Silver's Theorem, proved by Jack Silver in 1975 and presented that year at the International Congress of Mathematicians, states that if kappa is a singular cardinal of uncountable cofinality and the generalized continuum hypothesis holds for every infinite cardinal below kappa, then it also holds at kappa itself. The result was a surprise, since set theorists had expected forcing to show the generalized continuum hypothesis could fail at such a cardinal even when it held below it, and Silver's proof introduced the master condition technique that became a standard tool in later forcing arguments over large cardinals.
Facts
StatementIf kappa is a singular cardinal of uncountable cofinality and the generalized continuum hypothesis holds for every infinite cardinal below kappa, meaning 2 to the lambda equals lambda-plus for every infinite lambda less than kappa, then the generalized continuum hypothesis also holds at kappa itself, meaning 2 to the kappa equals kappa-plus. 1 Classification
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Source Jack Silver, Wikipedia
Sources
1. Jack Silver, Wikipedia
Work section
In his 1975 paper "On the Singular Cardinals Problem", Silver proved that if a cardinal κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+.
References section, Silver 1975 ICM citation
Silver, Jack (1975). "On the singular cardinals problem". In Proceedings of the International Congress of Mathematicians 1, pp. 265-268.
In Group: Set Theory, Section: Contributions
In his 1975 paper "On the Singular Cardinals Problem", Silver proved that if a cardinal κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+.
In Branch: Set Theory, Article text
Jack Howard Silver (23 April 1942-22 December 2016) was an American set theorist and logician at the University of California, Berkeley.
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