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A strictly finitary non-triviality proof for a paraconsistent system of set theory deductively equivalent to classical ZFC minus foundation

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The paraconsistent system CPQ-ZFC/F is defined. It is shown using strong non-finitary methods that the theorems of CPQ-ZFC/F are exactly the theorems of classical ZFC minus foundation. The proof presented in the paper uses the assumption that a strongly inaccessible cardinal exists. It is then shown using strictly finitary methods that CPQ-ZFC/F is non-trivial. CPQ-ZFC/F thus provides a formulation of set theory that has the same deductive power as the corresponding classical system but is more reliable in that non-triviality is provable by strictly finitary methods. This result does not contradict Gödel's incompleteness theorem because the proof of the deductive equivalence of the paraconsistent and classical systemss use non-finitary methods.
Original languageEnglish
Pages (from-to)581-598
Number of pages18
JournalArchive for Mathematical Logic
Volume39
Issue number8
DOIs
Publication statusPublished - 2000

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