[FOM] Explaining incompleteness
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Fri Oct 21 00:20:09 EDT 2005
((First, a correction: in my posting of 20.x.05, I said Gödel's proof
of the incompleteness of arithmetic provided a "Pi-1-1" example of an
unprovable sentence: I of course should have said "Pi-0-1". (I keep
doing that: the superscript denoting that it is blocks of FIRST order
quantifiers we are counting is 0, but this was at least the second
FoM posting where I wrote 1. I'm sorry.))
Second, about explanation.... I look forward to reading the
Tappenden paper that Richard Heck has referred to. There is an old
paper by Mark Steiner, "Mathematical explanation and scientific
knowledge" ("Nous" vol. 12 (1978)). My vague recollection is that a
one-line summary might be something like: an explanation has to refer
to an "essential" feature of the topic, and so to features that would
also decide similar questions in a range of variant situations.
Roger Bishop Jones, in the post that strated this string, mentions
two candidate "explanations" for the incompleteness of arithmetic:
Gödel's explanation in terms of the undefinability of truth (thanks,
Jeff Ketland, for the textual smoking gun!) and one turning on the
fact that arithmetic is undecidable and that proofs have to be
finite, effectively recognizable, "certificates" of theoremhood. It
is perhaps evidence that the "truth" explanation is "right"
("righter?") that it gives you incompleteness in two related
situations where certificates of "theoremhood" are NOT finite and
effectively recognizable. One is Jeroslow's notion of an
"experimental logic" which I mentioned in my previous post. (Here a
sentence is a theorem iff (very roghy) it is proven from assumptions
which will not subsequently be refuted; an infinite string of future
proofs would have to be examined to verify that the assumptions WILL
not be refuted.) Another is the case of Second-Order PA, formulated
with an Omega rule, treated in Rosser's classic "Gödel Theorems for
Nonconstructive Logics" ("JSL" v. 2 (1937)). (Here a "proof" is in
effect an infinite tree of formulas, and the notion of proof is
definable in the Second-Order language. The system is complete for
First-Order statements, but not for Second-Order.)
But I am not confident that I understand the notion of essentiality
Steiner appealed to.
--
Allen Hazen
Philosophy Department
University of Melbourne
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