[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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