FOM: Mathematicians' views of Goedel's incompleteness theorem(s)
John Case
case at eecis.udel.edu
Mon Dec 1 21:10:48 EST 1997
Vaughan's point about denial is a fun one. Couple other points follow.
Positive Note
Hmmm, some non-logician mathematicians seem to be aware that some sentences of
interest to them may be/are (sometimes provably) independent of all the
mathematical truths known to them. This is not, as I understand it, what was
the concern, but provides hope for them (if I'm right about them (-8 ).
Not So Positive Note
Many non-logician mathematicians don't really know what the Goedel theorems
say methinks. How many of them know that the problem is with recursive (and
some other) axiomatizations, not axiomatizations in general. It's easy to
completely axiomatize first (or second) order arithmetic. Just take as the
set of axioms all and only the true sentences. Of course this is not even
an arithmetical axiomatization. How many non-logician mathematicians, if told
this, would realize the point is it is hard to check proofs in non-recursive
(or non-r.e.) axiomatizations? Of course if humans aren't essentially discrete
machines, maybe (some of them) can check proofs in a system where the set of
axioms is not even arithmetical (but the rest of us would have trouble checking
those humans (-8 ). Empirically, even long _mechanical_ proof checking can be
tough for most of us humans (actually it seems to me pretty unlikely that any
of us humans surpass the mechanical). Anyhow, I suspect most non-logicians are
missing which kinds of axiomatizations are at stake and why that is
important (as are some/most/all? popular accounts of the Goedel theorems).
(-8 John
More information about the FOM
mailing list