# Forcing lightface definable well-orders without the CGH

Aspero, David, Holy, Peter and Luecke, Philipp (2015) Forcing lightface definable well-orders without the CGH. Annals of Pure and Applied Logic, 166 (5). 553–582. ISSN 0168-0072

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For any given uncountable cardinal $$\kappa$$ with $$\kappa^{{<}\kappa}=\kappa$$, we present a forcing that is $$<\kappa$$-directed closed, has the $$\kappa^+$$-c.c. and introduces a lightface definable well-order of $$H(\kappa^+)$$. We use this to define a global iteration that does this for all such $$\kappa$$ simultaneously and is capable of preserving the existence of many large cardinals in the universe.