Finding the limit of incompleteness I [article]

Yong Cheng
2020 arXiv   pre-print
In this paper, we examine the limit of applicability of Gödel's first incompleteness theorem (G1 for short). We first define the notion "G1 holds for the theory T". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which G1 holds. To approach this question, we first examine the following question: is there a theory T such that Robinson's R interprets T but T does not interpret R (i.e. T is weaker than R w.r.t. interpretation) and
more » ... G1 holds for T? In this paper, we show that there are many such theories based on Jeřábek's work using some model theory. We prove that for each recursively inseparable pair 〈 A,B〉, we can construct a r.e. theory U_〈 A,B〉 such that U_〈 A,B〉 is weaker than R w.r.t. interpretation and G1 holds for U_〈 A,B〉. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree 0< d<0^', there is a theory T with Turing degree d such that G1 holds for T and T is weaker than R w.r.t. Turing reducibility. As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which G1 holds.
arXiv:1902.06658v2 fatcat:4wyrwroil5h3hnhparlk64kd54