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AIC Seminar Series

Moderately powerful 1st order theories are inconsistent

Carl HewittMIT Professor Emeritus[Home Page]

Notice:  Hosted by Richard Waldinger

Date:  Tuesday, November 14th 2017 at 4:00pm

Location:  EK255 (SRI E building)  (Directions)


Recording available at:

Slides available at:


Carl Hewitt

In 1934, Church published an article proving that moderately powerful 1st order theories are inconsistent. In fact, a 1st order theory is inconsistent if it proves that its theorems are computationally enumerable, which is not difficult to prove. For a long time, no one paid much attention to Church’s result. However, the needs of Computer Science have renewed interest in Church’s Paradox. The upshot is that powerful mathematical theories of Computer Science must use higher order theories. Fortunately, higher order theories can be categorical, i.e., characterize mathematical objects up to a unique isomorphism. For example, natural numbers, real numbers, ordinals, sets, and Actors can be categorically axiomatized, which is impossible using 1st order logic. Some consequences of adopting higher order theories contradict currently strongly held dogmas concerning logical completeness and proof of formal consistency that were established using 1st order theories. For example, higher order theories of natural numbers, ordinals, sets, and Actors prove that every true proposition about such mathematical objects is in fact provable in the respective theory. Consequently, there are no inferentially undecidable propositions in the theories. Even more challenging to conventional dogma is that theorems of these higher order theories are not computationally enumerable.

Of course, the above theorems are at odds with the current understanding of Gödel’s 1931 incompleteness results. This talk discusses how the disparity arose and draws some conclusions. For example, must theories for Intelligent Applications operate on absolutely consistent information, as required by 1st order logic, or should they accommodate possibly inconsistent information without blowing up?

   Bio for Carl Hewitt

Carl Hewitt is an emeritus professor of computer science (MIT) who is best known for his work on the Actor model of computation, which is in widespread use in eBay, Microsoft, Twitter, etc. For the last decade, his work has been in the field of Inconsistency Robustness, which aims to provide practical rigorous foundations for systems dealing with pervasively inconsistent information. He is co-editor, with John Woods assisted by Jane Spurr, of the monograph “Inconsistency Robustness” (Vol. 52 of Studies in Logic).

Hewitt is currently Board Chair of the International Society for Inconsistency Robustness (iRobust™) and also Board Chair of Standard IoT™, an international standards organization for the Internet of Things, which is using the Actor Model to unify and generalize emerging standards for IoT. Also, he has been a Visiting Professor at Keio University and Stanford.

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