Proof that a topological space X is sequential if and only if iterating sequential closure stabilizes and it does at most at the ordinal $\omega_1$
I'm currently studying sequential spaces at an introductory level and came across a proposition in an article by A. V. Arhangel'skiĭ and S. P. Franklin ("Ordinal invariants for topological spaces&...
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75
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How to know the "size" of structures defined by transfinite recursion
I was doing a Models of Computation course and noticed that various sets (or classes?) were defined in a Haskell-like recursive notation, e.g. for some nonempty set $A$ the definition
$$
\begin{align}
...
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Alternative proof that if $\mathcal{M}$ is the σ-field generated by $\Lambda$, every $E\in\mathcal{M}$ is generated by countably many $A\in\Lambda$
Alternative Solution for Exercise in Folland's Real Analysis
I found a solution to Exercise 5 in Section 1 of Folland's Real Analysis (Folland, 1999, p.24) using transfinite induction, but I ...
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Maximum cardinality of a collection of sets of integers with pairwise finite intersections [duplicate]
Let $\mathcal{C}$ be a collection of subsets of $\mathbb{N}$ such that any two distinct $U,V\in\mathcal{C}$ have finite intersection $U\cap V$. Then what can be said about the cardinality of $\mathcal{...
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Why do we need recursion to define ordinal arithmetic?
On notation: $\def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\ZF{\textbf{ZF}} \def\Suc{\text{Suc}} \def\Lim{\text{Lim}} \def\f{\varphi} \def\x{\xi} \def\y{\psi} \def\ON{\text{ON}}$ let $\ON$ be the ...
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Clarifying a Subtlety in Primitive Recursion on the Ordinals
I understand the Primitive Recursion on $ON$, but I only just noticed a subtle point that occurs when actually defining objects via this recursion and I would like some help getting things straight.
...
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Is there a function $f :A^{<\mathbb{N}}\to E$ such that $f (s{}^\frown a)=h(f(s),a,|s|)$ for any $s\in A^{<\mathbb{N}}$ and $a\in A$?
Given a set $A$, define $A^{<\mathbb{N}}:=\cup _{n\in\mathbb{N}}A^n$ with $A^0:=\{\emptyset\} $ and $A^n$ being the $n$-fold cartesian product of $A$.
For any $s:=(s_0,\cdots,s_{n-1})\in A^n\...
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Any subset of a well-ordered set is isomorphic to an initial segment of this well-ordered set.
I wanted to prove the fact for which I have a sketch of proof: Let $(W,\leq )$ be a well-ordered set and $U$ be a subset of $W$. Then considering the restriction of $\leq $ to $U\times U$, we have ...
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Every real function is the sum of two bijections
Context. We are in $\mathsf{ZFC}$.
Problem. Given a function $f : \mathbb R \to \mathbb R$ prove that there exist two bijective functions $g$ and $h$ from $\mathbb R \to \mathbb R$ such that $f = g + ...
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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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Relativized extensionality $\implies$ existence of a transitive set isomorphic to the first one
Background. Given two sets $A$ and $B$, we say that $A$ is isomorphic to $B$ if there exists a bijection $f : A \to B$ such that $\forall x,y \in A \; y \in x \leftrightarrow f(y) \in f(x)$. We work ...
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Is there a smarter way to calculate $F(\omega^{15} + 7)$? (Ordinal arithmetic)
Given the function $F : \text{Ord} \to \text{Ord}$ definite by transfinite recursion as it follows:
$$F(0) = 0 \qquad F(\alpha + 1) = F(\alpha) + \alpha \cdot 2 + 1 \qquad F(\lambda) = \sup_{\gamma &...
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Is a totally ordered set consisting of discrete point sets containing point p a well-ordered set?
I have encountered a fully ordered set and want to determine whether it is well ordered.
Each element B in this totally ordered set A is a set and includes point p, which means that the elements in A ...
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2
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152
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Transfinite recursion to construct a function on ordinals
I am asked to use transfinite recursion to show that there is a function $F:ON \to V$ (here $ON$ denote the class of ordinals and $V$ the class of sets) that satisfies:
$F(0) = 0$
$F(\lambda) = \...
