If the concept of proof （including arithmetic proof） is syntactically restricted to closed sentences （or their Godel numbers）, then the standard accounts of Godel＇s Incompleteness Theorems （and Lob＇s Theorem） ...If the concept of proof （including arithmetic proof） is syntactically restricted to closed sentences （or their Godel numbers）, then the standard accounts of Godel＇s Incompleteness Theorems （and Lob＇s Theorem） are blocked. In these standard accounts （Godel＇s own paper and the exposition in Boolos＇ Computability and Logic are treated as exemplars）, it is assumed that certain formulas （notably so called ＂Godel sentences＂） containing the Godel number of an open sentence and an arithmetic proof predicate are closed sentences. Ordinary usage of the term ＂provable＂ （and indeed ＂unprovable＂） favors their restriction to closed sentences which unlike so-called open sentences can be true or false. In this paper the restricted form of provability is called strong provability or unprovability. If this concept of proof is adopted, then there is no obvious alternative path to establishing those theorems.展开更多
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
A further reformulation of Naive Set Comprehension related to that proposed in ＂Resolving Insolubilia： Internal Inconsistency and the Reform of Naive Set Comprehension＂ （2012） is possible in which contradiction i...A further reformulation of Naive Set Comprehension related to that proposed in ＂Resolving Insolubilia： Internal Inconsistency and the Reform of Naive Set Comprehension＂ （2012） is possible in which contradiction is averted not by excluding sets such as the Russell Set but rather by treating sentences resulting from instantiation of such sets as the Russell Set in their own descriptions as invalid. So the set of all sets that are not members of thernselves in this further revision is a valid set but the claim that that set is or is not a member of itself is not validly expressible. Such an approach to set comprehension results in a set ontology co-extensive with that permitted by the Naive Set Comprehension Principle itself. This approach （that may be called Revised Set Comprehension Ⅱ） has as strong a claim to consistency as that formulated in ＂Resolving Insolubilia： Internal Inconsistency and the Reform of Naive Set Comprehension.＂展开更多
文摘If the concept of proof （including arithmetic proof） is syntactically restricted to closed sentences （or their Godel numbers）, then the standard accounts of Godel＇s Incompleteness Theorems （and Lob＇s Theorem） are blocked. In these standard accounts （Godel＇s own paper and the exposition in Boolos＇ Computability and Logic are treated as exemplars）, it is assumed that certain formulas （notably so called ＂Godel sentences＂） containing the Godel number of an open sentence and an arithmetic proof predicate are closed sentences. Ordinary usage of the term ＂provable＂ （and indeed ＂unprovable＂） favors their restriction to closed sentences which unlike so-called open sentences can be true or false. In this paper the restricted form of provability is called strong provability or unprovability. If this concept of proof is adopted, then there is no obvious alternative path to establishing those theorems.
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.
文摘A further reformulation of Naive Set Comprehension related to that proposed in ＂Resolving Insolubilia： Internal Inconsistency and the Reform of Naive Set Comprehension＂ （2012） is possible in which contradiction is averted not by excluding sets such as the Russell Set but rather by treating sentences resulting from instantiation of such sets as the Russell Set in their own descriptions as invalid. So the set of all sets that are not members of thernselves in this further revision is a valid set but the claim that that set is or is not a member of itself is not validly expressible. Such an approach to set comprehension results in a set ontology co-extensive with that permitted by the Naive Set Comprehension Principle itself. This approach （that may be called Revised Set Comprehension Ⅱ） has as strong a claim to consistency as that formulated in ＂Resolving Insolubilia： Internal Inconsistency and the Reform of Naive Set Comprehension.＂
