[FOM] Yessenin-Volpin's consistency proof for ZF

Jean Paul van Bendegem jpvbende at vub.ac.be
Wed Oct 26 14:13:51 EDT 2005

 > In his paper "The Ultraintuitionistic Criticism and the
 > Antitraditional Program for Foundations of Mathematics",
 > Yessenin-Volpin claimed to have a consistency proof for ZF with any
 > finite number of inaccessible cardinals from his "ultra-intuitionistic"
 > standpoint. Does anyone know where he gave the details of the proof?

I know of three papers by Volpin, not in Russian. There is supposed to be a
typescript in Russian containing the proof, but I have never seen it. I did
write to him many years ago, but what I received was a list of publications,
not the papers.

YESSENIN-VOLPIN, A. S. : "Le programme ultra-intuitioniste des fondements
des mathématiques". In: Infinitistic Methods, Proceedings Symposium on
Foundations of Mathematics, Pergamon Press, Oxford, 1961, pp. 201-223.
YESSENIN-VOLPIN, A. S. : "The ultra-intuitionistic criticism and the
antitraditional program for foundations of mathematics". In: KINO, MYHILL &
VESLEY (eds.), Intuitionism & proof theory. North-Holland, Amsterdam, 1970,
pp. 3-45.
YESSENIN-VOLPIN, A. S. : "About infinity, finiteness and finitization". In
RICHMAN, F. (ed.), 1981, pp. 274-313.

Someone who has tried to make sense of the underlying logic of Volpin,
(though not the consistency proof) is David Isles.

David ISLES: "What Evidence is There That 2^65536 is a Natural Number?"
Notre Dame Journal of FormalLogic, Vol. 33, nr. 4, 1992, pp. 465-480.
David ISLES: "A Finite Analog to the Löwenheim-Skolem Theorem. Studia
Logica, Vol. 53, 1994, pp. 503-532.
And, if German is no problem, a nice historical situation of the work of
Volpin is the book by Ernst Welti. Interestingly or oddly enough (depending
on your taste), one of his supervisors was Paul Feyerabend.
Ernst WELTI, Die Philosophie des strikten Finitismus.
Entwicklungstheoretische und mathematische Untersuchungen über
Unendlichkeitsbegriffe in Ideengeschichte und heutiger Mathematik, Bern,
Peter Lang, 1987.

Jean Paul Van Bendegern 

More information about the FOM mailing list