# Tagged Questions

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

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### What are the most prominent uses of transfinite induction outside of set theory?

What are the most prominent uses of transfinite induction in fields of mathematics other than set theory? (Was it used in Cantor's investigations of trigonometric series?)
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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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### Is transfinite induction needed to remove all the elements from an uncountable set?

Given any uncountable set S, would I need to use transfinite induction to prove if I remove single elements recursively, I will be left with the empty set? It seems like this can be thought of as an ...
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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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### 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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### Is there continuous $f: [0, 1] \rightarrow [0, \infty)$ such that for all $x$ there is $y$ with $f(y) < f(x)$?

I think there isn't. Here's a sketch of a proof. I'm just not sure whether it really works because I'm not confident with the transfinite versions of the standard theorems about limits and convergent ...
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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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### Computing the rank funciton of the Well founded universe of sets

Definitions: $(1)$ We define the $V_\alpha$ function by transfinite recursion as: $V_0=\varnothing$; $V_{\alpha+1}=P(V_\alpha)$; Lim$(\lambda)\rightarrow V_\lambda=\bigcup_{\alpha<\lambda}V_\alpha$...
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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: can't 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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### 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 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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### 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 ...