# Tagged Questions

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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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### 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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### 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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### 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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### 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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### 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 ...
### 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?). ...
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 ...