The inconsistencies involved in the foundation of set theory were invariably caused by infinity and self-reference; and only with the opportune axiomatic restrictions could them be obviated. Throughout history, both concepts have proved to be an exhaustible source of paradoxes and contradictions. It seems therefore legitimate to pose some questions concerning their formal consistency. This is just the objective of this paper. Starting from an extension of Cantors paradox that suggests the inconsistency of the actual infinity, the paper makes a short review of its controversial history and proposes a new way of criticism based on w-order. Self-reference is also examined from a critique perspective which includes syntactic and semantic considerations. The critique affects the formal sentence involved in Godels first incompleteness theorem and its ordinary language interpretation.
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