The strong tree property and the failure of SCH
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Fontanella (J Symb Logic 79(1):193–207, 2014) showed that if \(\langle \kappa _n:n<\omega \rangle \) is an increasing sequence of supercompacts and \(\nu =\sup _n\kappa _n\), then the strong tree property holds at \(\nu ^+\). Building on a proof by Neeman (J Math Log 9:139–157, 2010), we show that the strong tree property at \(\kappa ^+\) is consistent with \(\lnot SCH_\kappa \), where \(\kappa \) is singular strong limit of countable cofinality.
KeywordsStrong tree property Singular cardinal hypothesis Large cardinals Compactness
Mathematics Subject Classification03E05 03E35 03E55
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