Formal Theories of Truth and Nonstandard Models of Arithmetic

W załączniku

In this dissertation we study certain model-theoretic properties of formal theories of truth in the context of the debate on the so-called de ationism about truth and some computability-theoretic properties of nonstandard models of arithmetic and their presentations in the context of the debate on the distinguishability of the intended model of arithmetic. Further, we investigate the relation between the arithmetical and the truth-theoretic structures of models of su ciently strong theories. Finally, we examine certain nitistic solutions to Yablo's antinomy under the modal interpretation of quanti ers. The main mathematical results of the thesis are that: The axiomatic theory TB of local disquotational truth is not semantically conservative over any complete extension of PA. Any model Mj= PA has an elementary (nonstandard) extension K which is expandable to a model of TB, but not recursively saturated. No nonstandard model of PA has a computable quotient presentation by a computably enu- merable equivalence relation, even in the restricted (but fully expressive) language f+; g with only addition and multiplication: there is no computable structure (N; ; ) and a computably enumerable equivalence relation E, which is a congruence with respect to this structure, such that the quotient (N; ; )=E is a nonstandard model of PA. No nonstandard model of arithmetic in the language f+; ; g has a computably enumerable quotient presentation by any equivalence relation, of any complexity. That is, there is no com- putably enumerable structure hN; ; ;Ei, where and are computable binary operations and E is a computably enumerable relation, and an equivalence relation E that is a congru- ence with respect to that structure, such that the quotient hN; ; ;Ei=E is a nonstandard model of arithmetic in the language f+; ; g. There is no computable structure hN; ; i and a co-computably enumerable equivalence relation E, which is a congruence with respect to this structure, such that the quotient hN; ; i=E is a nonstandard model of true arithmetic. There is no computable structure hN; ; i and a co-computably enumerable equivalence relation E, which is a congruence with respect to this structure, such that the quotient hN; ; i=E is a 1-sound nonstandard model of arithmetic, or even merely a nonstandard model of arithmetic with 00 in the Standard System of the model. A corollary of this is actually a strengthening of the form: no nonstandard model of arithmetic in the language f+; ; 0; 1;