Consistency And Interpretation Of Neo-Frege Arithmetic

If we want to probe the reason why there occurs the inconsistency in Frege's Grundge-setze, then we should not only find out the model in which we can avoid this phenomenon of consistency, but also the interpretability relating to this model of consistent arithmetic. Out of this train of thought, My work can be divided into these two parts:The first one is to prove the consistency in the first-order portion and second-order fragment of arithmetic, The other one is to prove the interpretability strength of these fragment of arithmetic. The major previous results on the interpretability strength of the subsystems of PA2, HP2, BLV2 can be described as follows. Frege once showed that PA2(?)HP2; Heck and Linnebo noted that Frege's proofs in fact show that?11-CA0(?)?11-HP0; Further, Boolos showed that the converse holds, so that one has?11-CA0??11-CA0; Heck then showed that ABL0(?)Q, Ganea and Visser independently showed that the converse holds, so that ABL0?Q.Burgess showed that AHP0(?)Q; Ferreira and Wehmeier showed that?11-BL0 is consistent. Recently, Walsh showed that?11-LB0+(?)?11-AC0 and that ACA0(?)?11-PH0. Over the past 30 years, philosophers have studied the systems closely related to subsystems of second-order arithmetic. These constructions use tools from computability theory, including:hyperarithmetic theory, computable model theory and reverse mathematics. The reason why we introduced Frege Arithmetic and Frege's The-orem is that, more than one century from now, informal arithmetic has almost without exception been given some Peano style axiomatization. These axiomatizations regard the natural numbers as finite ordinals, individuated by their position in an?-sequence. Frege's Theorem shows that an alternative and conceptually completely different ax-iomatization of arithmetic is possible, based on the idea that the natural numbers are finite cardinals, individuated by the cardinalities of the concepts whose numbers they are.