Yong Cheng (Wuhan University)

The limit of Gödel's first incompleteness theorem

Note date and time change: Fri 4pm

In this talk, we discuss the limit of the first incompleteness theorem (G1). It is well known G1 can be extended to both extensions and weak sub-systems of PA. We examine the question: are there minimal theories for which G1 holds. The answer of this question depends on how we define the notion of minimality. We discuss different answers of this question based on varied notions of minimality.

The notion of interpretation provides us a general method to compare different theories in distinct languages. We examine the question: are there minimal theories for which G1 holds with respect to interpretability. It is known that G1 holds for essentially undecidable theories, and there are no minimal essentially undecidable theories with respect to interpretability. G1 holds for effectively inseparable (EI) theories and the notion of effective inseparability is much stronger than essential undecidability. A natural question is: are there minimal EI theories with respect to interpretability? We negatively answer this question and prove that there are no minimal effectively inseparable theories with respect to interpretability: for any EI theory T, we can effectively find a theory which is EI and strictly weaker than T with respect to interpretability. Moreover, we prove that there are no minimal finitely axiomatizable EI theories with respect to interpretability.

Finally, we give a summary of the similarities and differences between logical incompleteness and mathematical incompleteness based on technical evidences and philosophical reflections.