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CATEGORIES:Midlands Logic Seminar
SUMMARY:Set theoretic principles required for categoricity
II - School of Maths\, University of Birmingham
DTSTART:20161122T160000Z
DTEND:20161122T180000Z
UID:TALK2320AT
URL:/talk/index/2320
DESCRIPTION:This is a continuation of the last talk in this se
ries and based on a paper by Simpson and Yokoyama
on the set theoretic principles required for categ
oricity of the axioms for the natural numbers. If
time permits we will also look at a follow-up pap
er by Kolodziejczyk and Yokoyama.\n\nDETAILS OF TH
E PREVIOUS TALK WILL NOT BE REQUIRED. \nAll welco
me.\n\nDedekind proved that two structures satisfy
ing the second order axioms for natural numbers ar
e canonically isomorphic. Obviously a key point in
the argument is that\, assuming two such structur
es are not isomorphic\, an inductive set must be c
onstructed in one structure that is not the whole
thing. Thus set theoretic principles are required
for categoricity results of this kind\, and these
arguments are sensitive to the set theoretic frame
work we are working in\, and in particular what se
t existence axioms are available.\n\nSo the questi
on is\, which set existence axioms are actually ne
eded? Simpson and Yokoyama investigated this quest
ion over the base theory that they call RCA0star.
The second half of this paper looks at different f
ormulations of the question where\, instead of Ded
ekind's system of (N\,S\,0) (a set\, a 1--1 succes
sor function and a number not in the image of S) t
hey look at structures based on order relations.
The answers are subtly different.\n\nA question le
ft open was whether or not there is some character
ization of the natural numbers by a second order s
entence which is provably categorical in RCA0star.
This was answered negatively by Kolodziejczyk and
Yokoyama.\n\nWe shall give a reading of these pap
ers. This will be a useful introduction to some of
the methods and results in so-called “reverse mat
hematics”.\n
LOCATION:Watson Building (Mathematics\, R15 on map) Room 31
0
CONTACT:Richard
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