# Forcing lightface definable well-orders without the CGH

David Aspero, Peter Holy, Philipp Luecke

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
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.