[FOM] Is Godel's Theorem surprising?
Mark van Atten
Mark.vanAtten at univ-paris1.fr
Sat Dec 9 05:01:11 EST 2006
(The following is an extract from the chapter 'Gödel's logic', by
Juliette Kennedy and me, for Elsevier's upcoming Handbook fo the History
of Logic)
Brouwer must have realized already around 1907 that one can diagonalize
out of formal systems. In his dissertation of that year he noted that
the totality of all possible mathematical constructions is `denumerably
unfinished'; by this he meant that `we can never construct in a
well-defined way more than a denumerable subset of it, but when we have
constructed such a subset, we can immediately deduce from it, following
some previously defined mathematical process, new elements which are
counted to the original set'. In one of the notebooks leading up to his
dissertation, Brouwer stated that `The totality of mathematical theorems
is, among other things, also a set which is denumerable but never finished'.
In 1928, Brouwer gave two lectures in Vienna: on March 10, on general
philosophy and intuitionistic foundations of mathematics; and on March
14, on the intuitionistic theory of the continuum. According to Hao
Wang, `it appears certain that Gödel must have heard the two lectures';
indeed, Gödel wrote to Menger on April 20, 1972 that `I only saw
[Wittgenstein] once in my life when he attended a lecture in Vienna. I
think it was Brouwer's'. An entry in Carnap's diary for December 12,
1929, states that Gödel talked to him that day `about the
inexhaustibility of mathematics (see separate sheet) He was stimulated
to this idea by Brouwer's Vienna lecture. Mathematics is not completely
formalizable. He appears to be right'. On the `seperate sheet', Carnap
wrote down what Gödel had told him:
`We admit as legitimate mathematics certain reflections on the grammar
of a language that concerns the empirical. If one seeks to formalize
such a mathematics, then with each formalization there are problems,
which one can understand and express in ordinary language, but cannot
express in the given formalized language. It follows (Brouwer) that
mathematics is inexhaustible: one must always again draw afresh from the
``fountain of intuition''. There is, therefore, no characteristica
universalis for the whole mathematics, and no decision procedure for the
whole mathematics. In each and every closed language there are only
countably many expressions. The continuum appears only in ``the whole of
mathematics'' ... If we have only one language, and can only make
``elucidations'' about it, then these elucidations are inexhaustible,
they always require some new intuition again.'
This record contains in particular elements from the second of Brouwer's
lectures, in which one finds the argument that Gödel refers to: on the
one hand, the full continuum is given in a priori intuition, while on
the other hand, it cannot be exhausted by a closed language with
countably many expressions.
This explains why Brouwer could say (e.g., to Freudenthal and to Wang)
that he was not surprised by Gödel's incompleteness theorems.
But of course, in the theorems that Gödel, after having been inspired by
Brouwer's lectures, eventually arrived at, he went considerably beyond
Brouwer. As Gödel stressed (without reference to Brouwer's lectures) in
his letter to Zermelo of October 12, 1931: "I would still like to remark
that I see the essential point of my result not in that one can somehow
go outside any formal system (that follows already according to the
diagonal procedure), but that for every formal system of metamathematics
there are statements which are expressible within the system but which
may not be decided from the axioms of that system, and that those
statements are even of a relatively simple kind, namely, belonging to
the theory of the positive whole numbers."
Mark van Atten.
--
IHPST (Paris 1/CNRS/ENS)
13 rue du Four, F-75006 Paris, France
tel ++ 33 (0)1 43 54 94 60
fax ++ 33 (0)1 43 25 29 48
http://www-ihpst.univ-paris1.fr
--
Ce message a ete verifie par MailScanner
pour des virus ou des polluriels et rien de
suspect n'a ete trouve.
More information about the FOM
mailing list