FOM: Mathematicians' views of Goedel's incompleteness theorem(s)

John Case case at
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

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