Diagonal cut

In the mathematical branch of set theory , the diagonal cut is a construction related to the average, and a family of sets can be assigned a new one, namely its diagonal cut. The elements of the diagonal section of the family are certain indices , which in turn belong to certain of the sets . The concept formation to be discussed here is therefore only meaningful if the indices themselves appear as elements of the sets, therefore sets of ordinal numbers indexed with ordinal numbers are considered. ${\displaystyle (X_{\alpha })_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle X_{\xi }}$

definition

Let it be a cardinal number and a family of sets . Then is called ${\displaystyle \kappa }$ ${\displaystyle (X_{\alpha })_{\alpha <\kappa }}$ ${\displaystyle X_{\alpha }\subset \kappa }$

${\displaystyle \Delta _{\alpha <\kappa }X_{\alpha }\,=\,\{\xi <\kappa ;\,\xi \in \bigcap _{\alpha <\xi }X_{\alpha }\}}$

the diagonal cut of the family . ${\displaystyle (X_{\alpha })_{\alpha <\kappa }}$

properties

The diagonal section contains exactly those elements of the diagonals that are also in the relation . ${\displaystyle R}$

The data of the above definition may be available. Of course, the average is contained in the diagonal section, that is, it is true , because is contained in each of the sets , especially in , and that is precisely the defining condition for belonging to . ${\displaystyle \bigcap _{\alpha <\kappa }X_{\alpha }\subset \Delta _{\alpha <\kappa }X_{\alpha }}$ ${\displaystyle \xi <\kappa }$ ${\displaystyle X_{\alpha }}$ ${\displaystyle \bigcap _{\alpha <\xi }X_{\alpha }}$ ${\displaystyle \xi }$ ${\displaystyle \Delta _{\alpha <\kappa }X_{\alpha }}$

If one sets , then is a falling function , where stands for the power set , that is, it follows from . By definition is equivalent to . On the Cartesian product, define the relation and the " diagonal " . Then, the diagonal cut exactly the set of ordinal numbers , for which the diagonal element in is: ${\displaystyle Y_{\xi }:=\bigcap _{\alpha <\xi }X_{\alpha }}$ ${\displaystyle \xi \mapsto Y_{\xi }}$ ${\displaystyle \kappa \rightarrow P(\kappa )}$ ${\displaystyle P}$ ${\displaystyle \xi <\eta }$ ${\displaystyle Y_{\xi }\supset Y_{\eta }}$ ${\displaystyle \xi \in Y_{\xi }}$ ${\displaystyle \xi \in \Delta _{\alpha <\kappa }X_{\alpha }}$ ${\displaystyle \kappa \times \kappa }$ ${\displaystyle R:=\{(\xi ,\eta );\,\xi \in Y_{\eta }\}}$ ${\displaystyle d:\kappa \rightarrow \kappa \times \kappa ,\,\xi \mapsto (\xi ,\xi )}$ ${\displaystyle \xi }$ ${\displaystyle (\xi ,\xi )}$ ${\displaystyle R}$

${\displaystyle \Delta _{\alpha <\kappa }X_{\alpha }=d^{-1}(R)=\{\xi <\kappa ;\,\xi \in Y_{\xi }\}}$.

Membership from to the diagonal average depends only on membership in the first . This becomes particularly clear in the following formula: ${\displaystyle \xi }$ ${\displaystyle X_{\alpha },\alpha <\xi }$

${\displaystyle \Delta _{\alpha <\kappa }X_{\alpha }=\bigcap _{\alpha }(X_{\alpha }\cup \{\xi ;\xi \leq \alpha \})}$

example

To demonstrate how the term presented here works, the following simple statement should be proven:

• It is a cardinal and an ordinal number is . Then applies ${\displaystyle \kappa }$ ${\displaystyle \alpha <\kappa }$ ${\displaystyle X_{\alpha }:=\{\xi <\kappa ;\,\alpha +1<\xi <\kappa \}}$

${\displaystyle \Delta _{\alpha <\kappa }X_{\alpha }=\{\xi <\kappa ;\,\xi {\text{ ist eine Limes-Ordinalzahl}}\}}$

Proof: " ": Is , so is , so for everyone . So it applies to all , hence is a Limes ordinal number . ${\displaystyle \subset }$ ${\displaystyle \xi \in \Delta _{\alpha <\kappa }X_{\alpha }}$ ${\displaystyle \xi \in \bigcap _{\alpha <\xi }X_{\alpha }}$ ${\displaystyle \xi \in X_{\alpha }}$ ${\displaystyle \alpha <\xi }$ ${\displaystyle \alpha <\xi }$ ${\displaystyle \alpha +1<\xi }$ ${\displaystyle \xi =\sup _{\xi <\alpha }\alpha }$

" ": Conversely, if a Limes ordinal number is, then for all and therefore what exactly is the defining condition for . ${\displaystyle \supset }$ ${\displaystyle \xi =\sup _{\alpha <\xi }\alpha }$ ${\displaystyle \alpha +1<\xi }$ ${\displaystyle \alpha <\xi }$ ${\displaystyle \xi \in \bigcap _{\alpha <\xi }X_{\alpha }}$ ${\displaystyle \xi \in \Delta _{\alpha <\kappa }X_{\alpha }}$

use

The diagonal cut is used particularly in the investigation of uncountable regular cardinal numbers . A filter on a cardinal number is called normal if it is closed with regard to the formation of diagonal cuts, that is, it is again an element of the filter if all of them are there. For example, the club filter is normal on an uncountable regular cardinal number. This fact is used, for example, in Fodor's theorem . ${\displaystyle \kappa }$ ${\displaystyle \Delta _{\alpha <\kappa }X_{\alpha }}$ ${\displaystyle X_{\alpha }}$

Diagonal union

The term dual to the diagonal cut is the diagonal union . Is a cardinal number and a family of sets , so is called ${\displaystyle \kappa }$ ${\displaystyle (X_{\alpha })_{\alpha <\kappa }}$ ${\displaystyle X_{\alpha }\subset \kappa }$

${\displaystyle \Sigma _{\alpha <\kappa }X_{\alpha }:=\{\xi <\kappa ;\,\xi \in \bigcup _{\alpha <\xi }X_{\alpha }\}}$

the diagonal union of the set family . ${\displaystyle (X_{\alpha })_{\alpha <\kappa }}$

The definition is designed in such a way that applies. ${\displaystyle \kappa \setminus \Sigma _{\alpha <\kappa }X_{\alpha }=\Delta _{\alpha <\kappa }(\kappa \setminus X_{\alpha })}$

literature

• Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin et al. 2003, ISBN 3-540-44085-2 , chapter 8.