# Questions tagged [transfinite-recursion]

Questions dealing with set-theoretic functions defined by transfinite recursion.

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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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### Recursive Definition of the Closure of a Set in a Family of Functions

For the sake of consistency, I will define some terms below the question, however, if you are familiar with the terms, that section can be skipped. I am attempting to prove this lemma, for my text ...
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### Question about the formulation of Kenneth Kunen's “Primitive Recursion on Ordinals” Theorem

In Kenneth Kunen's The Foundations of Mathematics, the subject of recursion (on ordinals) is introduced and is described using the following "recipe": $f(\xi)=G(f_{|\xi})$, where $G$ and $f$...
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### Why does creating the basis of any vector space using dense order instead of well order fail in general case?

This problem occurred to me when I was solving problem 112 in A. Shen and N. K. Vereshchagin, Basic Set Theory (AMS 2002)), just as in this post Can this proof of existence of a Hamel basis using ...
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### Continuous function by transfinite induction

Let $A$ and $B$ subsets of $\mathbb R$ such that both have cardinality $\mathfrak c.$ I want to construct a noncontact function $f\colon A\to B$ by transfinite induction such that $f$ is ...
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### About construct a function with specific property that miss a line

Recall some definitions: Definition. $A$ is algebraically independent if and only if for every $a\in A$ , $a\notin\mathbb Q(A\setminus\{a\}$). $\mathcal M$- the calss of functions $f$ for which the ...
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### Bernstein set and it proof.

Definition. A set $B\subset\mathbb R$ is called a Bernstein set if both $B$ and $\mathbb R\setminus B$ intersect each perfect set. I know the construct the of Bernstein set but I got confused with ...
1answer
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### Generalising the recursion theorem to well-founded posets

The recursion theorem states the following. Suppose $(A,\preccurlyeq)$ is a chain with order type $\omega$. Let $X$ be any set, let $x\in X$ and let $g\colon X\times A \to X$ be a function. Then ...
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### Equivalance relation for a mapping

I have an assignment to solve which is related to Equivalence relation of a mapping. The following picture will show you the content of the question. Actually, for the question a, it can be easily ...
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### What is the Turing degree of the set of True formula of Arithmetic whose order is an infinite ordinal

This question was originally posted as a part of this other question, but I was suggested to make a new question for this part. In the first question I asked about the Turing degree of the set of ...
1answer
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### Proving the cardinality of B<A (Terence Tao's Analysis I book Ex 8.5.15)

Let A and B be two non-empty sets such that A does not have lesser or equal cardinality to B. Using the principle of transfinite induction, prove that B has lesser or equal cardinality to A. (Hint: ...
2answers
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### Construction of an uncountable set using well ordering principle.

I was trying to prove that a metric space in which every uncountable set has a limit point is separable.Now I was trying to show that it is Lindelof.I proceed like this,If not lindelof,then there is ...
1answer
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### Using transfinite induction to split $R$ to continuum many pairwise disjoint subsets of $R$

I am looking for different ways to partition $R$. I know some like : (1) Define a relation as following $$x\sim y \ \text{iff} \ x-y\in\mathbb Q(x,y\in\mathbb R)$$. The equivalence classes have the ...
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### Continuous function with some properties plus everywhere surjective function must be everywhere surjective

Let me starts with some definitions : $f\colon \mathbb R\to \mathbb R$ is everywhere surjective if $f[I]=\mathbb R$ for every nonempty open interval $I.$ $f$ is Darboux function if $f$ satisfies the ...
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### How can I write a perfect set as $c$ many pairwise disjoint perfect sets

I have been thinking about this question but I have not get complete answer yet. the question is Let $P$ a nonempty perfect subsets of $\mathbb R$ then $P$ can be written as continuum many pairwise ...
2answers
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### Infinitary logic and Undecidability

Good day everybody, I would like to ask a question about undecidability. May I ask You, if we have some problem that is undecidable but true, for example if RH would be found out to be undecidable it ...
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### Primitive Recursive Functions

I would like to show that the function $f:\mathbb{N} \to \mathbb{N}$ defined by $f(n)=p_n$ where $p_n$ is the $n+1$ prime number is primitive recursive. The fact is that I just manage to show it is $μ$...
1answer
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### Definition of addition by transfinite recursion.

Transfinite recursion theorem from T.Jech Set Theory: Let G be a function (on V), then (2.6) below defines a unique function F on Ord such that $F (α) = G(F \restriction α)$ for each α. (2....
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### Characterize a $\sigma$-algebra generated by a family of sets $\sigma(C)=\mathcal{B}_{\Omega}$

Well, my intention is basically to understand what is done in the answer here, which I will summarize in the following... and discuss a little more (hence I made a new question). Problem: Given a ...
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### Computing these ordinal hyperoperations that are left/right associative

It is somewhat counterintuitive to want tetration to be right-associative over ordinals when ordinal arithmetic is left-associative. To merge these two, I simply took the supremum of the left- and ...
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### “Transfinite Peano Axioms”

Perhaps, the class of ordinals $\Omega$ can be axiomatised up to isomorphism by claiming it to be well-ordered such that for every subset $X\subseteq \Omega$ there exists a "succesor" ordinal $\sigma$ ...
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### $\omega$th iteration of Cayley-Dickson construction

The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D). The ...
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### How do Hrbacek and Jech derive the class function $F$ in Theorem 4.12 from Theorem 4.11? [duplicate]

My question is about a proof of Theorem 4.12 given in the text Introduction to Set Theory by Hrbacek and Jech. Let $V$ be the class of all sets, $\operatorname{Ord}$ be the class of all ordinals, and ...
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### Are these two parametric versions of Transfinite Recursion equivalent?

Although I'm able to prove $S_1\implies S_2$ with not much difficulty, I failed to prove $S_2\implies S_1$ after several attempts. In particular, I can not handle the case in which \$F(z,\alpha+1)=G_2(...