Two Models of Consistency: A New Foundational Landscape

We are pleased to announce an upcoming guest lecture by the prominent logician and computer scientist, Professor Sergei Artemov.

ABSTRACT

In the 1930s, Gödel’s Second Incompleteness Theorem was interpreted as the Unprovability of Consistency Thesis (UCT): “There exists no consistency proof of a system that can be formalized in the system itself.” This thesis profoundly impacted mathematics, philosophy, AI/ATP, and cognitive studies. In fact, Gödel only demonstrated the unprovability of a specific consistency formula; it was von Neumann who popularized the broader UCT view.

We show that the standard consistency formula for Peano Arithmetic (PA) was a non-equivalent redefinition of Hilbert's original concept. We introduce a provably equivalent, finite arithmetical representation of Hilbert’s consistency, and deliver its mathematical proof that is fully formalizable within PA. This renders the traditional interpretation of UCT false. We do not correct Gödel, but rather the dogmatic orthodoxy that grew in his shadow. The Gödelian model measures relative strength, while the Hilbertian model provides internal trust. They are complementary. Practically, these findings leave the proof-theoretic landscape intact while removing the perceived logical barriers to self-verification in philosophy, epistemology, and Automated Theorem Proving.

About the Speaker

Sergei Artemov is a highly distinguished Russian-American researcher widely recognized for his foundational contributions to mathematics, logic, and computer science. He currently holds the title of Distinguished Professor at the Graduate Center of the City University of New York (CUNY), where he founded and directs the research laboratory for logic and computation.