[FOM] Did Gödel believe V=L?

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Tue Jun 3 02:02:09 EDT 2003

   In an effort to extend Martin Davis's "honor roll" of logicians who have
proposed inconsistent systems, Panu suggests that at one stage Gödel
believed inconsistent things about sets.  (We leave to one side the episode
late in his life when he tried to refute the continuum hypothesis by
proposing a new axiom that implied it [1970a/b/c in v. III of G's
"Collected Works"]: by that age he was entitled to a mistake!)  One side of
the contradiction is the "Axiom" of constructibility, V=L, leading to the
question, did Gödel ever actually believe this?
   From the very strongly platonistic standpoint Gödel espoused in later
years (it can certainly be detected in his  philosophical papers of the
1940s), it seems to me that the Axiom of Constructibility ought to SEEM
unlikely: why SHOULD the ... conceptualistic? ... scheme of introducing new
sets by (parametric) explicit definition, even if pursued to infinity, be
expected yield ALL the inhabitants of Plato's heaven.  And Gödel (in
conversations with Wang, for one place) claimed that his "realist"
philosophical stance had informed his logical work from the beginning.
    But old men's recollections of their youthful beliefs are suspect, so
this certainly doesn't settle the question of what his attitude to V=L was
in the 1930s.
    As far as I know, he didn't anywhere say, in print, anything like "I
believe this axiom" or "this axiom seems likely to be true," but that
doesn't settle the question either: he WOULDN'T have said these things,
even if he believed them.  For whatever reason (cf. Feferman's 1984 essay
"Kurt Gödel: conviction and caution" in "Philosophia Naturalis" 21; repr.
in F's recent book), Gödel was very cautious in his public philosophical
pronouncements, very loath to say things-- like drawing a sharp distinction
between truth and provability-- that micht have gotten him accused of being
a metaphysician.  (And, of course, saying things like "it seems likely to
me..." would be out of keeping with the style of mathematical journals!)
    What he DOES say [1938 in "Collected Works" II]is that  V=L "added as a
new axiom seems to give a natural completion to the axioms of set theory,
in so far as it determines the vague notion of an arbitrary infinite set in
a defrinite way."  This, it must be admitted, SOUNDS like a commendation of
the axiom, but-- particularly in its historical context-- I don't think it
has to be understood that way.  I think Gödel was simply making a comment
about the nature, or "methodological" function, of the axiom, a comment
that was neutral as to its truth or plausibility.  It had been noted, and
discussed by philosophers of science, that a number of interesting sets of
axioms contained tow different kinds of axioms: ordinary ones, making
positive assertions about some system of objects, and then a "completing"
axiom saying, if effect, that the system of objects described was the
biggest (or smallest) possible model of the other axioms.   Examples; the
fifth Peano postulate, induction, says that the ONLY numbers are the ones
obtained by repeatedly going "+1" from the starting  number, so that the
system of natural numbers  is the smallest system satisfying the first four
postulates.  Hilbert's axioms for geometry end (in editions 2 through 6 of
his "Grundlagen der Geometrie") with a "completeness axiom" saying that the
system of geometrical objects (points, lines, planes) "constitue a system
of things which cannot be extended while maintaining simultaneously the
cited axioms," so that they form a maximal model of the earlier axioms.
Gödel was certainly familiar with this sort of thing.  His old friend
Carnap had (together with Friedrich Bachmann) published a study of this
sort of axiom, "Über extremalaxiome," two years before ("Erkenntnis," v. 6
(1936); Eng. tr. "On Extremal Axioms," in "History and Philosophy of
Logic," v. 2 (1981), pp. 67-85). Now V=L is certainly of this nature: if we
take the ordinals as given, the constructible sets (V) are the smallest
model of the other ZF axioms that contains those ordinals.  Gödel's
definition of the constructible hierarchy has not, historically, always
been found transparent: in saying that V=L was, in Carnap and Bachmann's
sense, an extremal axiom, Gödel was making an insightful and (perhaps)
helpful comment on how his model worked.  I think his choice of the word
"completion" in describing it was a deliberate reference to Hilbert's usage
(but that may just be me).
    So, did Gödel, in 1938, ***believe*** that  V=L?  I'm not sure.  (My
apologies, Panu, for responding to your post with such a rant.)
Allen Hazen
Philosophy Department
University of Melbourne
Interests: historico-conceptual pedantry
I think G's choice of the word "completion"

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