A Photo of Barkley Rosser

I stumbled upon the following photo accompanying an interview with Albert Tucker (Maurer 1985).



The 3rd man from the left in the first row is the father of Barkley Rosser, Jr.

I have tried reading Rosser (1936), but I did not really understand the proof. As I understand it, Rosser puts some theorems of the time together to alter a theorem of Gödel's so that it's statement seems more natural. It is a very concise paper. Gödel shows that ω-consistency implies the existence of undecidable propositions. Rosser discarded the ω; he showed that consistency implies the existence of undecidable propositions. I guess there can be consistent systems that are not ω-consistent. Consistency is a syntactical property, and it does not require intuitions about universal quantification over all natural numbers.

Definition: A system is ω-inconsistent if and only if there exists a proposition p(n), with free variable n, such that

1. p(0) is provable, p(1) is provable, p(2) is provable, and so on.
2. It is provable that not ( for all n, p(n))

Definition: A system is ω-consistent if and only if it is not ω-inconsistent.

References

• Stephen B. Maurer (1985). "Albert Tucker", in Mathematical People: Profiles and Interviews (Ed. by D. J. Albers and G. L. Alexanderson), Birkhauser
• Barkley Rosser (1936) "Extensions of some Theorems of Gödel and Church", The Journal of Symbolic Logic, 1 (3), (September): 87-91.