Silver's theorem

The set of Silver , named after Jack Silver , is a set from the set theory , which deals with possible generalizations of the continuum hypothesis is concerned. The generalized continuum hypothesis is independent of the usual axioms of set theory, that is, of ZFC , so it can neither be proven nor refuted there. The proposition to be discussed here provides a qualification for the invalidity of the generalized continuum hypothesis; it says that the smallest cardinal number for which the generalized continuum hypothesis is false cannot be a singular cardinal number with uncountable cofinality . This result was surprising, Silver himself writes:

This result is contrary to the previous expectations of nearly all set-theorists, including myself. (German: This result contradicts the earlier expectations of almost all set theorists, including myself.)

The proof methods also lead to a theorem about the singular cardinal number hypothesis , also known as Silver's Theorem.

formulation

The generalized continuum hypothesis states that for every infinite cardinal numbers applies. It is the cardinality of the power set of a set of cardinality and the successor cardinal of . The following theorem says that the property is retained for certain cardinal numbers if it is already valid for all smaller ones. ${\displaystyle 2^{\kappa }=\kappa ^{+}}$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa ^{+}}$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\kappa }=\kappa ^{+}}$

Silver's theorem : If a singular cardinal number is with and applies to all infinite cardinal numbers , then also applies . ${\displaystyle \kappa }$ ${\displaystyle \operatorname {cf} \kappa >\aleph _{0}}$ ${\displaystyle 2^{\lambda }=\lambda ^{+}}$ ${\displaystyle \lambda <\kappa }$ ${\displaystyle 2^{\kappa }=\kappa ^{+}}$

It is the Kofinalität of and the first infinite cardinal . ${\displaystyle \operatorname {cf} \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \aleph _{0}}$

The singular cardinal number hypothesis says that ( see also Gimel function ) for singular cardinal numbers with . It is also independent of ZFC and it follows from the generalized continuum hypothesis, so it is weaker than this. The following theorem applies to the singular cardinal number hypothesis: ${\displaystyle \kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\operatorname {cf} \kappa }<\kappa }$

Silver's theorem : The singular cardinal number hypothesis is already valid if it is valid for all singular cardinal numbers with countable cofinality.

For proof

Both theorems use a lemma about the continuation of the property in the sense that if this equation holds for a sufficiently large number of smaller cardinals than , then it holds for . The following technical statement is proven more precisely: ${\displaystyle \kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$

Let there be a singular cardinal number with and an ascending sequence of cardinal numbers indexed with ordinal numbers , so that applies ${\displaystyle \kappa }$ ${\displaystyle \operatorname {cf} \kappa >\aleph _{0}}$ ${\displaystyle (\kappa _{\alpha })_{\alpha <\operatorname {cf} \kappa }}$

1. ${\displaystyle \lambda ^{\operatorname {cf} \kappa }<\kappa }$ for all ${\displaystyle \lambda <\kappa }$
2. ${\displaystyle \textstyle \lim _{\alpha <\operatorname {cf} \kappa }\kappa _{\alpha }=\kappa }$(this is equivalent to ) ${\displaystyle \textstyle \sup _{\alpha <\operatorname {cf} \kappa }\kappa _{\alpha }=\kappa }$
3. ${\displaystyle \textstyle \sup _{\alpha <\beta }\kappa _{\alpha }=\kappa _{\beta }}$for all Limes ordinal numbers (such sequences are called normal) ${\displaystyle \beta <\operatorname {cf} \kappa }$
4. The crowd is stationary in ${\displaystyle \{\alpha <\operatorname {cf} \kappa ;\kappa _{\alpha }^{\operatorname {cf} \kappa _{\alpha }}=\kappa _{\alpha }^{+}\}}$ ${\displaystyle \operatorname {cf} \kappa }$

Then applies . ${\displaystyle \kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$

We dispense with the proof of this lemma, but we shall briefly explain how Silver's theorem about the continuum hypothesis results from this:

Let it be a singular cardinal number with and it applies to all cardinal numbers . To apply the above lemma we choose any normal sequence with Limes that exists according to the definition of the cofinality, and check the assumptions of the lemma one after the other. ${\displaystyle \kappa }$ ${\displaystyle \operatorname {cf} \kappa >\aleph _{0}}$ ${\displaystyle 2^{\lambda }=\lambda ^{+}}$ ${\displaystyle \lambda <\kappa }$ ${\displaystyle (\kappa _{\alpha })_{\alpha <\operatorname {cf} \kappa }}$ ${\displaystyle \kappa }$

Regarding 1. note that for all of the series, König's theorem , monotonic properties of the power of cardinal numbers, cardinal number arithmetic and the assumed continuum hypothesis were used for all smaller cardinal numbers. This chain of inequalities results in all cardinal numbers . Then also applies to all , because the power can at most be the same for all cardinal numbers because of the assumed continuum hypothesis , but equality cannot apply, since a singular cardinal number is not a successor cardinal number. The prerequisites 2. and 3. apply according to the choice of the normal sequence . Regarding 4. note that in the proof of 1. the equation was established for all cardinal numbers . Therefore , what applies to the stationarity in . ${\displaystyle \lambda <\lambda ^{\operatorname {cf} \lambda }\leq \lambda ^{\lambda }=2^{\lambda }=\lambda ^{+}}$ ${\displaystyle \lambda <\kappa }$ ${\displaystyle \lambda ^{\operatorname {cf} \lambda }=\lambda ^{+}}$ ${\displaystyle \lambda <\kappa }$ ${\displaystyle \lambda ^{\operatorname {cf} \kappa }<\kappa }$ ${\displaystyle \lambda <\kappa }$ ${\displaystyle <\kappa }$ ${\displaystyle \max\{\lambda ,\operatorname {cf} \kappa \}^{+}\leq \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle (\kappa _{\alpha })_{\alpha <\operatorname {cf} \kappa }}$ ${\displaystyle \lambda ^{\operatorname {cf} \lambda }=\lambda ^{+}}$ ${\displaystyle \lambda <\kappa }$ ${\displaystyle \{\alpha <\operatorname {cf} \kappa ;\kappa _{\alpha }^{\operatorname {cf} \kappa _{\alpha }}=\kappa _{\alpha }^{+}\}\supset \{\lambda ;\lambda <\operatorname {cf} \kappa \}}$ ${\displaystyle \operatorname {cf} \kappa }$

With that all requirements of the lemma are fulfilled, and it follows . Because as singular cardinal is a limit cardinal number that applies (see cardinal arithmetic ) and because of the assumption about is , for a total , which completes the proof. ${\displaystyle \kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$ ${\displaystyle \kappa }$ ${\displaystyle 2^{\kappa }=(2^{<\kappa })^{\operatorname {cf} \kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle \textstyle 2^{<\kappa }:=\sup _{\lambda <\kappa }2^{\lambda }=\sup _{\lambda <\kappa }\lambda ^{+}=\kappa }$ ${\displaystyle 2^{\kappa }=\kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$

Individual evidence

1. ^ Jack Silver: On the singular cardinals problem , Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Volume 1, pages 265-268, Canad. Math. Congress, Montreal, available online here ( Memento of the original dated November 13, 2013 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 56.8 MB) 
2. ↑ Thomas Jech : Set Theory , Springer-Verlag (2003), ISBN 3-540-44085-2 , Theorem 8.12
3. ↑ Thomas Jech: Set Theory , Springer-Verlag (2003), ISBN 3-540-44085-2 , Theorem 8.13