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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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On the proof that there exists a strictly increasing function $f : \text{cf}(A) \to A$ such that $\text{rng}(f)$ is cofinal in $A$
I am looking at the following theorem in a set of notes:
Theorem 12.48. Suppose that $(A, <)$ is a simple order with no largest element. Then there is a strictly increasing function $f : \text{cf}(...
- 5,164
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Recursive definition for the sum of a transfinite sequence of ordinals
A transfinite sequence is a function whose domain is an ordinal $\alpha$. Let $C$ denote the class of all transfinite sequences of ordinals, and let $\text{On}$ denote the class of ordinals.
Use the ...
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Exercise 6.4.2 Introduction to Set Theory by Hrbacek and Jech
This is exercise 6.4.2 from the book Introduction to Set Theory 3rd ed. by Hrbacek and Jech.
Using the Recursion Theorem 4.9 show that there is a binary operation $F$ such that
$(a)$ $F(x,1)=0$ for ...
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How to define ordinal addition
From Jech's Set Theory:
We shall now define addition, multiplication and exponentiation of ordinal numbers, using Transfinite Recursion.
Definition 2.18 (Addition). For all ordinal numbers $\alpha$
$...
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On the definition of ordinal addition using transfinite recursion
Am reading the textbook Introduction to Set Theory 3rd ed. by Hrbacek and Jech. In chapter 6 the authors present two versions of the transfinite recursion theorem:
Transfinite Recursion Theorem. Let $...
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On the proof of the parametric version of the transfinite recursion theorem
In the book Introduction to Set Theory 3rd ed. by Hrbacek and Jech one can find proofs of the following two theorems:
Transfinite Recursion Theorem. Let $G(x)$ be an operation. Then there exists an ...
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Why does proof of Zorn's lemma need to use the fact about ordinals being too large to be a set?
I'm not understanding why its necessary to invoke the knowledge about ordinals in order to prove Zorn's lemma.
The Hypothesis in Zorn's lemma is
Every chain in the set Z has an upper bound in Z
Then ...
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TREE(3) and the Goodstein sequence
TREE(3) is an extremely large number that requires ordinal arithmetic to prove it is finite. For what value of n would $G(n)>TREE(3)$? The length of the
Goodstein sequence $G(n)$ is how many ...
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Help with basic ordinal arithmetic: What is the supremum of the sequence $\omega^{2} + \omega$, $\omega^{2}*2 + \omega$, $\omega^{2}*3 + \omega$, ...?
I need some help with some basic ordinal arithmetic. I am trying to determine the supremum of the sequence $\omega^{2}*1 + \omega$, $\omega^{2}*2 + \omega$, $\omega^{2}*3 + \omega$, ... where the ...
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Best Recursive Subdivision Tiling Mapping Function
I am trying to create subdivision tilings inspired by the work of Brian Rushton (eg. page 77 of this paper). The challenge is to subdivide tiles according to a certain rule, and then apply this rule ...
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How to prove this alternative version of transfinite recursion
There are several formulations of transfinite recursion. I am interested in the following one.
Let $(V, \in)$ be a model of ZF. Let $g_1$ be a set and $G_2,G_3 : V \to V$ be two definable class ...
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Why doesn't transfinite recursion imply dependent axiom of choice
The theorem of transfinite recursion states the following (A quick introduction to basic set Theory).
Theorem 4.4 (Transfinite recursion). For every ordinal $\kappa$, set $A$, and map $^3 F: \...
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How to invoke transfinite recursion
Transfinite recursion states that
For any $F:\mathbf{V}\to \mathbf{V}$, there exists a unique $G:\mathbf{ON}\to\mathbf{V}$ such that $\forall\alpha[G(\alpha)=F(G\vert_\alpha)]$
where $\mathbf{V}$ ...
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How to calculate multiplication of transfinite nimbers with a Cantor normal form
I failed to calculate nimber multplication in the form of $[\omega^\alpha]*[\omega^\beta]$, according to the "mex" definition.
The cases when $\alpha<3,\beta<3$ are easy, while $[\...
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$\mathbb{R}_{\rm Sorg.}$ is paracompact but $\mathbb{R}_{\rm Sorg.} \times \mathbb{R}_{\rm Sorg.}$ is not
I’m trying to solve this problem: $\def\bbR{\mathbb{R}} \def\RSorg{\bbR_{\rm Sorg.}} \def\calR{\mathcal{R}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \DeclareMathOperator{\range}{range}$
Prove ...
