[FOM] Ultrafinitism - help!

Nick Nielsen john.n.nielsen at gmail.com
Tue Jan 22 23:26:38 EST 2013

I believe that the classic paper of the genre is "The
Ultra-Intuitionistic Criticism and the Antitraditional Program for
Foundations of Mathematics" by  A.S. Yessenin-Volpin.

Yessenin-Volpin was an interesting figure, and some of his papers were
once available on the internet, but I didn't save a copy when I saw
them online (which I now regret) and they have since disappeared.

While the line between "strict finitism" and "ultrafinitism" has not
been strictly drawn, a good source for strict finitism is Strict
finitism: An examination of Ludwig Wittgenstein's remarks on the
foundations of mathematics, by Charles F. Kielkopf.

Following Kielkopf, one could go in a Wittgensteinian direction with
sources, but I get the impression that that's not exactly what you're
looking for.

Interestingly, Zermelo uses "ultrafinitism" as more or less a synonym
for the transfinite:

“Scientific reactionaries and anti-mathematicians have so eagerly and
lovingly appealed to the ‘ultrafinite antinomies’ in their struggle
against set theory. But these are only apparent ‘contradictions’, and
depend solely on confusing set theory itself, which is not
categorically determined by its axioms, with individual models
representing it.”

Zermelo, Ernst, “On Boundary Numbers and Domains of Sets” (1930),
collected in Ewald, William, From Kant to Hilbert: A Source Book in
the Foundations of mathematics, Oxford: Clarendon Press, 1996, p.

Best wishes,


On Tue, Jan 22, 2013 at 4:24 AM, <T.Forster at dpmms.cam.ac.uk> wrote:
> I have a student here who wants to write a fourth year (Part III) essay on ultrafinitism, a topic of which my knowledge is, as it were, *ultra*finite. Within 10 days i have to supply the faculty office a couple of paragraphs saying what the student should read and what sort of topics the essay should cover, which results should be proved, and so on.
>     Any enthusiasts out there who can get me out of this hole?
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