This paper seeks to identify the minimal restrictions that need to be placed on the naive comprehension principle to avoid inconsistency in set theory. Analysis of the logical antinomies shows that at the root of inco...This paper seeks to identify the minimal restrictions that need to be placed on the naive comprehension principle to avoid inconsistency in set theory. Analysis of the logical antinomies shows that at the root of inconsistency in naive set theory are certain ＂self contradictory＂ predicate functions in extensional set descriptions containing the matrix ＂-（x∈y）＂ （or ＂-（x∈x）＂） rather than ＂size,＂ vicious circularity, or self-reference. A reformed set comprehension system is proposed that excludes extensional set descriptions that conform to the formula, （Vx） （Зy） （x∈y →P （x）） （3u） （u∈y→（u∈y））, from comprehension and otherwise preserves the ontology of na＇fve set theory. This reform avoids the paradoxes by scrutiny of a set＇s description without recourse to type or other constructivist limitations on self-membership and has the most liberal rules for set formation conceivable including self-membership. The intuitive appeal for such an approach is compelling because as a revision of na＇fve set theory, it allows all possible set descriptions that do not lead to inconsistency.展开更多
文摘This paper seeks to identify the minimal restrictions that need to be placed on the naive comprehension principle to avoid inconsistency in set theory. Analysis of the logical antinomies shows that at the root of inconsistency in naive set theory are certain ＂self contradictory＂ predicate functions in extensional set descriptions containing the matrix ＂-（x∈y）＂ （or ＂-（x∈x）＂） rather than ＂size,＂ vicious circularity, or self-reference. A reformed set comprehension system is proposed that excludes extensional set descriptions that conform to the formula, （Vx） （Зy） （x∈y →P （x）） （3u） （u∈y→（u∈y））, from comprehension and otherwise preserves the ontology of na＇fve set theory. This reform avoids the paradoxes by scrutiny of a set＇s description without recourse to type or other constructivist limitations on self-membership and has the most liberal rules for set formation conceivable including self-membership. The intuitive appeal for such an approach is compelling because as a revision of na＇fve set theory, it allows all possible set descriptions that do not lead to inconsistency.
