Singular Cardinal Number Hypothesis

The singular cardinal number hypothesis, also abbreviated as SCH after the English term singular cardinals hypothesis , is a statement that is independent of the usual axioms of set theory and can therefore neither be proven nor refuted. It appears in the context of the investigations into the continuum hypothesis .

formulation

The singular cardinal number hypothesis is the following: If the inequality holds for an infinite cardinal number , then . ${\displaystyle \kappa }$ ${\displaystyle 2^{\operatorname {cf} \kappa }<\kappa }$ ${\displaystyle \kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$

Where is the cofinality of and the successor cardinal number of . According to Cantor's theorem, is always . From therefore follows , that is, the cardinal number that satisfies must be singular . Therefore, in the above formulation only one statement is made about singular cardinal numbers, which explains the name singular cardinal number hypothesis . According to König's theorem, is always such that is the minimum possible value for . The above statement means that cardinal numbers are assumed to have the lowest possible value. ${\displaystyle \operatorname {cf} \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa ^{+}}$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\kappa }>\kappa }$ ${\displaystyle 2^{\operatorname {cf} \kappa }<\kappa }$ ${\displaystyle \operatorname {cf} \kappa <\kappa }$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\operatorname {cf} \kappa }<\kappa }$ ${\displaystyle \kappa ^{\operatorname {cf} \kappa }>\kappa }$ ${\displaystyle \kappa ^{+}}$ ${\displaystyle \kappa ^{\operatorname {cf} \kappa }}$ ${\displaystyle \kappa ^{\operatorname {cf} \kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\operatorname {cf} \kappa }<\kappa }$

SCH follows from GCH

Since and because of the above-mentioned König theorem applies . If the generalized continuum hypothesis GCH (English: Generalized Continuum Hypothesis) holds, then it always follows from the above inequality , that is, the conclusion in the SCH holds regardless of any assumptions. In particular, the singular cardinal number hypothesis follows from the generalized continuum hypothesis. ${\displaystyle \operatorname {cf} \kappa \leq \kappa }$ ${\displaystyle \kappa <\kappa ^{\operatorname {cf} \kappa }\leq \kappa ^{\kappa }=2^{\kappa }}$ ${\displaystyle 2^{\kappa }=\kappa ^{+}}$ ${\displaystyle \kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$

But there are also models with and SCH, in which the continuum hypothesis is violated, but the singular cardinal number hypothesis still applies. ${\displaystyle 2^{\aleph _{0}}=\aleph _{2}}$

See also

• In cardinal number arithmetic , the assumption of SCH leads to simplified formulas for raising cardinal numbers to the power.
• According to one of Silver's theorem , it is sufficient to consider the case in the singular cardinal number hypothesis . ${\displaystyle \operatorname {cf} \kappa =\aleph _{0}}$

Individual evidence

1. ^ Thomas Jech : Set Theory , Springer-Verlag (2003), ISBN 3-540-44085-2 , I.5, page 58
2. ^ Matteo Viale: The proper forcing axiom and the singular cardinal hypothesis , J. Symbolic Logic, Volume 71 (2): Pages 473-479 (2006)