# Tagged Questions

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

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### Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for ...
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### Proving that Monotone Convergence implies Least Upper Bound in $\mathbb{R}$.

I tried proving that Every bounded increasing sequence converges in $\mathbb{R}$. implies that $\mathbb{R}$ has the least upper bound. Here, $\mathbb{R}$ is taken as an ordered field which ...
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### nodes equation cant find formula.

Given the level $N$ at which a node $X$ is located in a binary tree, to search for node $X$ according to level-order traversal, we can use the knowledge of level N where $X$ is located to narrow our ...
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### How to iterate a function 8 times about a given interval of x in a Discrete Dynamical System

This is Dynamical Systems, specifically a discrete system. We are using L and R as in Left and Right such as: L=[0,0.5] R=(0.5,1] and LL=[0,0.25] LR=(0.25,0.5] and so on like that. We keep ...
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### closed form iterated logarithms

Is there any way that we could bound the following sum by a closed form expression $\sum_{i=1}^{\log^* N} \log^{(i)}N$ where $\log^{(i)}$ is the $\log$ function iterated $i$ time? Thanks
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### Can this proof of existence of a Hamel basis using transfinite recursion be shortened/simplified?

This is (I hope) a solution to Problem 112 in A. Shen and N. K. Vereshchagin, Basic Set Theory (AMS 2002). It is - I thought! - a semi-routine exercise, part of whose purpose is to enlighten the ...
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### What does $\upharpoonright$ in $G(F\upharpoonright\alpha)$ mean?

More formally, we can state the Transfinite Recursion Theorem as follows. Given a class function $G\colon V\to V$, there exists a unique transfinite sequence $F\colon\mathrm{Ord}\to V$ (where ...
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### Proof of the Principle of Transfinite Induction for On

Principle of Transfinite Induction for On: If $C\neq\varnothing, C\subseteq$ On then $\exists\alpha\in C\forall\beta\in C[\alpha\in\beta\vee\alpha=\beta]$ Proof: (1) As $C\neq\varnothing$, let ...
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### Proving $V=V_{\sf On}$ in $\sf Z+Reg$

Define the function $$V_0=\emptyset,\qquad V_{\alpha+1}={\cal P}(V_\alpha),\qquad V_{\delta}=\bigcup_{\beta<\delta}V_\beta.$$ Since we are working in $\sf Z$ (i.e. $\sf ZF$ without the axiom of ...
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### Uniqueness in transfinite recursion.

Theorem of transfinite recursion: Given a well-ordered set $A$ let $\varphi(g,y)$ be a ZF formula such that for every $a \in A$ and every function $g$ with domain $I_a$ (where $I_a$ is the initial ...
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### Translation invariance of sets implies translation invariance of their generated sigma algebra?

If we have a collection of translation invariant sets (i.e. if $A$ is in the collection, then $A+x$ is in it too) - assume we have some notion of addition (e.g in a vector space). Is the generated ...
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### Initial Segment Order Isomorphic to the Ordinal Numbers

Prove that every well-ordered proper class has an initial segment order isomorphic to the ordinal numbers, ON. I have a plan to prove this but it uses a recursive definition and induction which I do ...
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### Transfinite recursion and Replacement

I have a question about the use of Replacement when proving the transfinite recursion theorem. It seems that the crucial use of Replacement is made in the step involving the set of all partial ...
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### Principle of Proof by Induction on a Well-ordering

Let $(X,\leq)$ be a woset (well ordered set). Let $E$ be a subset of $X$ such that: (i) the smallest element of $X$ is a member of $E$ (ii) for any $x\in X$, if $y<x\rightarrow y\in E$, then ...
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### Transfinite induction under an ordinal $\delta$

Can anyone gove me a hint or a direction to the proof of this statement? Let $F:V \rightarrow V$ be an operation, and $\delta$ an ordinal. Prove, that there exists a unique function $g$ with domain ...
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### Definition by Recursion

I just started studying logic, not as a course at a university, but as pastime. Since I do not study logic at an institution I use many different textbooks, including Enderton's $A$ $Mathematical$ ...
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### How to prove an extension of ZFC is conservative

Working in ZFC. I've defined a function-like binary predicate $R$ on a proper class. It has to be recursive; i.e. $R(a,b)$ must usually depend on one or more $R(c,d)$ for some $c$s and $d$s ...
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### Good resource to learn transfinite induction and/or recursion?

I'm currently reading John H. Conway's On Numbers and Games, but without a good understanding of transfinite induction and/or recursion, progress is very slow. What's a good resource for learning ...
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### Which ordinals embed in $2^\omega$ ordered by inclusion?

Which ordinals can be embedded in the power set of $\omega$ ordered by inclusion? I see that $\omega\cdot\omega$ can (and therefore anything less than that): we can partition $\omega$ into $\omega$ ...
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### Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
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### Constructing a Bernstein set in a Polish space

I just read the proof of how to construct two Bernstein sets in a Polish space $X$ using the facts that there are at most $2^{\aleph_0}$ many closed subsets if $X$ is Polish and that every closed ...
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### Question on the use of a parametric version of Transfinite Recursion Theorem in Introduction to Set Theory 3rd ed. by Hrbacek and Jech

My question concerns a proof given on page 118 in the text Introduction to Set Theory 3rd ed. by Hrbacek and Jech. The authors on page 117 prove a version of the transfinite recursion theorem ...
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### Equivalent statement of transfinite/ordinal recursion

I am trying to prove that the "standard" statement of transfinite/ordinal recursion: "Suppose $G$ is a definite operation on partial functions on ordinals. Then there is a unique definite operation ...
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### When to use transfinite induction?

How do we know when we are allowed to use transfinite induction in a proof ? Edit : considering the replies i should say the following Consider an infinite sum of fractions. By induction we can ...
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### Transfinite induction vs transfinite recursion

Let $\mathfrak{A}$ is a well-ordered set. Let $f$ is a function which maps a start segment $\{ i\in\mathfrak{A} \,|\, i<c \}$ of a transfinite sequence of elements of atomic posets into an element ...
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### Epsilon induction

I recently came upon this technique called epsilon induction, and was searching for some proof using the same. But I saw no such proof. Does someone know of any proof using this technique?
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### Well ordering and maximal principle

I have to prove that given any poset $(P,\preceq)$ there exists a chain $S$ such that it is maximal (meaning that if $S\subseteq S'$ then $S=S'$). The book contains a proof using the axiom of ...
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### Showing the existence of the $\omega_\omega$-th power set of $\omega$

Can you tell me if my answer is correct: Show that the set $P^{\omega_\omega}(\omega)$ exists. My answer: Let $P^0 (\omega) = \omega$, $P^{\alpha + 1}(\omega) = P(P^\alpha (\omega))$ and for a ...
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### Injection from the set of countable ordinals $\Omega$ into $\mathbb{R}$

I'm reading through this and I'd like to define an injective function from the set of countable ordinals $\Omega$ into $\mathbb{R}$ using transfinite induction (or maybe transfinite recursion?). ...
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### Need help with Recursion Theorem (Set Theory)

The recursion theorem In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set $X$, an element $a$ of $X$ and a function $f\colon X \to X$, the theorem ...
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### Recursive Mapping

I was wondering about what is the general definition of a recursive mapping between any two sets. Is there a condition for a mapping to be able to be written in recursive form? Is the following claim ...