SELECTED LOGIC EXERCISES
1. Metamathematics
Exercise 1.1. Suppose T has arbitrarily large finite models. Show that T has an infinite model.
Exercise 1.2. Let L = {=, ·, <, 0, 1}. Prove that there is M |= Th(N) with a1 , a2 , . . . ∈ M st a1 >M
a2 >M > a3 >M . . ..
Exercise 1.3. Show that there are non-Archimedian fields elementarily equivalent to the real numbers.
Exercise 1.4. Let L = {<} and T an L-theory, with infinite models, extending the theory of linear orders.
Show that there is M |= T and an order preserving embedding σ : Q → M.
Exercise 1.5.
(a) Prove that the class of connected graphs is not an elementary class.
(b) Prove that the class of torsion abelian groups is not an elementary class.
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SELECTED LOGIC EXERCISES
(c) Prove that the class of torsion groups is not an elementary class.
(d) Prove that the classes of finite groups and finite abelian groups are not elementary classes.
(e) Prove that the class of cyclic groups is not an elementary class.
(f) Prove that the class of all algebraic extensions of Q is not an elementary class.
Exercise 1.6.
(a) Prove the the class of torsion free abelian groups is not finitely axiomatizable.
(b) Prove that the class of fields of characteristic zero is not finitely axiomatizable.
2. Metamathematics 2
Exercise 2.1. Let L = {w ∈ {0, 1}ω : |w| = n2 for some k ∈ Z}. Prove that L is not a regular language.
3. Model Theory
Exercise 3.1. Let M be an L-structure, A ⊆ M , and p ∈ SnM (A). Prove that there is an elementary
extension N of M such that |N | = |M | + ℵ0 and p is realized in N .
Exercise 3.2. Let T be a complete theory with infinite models in a countable language.
(a) Let M |= T . Prove that there is an elementary extension N of M such that N is ℵ0 -homogeneous and
|M | = |N |.
(b) Suppose M |= T is ℵ0 -homogeneous and realizes all types in Sn (T ). Prove that M is ℵ0 -saturated.
(c) Prove that if |Sn (T )| ≤ ℵ0 for all n, then T has a countable saturated model.
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Exercise 3.3. Let T be a complete theory with infinite models in a countable language. Prove that M |= T
is prime if and only if it is countable and atomic.
Exercise 3.4. Prove that the following are equivalent.
(i) |Sn (T )| < ℵ0 for all n < ω.
(ii) Every type in Sn (T ) is isolated for all n < ω.
(iii) T is ℵ0 -categorical.
Exercise 3.5. Suppose L is a finite language with no function symbols and T is an L-theory with quantifier
elimination. Prove that T is ℵ0 -categorical.
Exercise 3.6.
(a) For any cardinal κ, prove that there is a model M |= DLO such that |M| > κ and there is a dense
subset A ⊆ M with |A| = κ.
(b) Show DLO is not κ-stable for any infinite κ.
Exercise 3.7. Let L + {U, <}, where U is a unary predicate. Let T be the L-theory extending DLO where
U picks out a subset that is dense and has a dense complement. Let M |= T and A = U M . Show that there
is no prime model over A.
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SELECTED LOGIC EXERCISES
Exercise 3.8. Suppose that M is κ-saturated, and (ϕi (v̄) : i ∈ I) and (θi (v̄) : j ∈ J) are sequences of
LM -formulas such that |I|, |J| < κ and
!
M |=
_
ϕi (v̄) ↔ ¬
i∈I
_
θj (v̄) .
i∈J
Show that there are finite sets I0 ⊆ I and J0 ⊆ J such that
M |=
_
ϕi (v̄) ↔
i∈I
_
ϕi (v̄) and M |=
i∈I0
_
θj (v̄) ↔
j∈J
_
θj (v̄).
i∈J0
Exercise 3.9. Suppose M |= T is κ-saturated and A ⊆ M with |A| < κ. For a ∈ acl(A), let irr(a/A) be an
LA -formula with a as a solution and minimal number of solutions.
Fix b ∈ acl(A) and ϕ(x) = irr(b/A).
(a) If θ(x) ∈ LA such that M |= θ(b), then ϕ(M) ⊆ θ(M).
