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SUMMARY:Consistency\, a catchword making the second incompleteness theorem
more spectacular than the first: Comments on a comment by Georg Kreisel
- Doukas Kapantais (Athens)
DTSTART;VALUE=DATE-TIME:20170213T163000Z
DTEND;VALUE=DATE-TIME:20170213T183000Z
UID:https://talks.ox.ac.uk/talks/id/7af4865c-52bf-4a29-bb81-bb57ed6042b3/
DESCRIPTION:As Gödel himself stressed\, back in 1931\, his second theorem
is irrelevant to any sensible consistency problem. In any case\, if ConF
is in doubt\, why should it be proved in F (and not in an incomparable sys
tem)? […] He knew only too well the publicity value of this catch word [
i.e. “consistency”]\, which –contrary to his own view of the matter
– had made his second incompleteness theorem more spectacular than the f
irst.” \nGeorg Kreisel\, 1980\, “Kurt Gödel. 28 April 1906-14 January
1978”\, Biographical Memoirs of the Fellows of the Royal Society\, 26:
174.\n\nI will comment on this passage in relation to the several projects
of creating formal theories of arithmetic which\, unlike Peano Arithmetic
\, could possibly prove their own consistency. I distinguish between thos
e formulas of a formal theory F which\, under their canonical interpretati
on\, (i) carry the information that F is consistent and (ii) those that do
not carry the information that F is consistent. ConF trivially belongs to
(ii). Assume that we believe in F’s soundness\, and thereby its consis
tency. If a formula does not carry the information that F is consistent\,
it is a sensible project to try to prove/disprove this formula in F: one b
elieves that F always tells the truth\, and so\, one will believe F’s ve
rdict on this formula\, which says something different from the things one
already believes in. On the other hand\, if the formula carries the info
rmation that F is consistent\, and we already believe that F is sound\, an
d thereby consistent\, F’s possibly affirmative verdict is of purely alg
orithmic interest. We would believe the formula\, but only because we alre
ady believe in F’s soundness\, and thereby its consistency. Finally\, i
f we do not already believe in the soundness of F\, F’s potentially affi
rmative verdict on any formula belonging to (ii)\, would\, in itself\, hav
e no epistemological value whatsoever with regard to F’s consistency. I
will elaborate on this argument and apply it to arithmetics using Rosser
provability.\n\nSpeakers:\nDoukas Kapantais (Athens)
LOCATION:Radcliffe Humanities (Ryle Room\, First Floor)\, Woodstock Road O
X2 6GG
URL:https://talks.ox.ac.uk/talks/id/7af4865c-52bf-4a29-bb81-bb57ed6042b3/
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DESCRIPTION:Talk:Consistency\, a catchword making the second incompletenes
s theorem more spectacular than the first: Comments on a comment by Georg
Kreisel - Doukas Kapantais (Athens)
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