# Questions tagged [transfinite-recursion]

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

152 questions
Filter by
Sorted by
Tagged with
1 vote
45 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 ...
• 4,680
1 vote
70 views

• 4,680
95 views

• 1,546
65 views

### 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 $\mathbf{V}$ ...
• 475
155 views

• 2,391
1 vote
79 views

### 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, ...
• 3,539
212 views

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 ...
1 vote
103 views

128 views

### 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 ...
• 1,205
45 views

### 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 ...
• 4,525
78 views

### 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 ...
• 1,859
1 vote
88 views

### 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 ...
• 3,781
100 views

128 views

### 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 ...
• 1,075
79 views

### Meager set union Bernstein set with empty interior

Fix a meager set $M\subset \Bbb R$. of course $\text{int} M=\emptyset$. I want to construct a Bernstein set $B\subset \Bbb R$ such that $$\text{int}(M\cup B)=\emptyset \tag{1}$$ We know it is true ...
• 2,391
104 views

### Construct an additive group by transfinite induction

I know one way to construct a Bernstein set that is an additive group.Here is the way that I know. $\{P_\xi\colon \xi<\mathfrak c\}$ all nonempty perfect subsets of $\Bbb R.$ Choose, by recursion ...
• 2,391
1 vote
106 views

• 4,646
292 views

### Set of cardinality continuum contains a perfect set

I want to make sure if my claim is true or false : Every subset of $\mathbb{R}$ of cardinality continuum contains a perfect set A perfect set is a closed set with no isolated point. It might be there ...
• 2,391
295 views

### Recursion theorem for ordinals proof

I'm trying to understand the proof of the recursion principle of ordinals, the theorem is: The proof of this theorem uses this other theorem: The proof is pretty long (I'm sorry) so I will try to ...
1 vote
120 views

### Partition $\mathfrak c$-dense set to $\mathfrak c$- many dense set.

A subset of $A\subset\mathbb R$ is called $\mathfrak c$-dense if $|A\cap I|=\mathfrak c$ for any open interval $I\subset\mathbb R.$ Then, there is a partition for $A$ to continuum many dense set. ...
• 2,391
427 views

### $\mathbb R$ as continuum many of pairwise disjoint of Bernstein sets

The set $B\subset\mathbb R$ is called Bernstein set if neither $B$ nor $\mathbb R\setminus B$ contains any perfect sets. Theorem: $\mathbb R$ can be written as continuum many of pairwise disjoint ...
• 2,391
160 views

### family of pairwise disjoint sets in the complement of meager. set

Let $M$ be a meager subset of $\mathbb R$. I want to construct the following family in $\mathbb R\setminus M$ $$F:= \{A_{r}^\xi\colon r\in\mathbb R \ \&\ \xi<c\}$$ Such that all elemnets of $F$...
• 2,391
For $p,q\in \mathbb{Q} :p<q$ we define $(p,q) :=\{ x\in \mathbb Q : p<x<q \}$ and we want to define an order like this $$(p,q) < (t,l) \iff p<q$$ and I want to prove that there is an ...