This depends on what you mean by `infinity'. For example, if you are talking about transfinite ordinals, `infinity' might mean omega (also written omega_0), the first transfinite ordinal. In that case, `infinity plus one' would mean omega + 1, which is well-defined, and is distinct from omega (omega +1 = omega union { omega }). It has the same cardinality, yes, but ordinals are not cardinals.