König theorem (set theory)

The set of King is a set of the set theory , that of the Hungarian mathematician Julius King was discovered 1905th The theorem is a strict inequality between two cardinal numbers .

statement

For a family of cardinal numbers, the sum of these cardinal numbers is the power of the disjoint union of sets of power , ${\displaystyle \langle \kappa _{i}\mid i\in I\rangle }$ ${\displaystyle \kappa _{i}}$

${\displaystyle \sum _{i\in I}\kappa _{i}=\vert \bigcup _{i\in I}M_{i}\vert ,}$

and the product the thickness of the Cartesian product ,

${\displaystyle \prod _{i\in I}\kappa _{i}=|\prod _{i\in I}M_{i}|=\vert \{f\colon I\to \textstyle \bigcup _{i\in I}M_{i}\mid \forall i\in I\ f(i)\in M_{i}\}\vert .}$

Here the pairwise disjoint sets are with , for example . The well-definedness of both operations follows from the axiom of choice . ${\displaystyle M_{i}}$ ${\displaystyle \vert M_{i}\vert =\kappa _{i}}$ ${\displaystyle M_{i}=\kappa _{i}\times \{i\}}$

König's theorem now says:

For two cardinal sequences and with for all the following applies: ${\displaystyle \langle \kappa _{i}\mid i\in I\rangle }$ ${\displaystyle \langle \lambda _{i}\mid i\in I\rangle }$ ${\displaystyle \kappa _{i}<\lambda _{i}}$ ${\displaystyle i\in I}$

${\displaystyle \sum _{i\in I}\kappa _{i}<\prod _{i\in I}\lambda _{i}}$.

proof

Be , two families of pairwise disjoint sets with . Without loss of generality, one can assume that . It must be shown that there is an injective , but no bijective, mapping ${\displaystyle \langle X_{i}\mid i\in I\rangle }$ ${\displaystyle \langle Y_{i}\mid i\in I\rangle }$ ${\displaystyle \vert X_{i}\vert =\kappa _{i}<\lambda _{i}=\vert Y_{i}\vert }$ ${\displaystyle X_{i}\subsetneq Y_{i}}$

${\displaystyle \Phi \colon \bigcup _{i\in I}X_{i}\to \prod _{i\in I}Y_{i}=\{f\colon I\to \textstyle \bigcup _{i\in I}Y_{i}\mid \forall i\in I\ f(i)\in Y_{i}\}}$

For each is an element from . Be . Then there is a definite with . Let the function ${\displaystyle i\in I}$ ${\displaystyle \alpha _{i}}$ ${\displaystyle Y_{i}\setminus X_{i}}$ ${\displaystyle \textstyle x\in \bigcup _{i\in I}X_{i}}$ ${\displaystyle j\in I}$ ${\displaystyle x\in X_{j}}$ ${\displaystyle \textstyle f:=\Phi (x)\in \prod _{i\in I}Y_{i}}$

${\displaystyle f(i)={\begin{cases}x,&i=j\\\alpha _{i},&i\neq j\end{cases}}}$.

Then is injective. ${\displaystyle \Phi }$

Let an arbitrary such mapping be given. For define as an element . Then at the point is different from all images of from . Since this applies to everyone , is not surjective and therefore not bijective. ${\displaystyle \Phi }$ ${\displaystyle i\in I}$ ${\displaystyle f(i)}$ ${\displaystyle Y_{i}\setminus \{\Phi (x)(i)\vert x\in X_{i}\}}$ ${\displaystyle f}$ ${\displaystyle i}$ ${\displaystyle \Phi }$ ${\displaystyle X_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle \Phi }$

Inferences

Further inequalities can be derived directly from König's theorem ( and assume cardinal numbers): ${\displaystyle \kappa }$ ${\displaystyle \lambda }$

• If we denote the confinality of , then holds for infinite . ${\displaystyle \operatorname {cf} (\kappa )}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa ^{\operatorname {cf} (\kappa )}>\kappa }$
• for and infinite . ${\displaystyle \kappa >1}$ ${\displaystyle \lambda }$ ${\displaystyle \operatorname {cf} (\kappa ^{\lambda })>\lambda }$

literature

• Jech, Thomas: Set Theory , Springer-Verlag Berlin Heidelberg (2006), ISBN 3-540-44085-2 .
• König, Julius: To the continuum problem , Mathematische Annalen 60 (1905), 177-180.