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Can I add this assumption for arbitrary family?
Perhaps, my question is straightforward but I want to make sure. let's consider,
$\mathcal F=\{f_{\xi}\colon \xi<\mathfrak c\}$ family of functions from $\mathbb R\to\mathbb R$. If want to prove ...
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Construct a function that will be disjoint with continuum many lines
The symbol $\mathbb L$ will
stand for the family of all lines in the plane that are neither horizontal nor vertical. Also, we put $\mathbb L_0:=\{\ell\in\mathbb L\colon \ell(0)=0\}$
lemma.
Let $\...
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Transfinite Recursion Theorem - Particular case - Enderton
I have the following theorem for any formula $\gamma(x,y)$:
Theorem of Transfinite Recursion: Given a well-ordered set $A$ such that for any $f$ there is a unique $y$ such that $\gamma(f,y)$ holds, ...
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I have some questions about the Ross-Littlewood Paradox
TLDR at the end.
Hi, I recently saw this comment given by "completely-ineffable" on the r/badmathematics subreddit. And I just wanted to make sure if I understand it correctly and wanted to ...
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1
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170
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Ordinal notations and the Church-Kleene Ordinal
According to Kleene, an $r$-system is a pair $(S, |\cdot|)$ where $S\subseteq\mathbb{N}$ and $|\cdot|$ maps $S$ to countable ordinals, such that
There is a partial recursive $K(x)$, such that $K(x) = ...
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Proving a recursively defined mapping is right-unique and left-total [duplicate]
The question was originally written in German, but from what I understand, the gist is, that I'm supposed to show that the mapping is both left-total and right-unique.
Let A be a set, $a \in A$ and $h:...
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How to more rigorously formalise “value of (Weaver's) P-name” in set theory forcing (recursion)?
Background
I have been reading some introductory material on forcing, specifically Nik Weaver's Forcing for mathematicians. What Weaver calls a “$P$-name”, I will call “Weaver's $P$-name” because it ...
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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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The set is meager if it has a cover of clopen meager sets
Let $X$ be a topological space such that there exists a collection of meager clopen sets $(C_i)_{i \in I}$ such that $X = \bigcup_{i \in I} C_i$. I want to prove that $X$ is then meager itself.
As ...
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Comparability theorem for well ordered sets using transfinite recursion
Similar questions have already been asked here and here. But I am asking for verification of my proof. Halmos leaves out an "easy" transfinite induction argument, which I have struggled to ...
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Is my simplified proof of Tychonof theorem correct? [closed]
I have read the proof of Tychonof theorem from here.
Let $(E_\lambda)_{\lambda \in \mathfrak a}$ be an arbitrary collection of compact topological spaces. We endow $E := \prod_{\lambda \in \mathfrak ...
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How to define in ZFC transfinite hierarchies of proper classes (with an example usefull for proof theory)
I was studying some Proof Theory in Pohlers' Proof Theory: first step into impredicativity , and I found myself facing a Set Theory's problem.
We need to define this transfinite hierarchy of proper ...
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Does there exist a Bernstein set such that a finite sum still Bernstein?
Recall that $B\subset\Bbb R$ is a Bernstein set if $B\cap P\neq\emptyset \neq P\setminus B$ for every perfect set $P\subset\Bbb R.$ It can be constructed by an easy transfinite induction. Moreover, ...
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141
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About Construction a one-to-one function from $(a,b)$ onto $[a,b]$
This question asked to construct one-to-one function from $(a,b)$ onto $[a,b]$. I know there is a function but it seems the question to define this function explicitly. How this can be done?
Edit I ...
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Is there a generalization of transfinite recursion that allows defining proper classes?
Transfinite recursion lets one define a sequence of sets $S_\alpha$, for $\alpha$ an ordinal. My question is whether it is possible to generalize this recursion to allow $S_\alpha$ to be classes, not ...
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Classifying Vector Spaces without AC
Using the axiom of choice we can give a simple classification of all vector spaces over a given field $K$ up to isomorphism: Any $K$-vector-space $V$ is just isomorphic to $\bigoplus_{i\in B}K$ where $...
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