新弗雷格算术的一致性和解释性

【作者】 薄谋；

【导师】 刘放桐；

【作者基本信息】 复旦大学 ， 外国哲学， 2011， 博士

【摘要】 如果真要探究弗雷格算术系统中的非一致性根源,那么,不仅需要找到算术片段一致性的模型,而且也要找出算术片段的可解释性。因此,本文的主线索有两条：一条是证明弗雷格算术一阶片段和二阶片段的一致性；另一条是证明诸片段的可解释性。但是,两者并不是泾渭分明的,而是经常交叉在一起的。关于PA2,HP2,BLC2子系统和一致性和解释性,我们取得的主要成果有：Frege本人实际上阐明PA2(?)HP2；Heck和Linnebo阐明Π11-CA0(?)Π11-HP0；Boolos阐明Π11-HP0(?)Π11-CA0；同时得到Π11-CA0≡Π11-CA0；Heck｜阐述ABL0(?)Q；Ganea和Visser｜阐述Q(?)ABL0；同时得到ABL0≡Q；Burgess｜阐述AHP0(?)Q；Ferreira和Wehmeier阐述△11-BL0是一致的；对此两人证明的微小改进表明∑11-LB0是一致的；对整个证明的观察会表明∑11-LB0(?)Π11-CA0。Walsh用超算术理论证明∑11-LB0+(?)Σ11-AC0；根据递归饱和域,最终证明ACA(?)∑11-PH0.在对弗雷格算术片段的一致性证明过程中,Burgess使用了有穷论的证明论方法；而Heck等人采用了无穷论的模型论方法。而在采用模型论证明的过程中,Heck采用了变元-约束项-形成算子,而Wehmeier采用了△11-概括公式。由此,近30年来,弗雷格数学哲学研究主要采用了如下三种方法：超算术理论,计算模型理论和逆数学理论。弗雷格算术是由二阶逻辑和休谟原则构成的：而弗雷格定理阐述的是,二阶皮亚诺算术的所有公理都是可以从弗雷格算术和FD中推导出来的。引出弗雷格算术的意义在于,近一个世纪以来,非形式算术几乎无一例外地都被赋以某种Peano-Dedekind式公理化形式。这些公理化形式把自然数认作有穷序数,通过它们在ω-序列中所处的位置而得以个体化。然后,弗雷格定理表明,一个可替代的和概念上完全不同的算术公理化形式也是可能的,而基本的思路就是自然数是有穷基数,通过对概念取数的方式即概念数的基数性而得以个体化。更多还原

【Abstract】 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 thatΠ11-CA0(?)Π11-HP0; Further, Boolos showed that the converse holds, so that one hasΠ11-CA0≡Π11-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 thatΔ11-BL0 is consistent. Recently, Walsh showed thatΣ11-LB0+(?)Σ11-AC0 and that ACA0(?)Σ11-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 anω-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.更多还原