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 ,
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{\ displaystyle \ langle \ kappa _ {i} \ mid i \ in I \ rangle}
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{\ displaystyle \ kappa _ {i}}
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M.
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{\ displaystyle \ sum _ {i \ in I} \ kappa _ {i} = \ vert \ bigcup _ {i \ in I} M_ {i} \ vert,}
and the product the thickness of the Cartesian product ,
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{\ 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 .
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{\ displaystyle M_ {i}}
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{\ displaystyle \ vert M_ {i} \ vert = \ kappa _ {i}}
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{\ displaystyle M_ {i} = \ kappa _ {i} \ times \ {i \}}
König's theorem now says:
For two cardinal sequences and with for all the following applies:
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{\ displaystyle \ langle \ kappa _ {i} \ mid i \ in I \ rangle}
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{\ displaystyle \ langle \ lambda _ {i} \ mid i \ in I \ rangle}
κ
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{\ displaystyle \ kappa _ {i} <\ lambda _ {i}}
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{\ displaystyle i \ in I}
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κ
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{\ 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
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{\ displaystyle \ langle X_ {i} \ mid i \ in I \ rangle}
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{\ displaystyle \ langle Y_ {i} \ mid i \ in I \ rangle}
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{\ displaystyle \ vert X_ {i} \ vert = \ kappa _ {i} <\ lambda _ {i} = \ vert Y_ {i} \ vert}
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{\ displaystyle X_ {i} \ subsetneq Y_ {i}}
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{\ 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
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{\ displaystyle i \ in I}
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{\ displaystyle \ alpha _ {i}}
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{\ displaystyle Y_ {i} \ setminus X_ {i}}
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{\ displaystyle \ textstyle x \ in \ bigcup _ {i \ in I} X_ {i}}
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{\ displaystyle j \ in I}
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{\ displaystyle x \ in X_ {j}}
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{\ displaystyle \ textstyle f: = \ Phi (x) \ in \ prod _ {i \ in I} Y_ {i}}
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{\ displaystyle f (i) = {\ begin {cases} x, & i = j \\\ alpha _ {i}, & i \ neq j \ end {cases}}}
.
Then is injective.
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{\ 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.
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{\ displaystyle \ Phi}
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{\ displaystyle i \ in I}
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{\ displaystyle f (i)}
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{\ displaystyle Y_ {i} \ setminus \ {\ Phi (x) (i) \ vert x \ in X_ {i} \}}
f
{\ displaystyle f}
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{\ displaystyle i}
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{\ displaystyle \ Phi}
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{\ displaystyle X_ {i}}
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{\ displaystyle i \ in I}
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{\ displaystyle \ Phi}
Inferences
Further inequalities can be derived directly from König's theorem ( and assume cardinal numbers):
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{\ displaystyle \ kappa}
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{\ displaystyle \ lambda}
If we denote the confinality of , then holds for infinite .
cf.
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{\ displaystyle \ operatorname {cf} (\ kappa)}
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{\ displaystyle \ kappa}
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{\ displaystyle \ kappa}
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cf.
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{\ displaystyle \ kappa ^ {\ operatorname {cf} (\ kappa)}> \ kappa}
for and infinite .
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{\ displaystyle \ kappa> 1}
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{\ displaystyle \ lambda}
cf.
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{\ 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.
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