Model Theory 2013–14
Week 2
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Notation and conventions. Let A be a set. We let A∗ = n∈N An .
Let ϕ be a formula. We write ϕ(x1 , . . . , xn ) to indicate that the set of
free variables of ϕ is included in {x1 , . . . , xn }.
Let ϕ(x̄) be a formula , where x̄ = (x1 , . . . , xn ). When we write A |= ϕ[ā],
for some a ∈ A∗ , we always assume that x̄ and ā have the same length.
Same conventions apply to terms.
1. Let M be a L- structure and let ∅ 6= N ⊆ M. Recursively define
(Nk )k∈N as follows:
N0 = N ;
Nk+1 = {tM [m̄] : t is an L-term and m̄ ∈ Nk∗ }.
Show that
(a) Nk ⊆ Nk+1 ⊆ M, for all k ∈ M;
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(b) k∈N Nk is the domain of a substructure of M (call it hN iM );
(c) for every L-structure A such that N ⊆ A, hN iM ⊆ A.
Comment. (c) tells us that hN iM is the substructure of M generated
by N ).
2. Let Q = (Q, <) and R = (R, <).
(a) Prove the following: for all n ∈ N, all q1 < q2 < · · · < qn in Q
and all r ∈ R there exists an automorphism f of R such that
f (qi ) = qi , for all 1 ≤ i ≤ n, and f (r) ∈ Q.
Hint. Draw a picture. Assume q1 < r < q2 (the other cases
are similar). Prove the existence of an automorphism with the
required properties by actually constructing one.
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(b) Recall that if f : A → B, then, for all formulas ϕ(x̄) and all
tuples ā ∈ A∗ of the right length, A |= ϕ[ā] ⇔ A |= ϕ[f (ā)].
Use (2a) to show that Q and R satisfy condition 2. of the criterion
for elementary substructure and derive that Q 4 R.
(c) Finally, conclude that Q ≡ R. Clearly, Q ∼
6 R.
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3. Let L = {+, −, 0} be the language of (abelian) groups. Show that no
substructure of the group Z = (Z, +, −, 0) is an elementary substructure.
4. Exercises 3.5.1, 3.5.2 in Rothmaler, Introduction to Model Theory.
5. (a) Let A be a finite L- structure (i.e. |A| is finite). Let Γ(x) be a set
of L-formulas (in the free variable x) which is finitely satisfiable
in A (this means that for every finite subset Γ0 (x) of Γ(x) there
exists a ∈ A such that A |= Γ0 [a]. Show that Γ(x) is satisfiable
in A (i.e. there exists a ∈ A such that A |= Γ[a]).
(b) Show that the above conclusion does not hold, in general, for an
infinite structure A.
Hint Let N = (N, s, 0, <), where s : N → N is the successor
function.
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