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We show that Shelah cardinals are preserved under the canonical $${{\mathrm{GCH}}}$$GCH forcing notion. We also show that if $${{\mathrm{GCH}}}$$GCH holds and $$F:{{\mathrm{REG}}}\rightarrow {{\mathrm{CARD}}}$$F:REGÅºCARD is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $$\forall \kappa \in {{\mathrm{REG}}},~ 2^{\kappa }=F(\kappa )$$ÅºÅºÅºREG,2Åº=F(Åº). This gives a partial answer to a question asked by Cody (Arch Math Logic 52(5---6):569---591, 2013) and independently by Honzik (Acta Univ Carol 1:55---72, 2015). We also prove an indestructibility result for Shelah cardinals.
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