We show that Shelah cardinals are preserved under the canonical
GCH forcing notion. We also show that if
GCH holds and
F:REGźCARD 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
źźźREG,2ź=F(ź). 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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