2
votes
1answer
53 views

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 ...
2
votes
1answer
64 views

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 ...
3
votes
2answers
123 views

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 ...
3
votes
3answers
113 views

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 $\omega$ ...
7
votes
2answers
161 views

Does this require transfinite induction?

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 ...
2
votes
1answer
29 views

Well-founded part of a graph

Let (A,R) be a graph. Define by transfinite recursion: $ W_{0}=\emptyset \\ W_{\alpha+1}=\{a \in A : ext_{R}(a) \subseteq W_{\alpha +1}\} \\ W_{\alpha}=\cup_{\beta < \alpha} W_{\beta} \text{if ...
4
votes
2answers
201 views

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)\; |\; ...
4
votes
2answers
142 views

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 ...
2
votes
1answer
118 views

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 ...
0
votes
1answer
203 views

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 ...
-1
votes
5answers
400 views

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 ...
7
votes
1answer
248 views

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?
2
votes
1answer
149 views

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 ...
3
votes
1answer
137 views

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 ...
2
votes
2answers
345 views

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?). ...
8
votes
1answer
224 views

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 ...