the Barth sextic, which is more than the maximum of 52 claimed by Severi. Rokhlin invariant. Rokhlin incorrectly claimed in 1951 that the third stable stem of the homotopy groups of spheres is of order 12. In 1952 he discovered his error: it is in fact cyclic of order 24. The difference is crucial as it results in the existence of the Rokhlin invariant, a fundamental tool in the theory of 3- and 4-dimensional manifolds. In 1961, Jan-Erik Roos published an incorrect theorem about the vanishing of the first derived functor of the inverse limit functor under certain general conditions. However, in 2002, Amnon Neeman constructed a counterexample. Roos showed in 2006 that the theorem holds if one adds the assumption that the category has a set of generators. The Schur multiplier of the Mathieu group M22 is particularly notorious as it was miscalculated more than once: first claimed it had order 3, then in a 1968 correction claimed it had order 6; its order is in fact (currently believed to be) 12. This caused an error in the title of Janko's paper A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroup on J4: it does not have the full covering group as a subgroup, as the full covering group is larger than was realized at the time. The original statement of the classification of N-groups by Thompson in 1968 accidentally omitted the Tits group, though he soon fixed this. In 1967 Reinhardt proposed Reinhardt cardinals, which Kunen showed to be inconsistent with ZFC in 1971, though they are not known to be inconsistent with ZF. Per Martin-Löf's original version of intuitionistic type theory proposed in 1971 was shown to be inconsistent by Jean-Yves Girard in 1972, and was replaced by a corrected version. In 1975, Leitzel, Madan, and Queen incorrectly claimed that there are only 7 function fields over finite fields with genus > 0 and class number 1, but in 2013 Stirpe found another; there are in fact exactly 8. Busemann–Petty problem. Zhang published two papers in the Annals of Mathematics in 1994 and 1999, in the first of which he proved that the Busemann–Petty problem in R4 has a negative solution, and in the second of which he proved that it has a positive solution. Algebraic stacks. The book on algebraic stacks mistakenly claimed that morphisms