A Note on Seminar Logic and Analysis
Model Theory of Arithmetic and Cuts (Speaker: Tin Lok Wong. Friday 20 April, 2012) Naohi Eguchi Mathematical InstituteTohoku University, Japan
[email protected] April 20, 2012. Updated on May 7, 2012.
Backgrounds.
Abraham Robinson's works during 1950s1970s.
De nition 1. A model M of a theory T is called existentially closed (e.c. for short) if for any model N of T , any ∆0 -formula ϕ and for any constants c > M , M à §xϕ x, c holds whenever M b N and N à §xϕ x, c hold. Ñ
Ñ
Example 2.
Ñ
Ñ
Ñ
Ñ
Ñ
1. Any algebraically closed eld is an e.c. model of the theory of elds. (Hilbert, 1893)
2. The structure R, 0, 1, , , @ is an e.c. model of the theory of ordered elds. (Tarski, 1948)
Fact 3 (Rabin [3]). No model of PA is e.c.. Proof Sketch.
Let
M
be a model of
PA.
Then by Gödel's second incompleteness theorem
M
of PA ConPA such 0 1 in It is not di cult to check that x is (the Gödel number of ) a proof of 0 1 in PA can be expressed by a ∆0 -formula. On the other hand M ~ “There exists a proof of 0 1 in PA . Hence M is not e.c.. (and standard model-theoretic arguments) there exists a model
that
M b M .
Hence
M
à“
PA .
There exists a proof of
à
LA be the language of PA. To solve the problem, for each LA -formula θ we add a new function symbol Fθ to LA . De nition 4. For each LA -formula θ the function Fθ is de ned by From this fact it natural to ask what is wrong with
Ñ F θ x
An expanded language
Ñ y 1 µy θx, 0
¢ ¨ ¨ ¦ ¨ ¨ ¤
PA.
Let
if §yθ x, y holds, otherwise.
LSk is de ned by LSk LA 8
Ñ
Fθ S θ
is a
LA -formula
.
Seminar Logic and Analysis - Ghent University: http://cage.ugent.be/zwc/loganseminar.en.html
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De nition 5.
PA The de ning axiom for Fθ S θ is a
PASk
LA -formula .
Fact 6. Every model of PASk is an e.c. model of PASk . De nition 7. Let M à PA. 1. A
cut
2. An
of M is a proper nonempty initial segment of M .
We write
N
cut of M .
LA -formula θ.
PA.
to denote the standard model of
Exercise 8. If Proof.
of M is a cut that is closed under Fθ for all
elementary cut
N and M are elementary equivalent and N ~ M , then N is an elementary
à
N M and N ~ M . Then M PA since N M . On the other hand M is a nonstandard model of PA since N ~ M . Hence N is a proper initial segment of M , i.e., N is a cut of M . It remains to show that N is closed under Fθ for any LA -formula θ. Let θ be an LA -formula. It su ces to show that N is a model of the following statement. Suppose that
¦x§y Ñ
§zθ x, z ¦z θ x, z Ñ
Ñ y 1 , ¦y @ y 1 θ x, Ñ y 0 @ y , θx, .,. y 0
Ñ
We show the following statement is provable in
¦x¦u §y B u Ñ
Reason in
PA.
§z B u θ x, z
By induction on
u
Ñ
§y0 B u.
De ne
y
u
0,
y
by
by ¢ ¨ ¨ ¦ ¨ ¨ ¤
y It is not di cult to check that
PA.
we show a statement
we can de ne
y0 u1
if
§y B u Φ x, y, u
Ñ
y
0.
Suppose that
§z B u θ x, z
PA
PASk
8 8 8
holds, where
Ñ
Ñ y 0 , u Φx,
holds for some
holds,
otherwise.
Ñ y, u 1 Φx,
theory
Ñ y , ¦y @ y θ x, Ñ y . θx,
holds.
De nition 9. Let us de ne an expanded language LSk by LSk LSk 8 predicate symbol I. Then the
.
Ñ y , ¦y @ y θ x, Ñ y . θx,
Ñ y, u §z B uθ x, Ñ z Φx,
For the base case
PA
of elementary cut
I with a new unary
is de ned by
0 > I , §xx >~ I ¦x > I¦y > ~ Ix @ y Ñ > IFθ x Ñ > I S θ is a ¦x
LA -formula.
.
Theorem 10 (Kaye-Wong [1, 2]).
1. Every countable arithmetically saturated model of PA has an elementary generic cut.
2. If I is an elementary generic cut of M
à PA, then
2
M, I is an e.c. model of PA .
References [1] Richard Kaye and Tin Lok Wong. Truth in Generic Cuts.
Logic, 161(8):9871005, 2010.
Annals of Pure and Applied
[2] Richard Kaye and Tin Lok Wong. The Model Theory of Generic Cuts. 2011. Preprint, 13 pages. Available at
http://cage.ugent.be/wtl/papers/genmodel.pdf.
[3] Michael O. Rabin. Diophantine Equations and Non-standard Models of Arithmetic. In
Proceedings of the 1960 International Congress on Logic, Methodology, and the Philosophy of Science, pages 151158. Stanford Univ. Press, Stanford, Calif., 1962.
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