Easton's theorem

The set of Easton , named after William Bigelow Easton , is a mathematical theorem in the field of set theory . The generalized continuum hypothesis , which in set theory ZFC, ie in the Zermelo-Fraenkel set theory with the axiom of choice can not disprove neither prove, says that the cardinality of the power set of a cardinal , always with the successor cardinal number of matches. To demonstrate the unprovability, Paul Cohen had constructed a model in which this hypothesis is false. In addition, Easton's theorem states that the generalized continuum hypothesis for regular cardinal numbers can be violated in almost any way. ${\displaystyle 2^{\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa ^{+}}$ ${\displaystyle \kappa }$

Formulation of the sentence

Let it be the class of all cardinal numbers and the subclass of the regular cardinal numbers. Let us also assume a function with the following properties: ${\displaystyle K}$ ${\displaystyle R}$ ${\displaystyle F:R\rightarrow K}$

• ${\displaystyle F}$is monotonous, that is, for . ${\displaystyle F(\kappa )\leq F(\lambda )}$ ${\displaystyle \kappa <\lambda }$
• The confinality of is really greater than , that is, for everyone . ${\displaystyle F(\kappa )}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa <\operatorname {cf} (F(\kappa ))}$ ${\displaystyle \kappa \in R}$

Then there is a ZFC- model with all . ${\displaystyle 2^{\kappa }=F(\kappa )}$ ${\displaystyle \kappa \in R}$

Remarks

The theorem was proved by Easton in 1970 using generalized forcing methods.

The continuum function is trivially monotonic and, according to a conclusion from König's theorem, also satisfies the inequality . That is all that can be said about the continuum function at regular places in ZFC, because according to the above theorem of Easton there are ZFC models for every function that fulfills these two conditions for regular cardinal numbers, in which the continuum function is exactly this function . In this sense, the generalized continuum function can be almost arbitrarily wrong. ${\displaystyle \kappa \mapsto 2^{\kappa }}$ ${\displaystyle \kappa <\operatorname {cf} (2^{\kappa })}$

Even the simple continuum hypothesis, in Aleph notation , can be arbitrarily wrong. According to Easton's theorem, for every cardinal number with uncountable affinity there are ZFC models in which holds. For example, the equations are relatively consistent . ${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\aleph _{0}}=\kappa }$ ${\displaystyle 2^{\aleph _{0}}=2^{\aleph _{1}}=2^{\aleph _{2}}=\aleph _{100}}$

According to Silver's theorem , the smallest cardinal number for which the equation is violated is not a singular cardinal number with uncountable affinity. Easton's theorem cannot therefore be extended to singular cardinal numbers. ${\displaystyle 2^{\kappa }=\kappa ^{+}}$

Individual evidence

1. ↑ Thomas Jech : Set Theory , Springer-Verlag (2003), ISBN 3-540-44085-2 , Theorem 15.18
2. ^ WB Easton: Powers of regular cardinals , Annals of Mathematical Logic (1970), Volume 1, pages 139-178
3. ↑ Winfried Just, Martin Weese: Discovering Modern Set Theory. I. The Basics , Graduate Studies in Mathematics, Volume 8, American Mathematical Societey (1996), ISBN 0-821-80266-6 , page 183