Questions dealing with set-theoretic functions defined by 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 ...
8
votes
1answer
227 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 ...
7
votes
2answers
173 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 ...
7
votes
1answer
262 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?
5
votes
2answers
495 views

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 ...
4
votes
2answers
156 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 ...
4
votes
1answer
122 views

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 $A$ $Mathematical$ ...
4
votes
1answer
70 views

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 ...
4
votes
2answers
211 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)\; |\; ...
3
votes
3answers
121 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 $\omega$ ...
3
votes
1answer
139 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 ...
3
votes
2answers
140 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 ...
2
votes
2answers
362 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?). ...
2
votes
1answer
152 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 ...
2
votes
1answer
16 views

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 ...
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 ...
2
votes
1answer
65 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
104 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 ...
2
votes
1answer
129 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 ...
1
vote
2answers
291 views

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 ...
1
vote
2answers
37 views

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 ...
0
votes
1answer
25 views

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 ...
0
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1answer
110 views

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 ...
0
votes
1answer
217 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 ...
0
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1answer
30 views

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 ...
0
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1answer
92 views

Theorem unique bijection between two well-ordered sets satisfying certain conditions

I have some problems understanding the proof of the following theorem. Given two well-ordered sets $(X, \le _X), \ (Y, \le _Y)$ there exists exactly one partial function $f(X, Y)$ such that: 1)$f: \ ...
-1
votes
5answers
416 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 ...