# Tagged Questions

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

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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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### 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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### 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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### 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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### 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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### 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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### Is it possible to extend the set-inclusion order of a power set to a well-ordering?

The original aim is to define recursively a function on the power set of a set such that the functional value of a subset is determined by those of its proper subsets. Thank you.
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### uncountable repetitions

I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence: Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be {...
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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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### 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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### Application of Transfinite Induction

Our teacher gave us for practice to prove some properties of $V(\alpha)$ defined as V(0) = \emptyset,\; V(S(\alpha)) = \mathcal{P}(V(\alpha)),\; Lim(\alpha): V(\alpha) = \bigcup\{V(\beta)\; |\; \...
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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?). ...
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 ...