Cutting back by considering rank

The pruning by Rank viewing (or truncation by Rank viewing or localization by Rank viewing ) is in the set theory used and Tarski and Scott method proposed in 1955, how to limit the study of a class of studying its subsets.

To achieve this, one defines the subclass for a class if the rank function is. The existence of the rank function is either secured by a special axiom or proved with the help of the foundation and replacement axioms. If is, then is a set whose rank is at most . ${\displaystyle K\subseteq {\textrm {V}}=\{x|\exists y(x\in y)\}}$ ${\displaystyle K_{min}\equiv \tau (K):=\{x|x\in K;\forall y\in K(\rho (x)\leq \rho (y))\}}$ ${\displaystyle \rho :{\textrm {V}}\rightarrow {\textrm {On}}}$ ${\displaystyle \xi ={\textrm {min}}\{\rho (x)|x\in K\}}$ ${\displaystyle \tau (K)}$ ${\displaystyle \xi +1}$

By cutting back by considering the rankings, the following theorems can be proven:

• For each relation one exists predecessor small part relative to the same domain. ${\displaystyle R}$ ${\displaystyle S}$
• For each relation there is a partial relation with the same value range, the inverse relation of which is previously small . ${\displaystyle R}$ ${\displaystyle S}$
• If every non-empty set has a -smallest element, then every non-empty class also has a -smallest element and the following applies to every set-theoretical formula : (generalization of the induction principle). ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \Phi }$ ${\displaystyle (\forall x(\forall yRx\Phi (y)\Rightarrow \Phi (x)))\Rightarrow \forall z\Phi (z)}$
• For every set and a finite number of relations, there is an almost closed set for every one . ${\displaystyle A}$ ${\displaystyle R_{1},...,R_{k}}$ ${\displaystyle i\in \{1,...,k\}}$ ${\displaystyle R_{i}}$ ${\displaystyle B\supseteq A}$
• For every reflexive transitive symmetric relation there is a function . ${\displaystyle R}$ ${\displaystyle F\colon \forall x\forall y(F(x)=F(y)\Leftrightarrow xRy)}$

Web links

• Wolfram Pohlers: Set theory (PDF) , University of Münster, Institute for Mathematical Logic and Basic Research, lecture notes, SS 1994

Individual evidence

1. ↑ In English: Cutting Down Classes to Sets , also known as Scott's trick .
2. ↑ Tarski A., General principles of induction and resursion; The Notation of Rank in axiomatic set theory and some of its applications , 1955, Bull. Amer. Math., 61 , pp. 442-443
3. ^ Deiser O., Introduction to Set Theory , Springer, 2004, ISBN 978-3-540-20401-5 , 2.6, 2.8
4. ^ ^{A b} Levy A., Basic Set Theory , Springer, 1979, ISBN 3-540-08417-7 , II.7
5. ↑ Gloede, Klaus: Script for the lecture set theory . SS 2004. University of Heidelberg, Mathematical Institute, p. 181 . PDF , At DOCZZ , At Yumpu . Here page 62
6. ^ Zuckerman M., Sets and Transfinite Numbers , Macmillian Publishing Co., 1974, ISBN 0-02-432110-9 , 6.1