Alex Kiselev claims to have shown “There are no weakly inaccessible cardinals” in Zermelo-Fraenkel set theory (ZF). This would have the consequence that strongly inaccessible cardinals don’t exist either and so on for all the other large cardinals. Martin Davis on the FOM list cautions that the claim is “highly dubious”.
Here are links to Kiselev’s papers:
Link to the FOM list entry: http://www.cs.nyu.edu/pipermail/fom/2011-August/015694.html