Note: This implies that if ϕ1 , ϕ2 are two candidates for irr(b/A) then ϕ1 (M) = ϕ2 (M), so irr(b/A) is
well-defined up to equivalence.
(b) If M |= ϕ(c) then c ∈ acl(A) and ϕ = irr(c/A).
Exercise 3.10. Suppose M |= T is κ-saturated and A ⊆ M with |A| < κ. For a ∈ M, let O(a/A) be the
orbit of a under Aut(M/A).
(a) Suppose a ∈ M. Prove that O(a/A) is the set of realizations of tp(a/A).
(b) Suppose a ∈ M. Prove that a ∈ acl(A) if and only if O(a/A) is finite.
(c) Suppose a ∈ M. Prove that a ∈ acl(A) if and only if tp(a/A) has finitely many realizations.
(d) Suppose a ∈ acl(A). Prove that O(a/A) = irr(a/A)(M).
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(e) Suppose a ∈ acl(A). Prove that tp(a/A) is isolated in SnM (A).
Exercise 3.11. Suppose p ∈ SnM (A) is isolated in SnM (A) by ϕ(x) ∈ LA . Then ϕ(M) is the set of
realizations of p in M.
Exercise 3.12. Suppose T is a complete theory in a countable language with infinite models and M |= T
is saturated. Let A ⊂ M with |A| < |M|. Prove that if p ∈ SnM (A) is not algebraic, then p has |M|-many
realizations in M.
Exercise 3.13. Let M ≺ N be L-structures.
(a) Suppose ϕ(x) is a LM -formula such that ϕ(M) is finite. Prove that ϕ(M) = ϕ(N ).
(b) Prove that aclN (M ) = M .
Exercise 3.14. Suppose T has definable Skolem functions. Prove that acl(A) = dcl(A) for any set A.
Exercise 3.15. Let M be a κ-saturated L-structure. We write A ⊂ M to mean A ⊆ M with |A| < κ. A
subset A ⊆ M is type-definable if it is the intersection of less than κ many definable sets.
(a) Prove that if B ⊆ Mn is type-definable and co-type-definable, then it is definable.
(b) Let A ⊂ M. Prove that if X ⊆ Sn (A) is clopen then it is a basic clopen set of the form [ϕ(x, a)] for
some a ∈ A.
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SELECTED LOGIC EXERCISES
(c) Let A ⊂ M and suppose B ⊆ Mn is definable and invariant over A, i.e. σ(B) = B for all σ ∈ Aut(M/A).
Prove that B is definable over A.
(d) Let A ⊂ M and suppose p(x, y) ∈ S(A), where x and y are tuples of variables (with y possibly infinite).
Prove that
{a ∈ M|x| : ∃b ∈ M|y| , M |= p(a, b)}
is a type-definable subset of M|x| .
(e) A subset G ⊆ M is a definable group if G is a definable subset, which is a group under a definable
binary operation. Suppose G is a definable group and H ≤ G is a definable subgroup of G. Prove that
[G : H] is either finite or at least size κ.
(f ) A subset G ⊆ M is a type-definable group if G is a type-definable set, which is a group under a binary
operation with type-definable graph. Prove that if G is a definable group and H ≤ G is a type-definable
T
subgroup, then G = i∈I Hi , where |I| < κ and Hi ≤ G is type-defined by a countable set of formulas.
Exercise 3.16. For any A ⊂ M, prove there is some λ such that for any linear order of cardinality λ and
any sequence (ai )i∈I , there is a sequence (bi )i<ω , indiscernible over A, such that for all j1 < . . . < jn < ω
there are i1 < . . . < in in I with ai1 . . . ain ≡A bj1 . . . bjn .
Exercise 3.17. Given an infinite linear order I, a sequence (bi )i∈I , a set A ⊂ M, and an infinite linear order
J, prove there is a sequence (cj )j∈J , which is indiscernible over A and satisfies EM ((bi )i∈I /A).
Exercise 3.18. Let A ⊂ M and (bi )i<ω2 be indiscernible over A. Suppose there are a ∈ M and r(x, y), s(x, y) ∈
S(A) such that

 r(x, y)
0≤i<ω
tp(a, bi /A) =
.
 s(x, y) ω ≤ i < ω2
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Prove that there is (ai , ci )i<ω , indiscernible over M , such that

 r(x, y) i ≤ j
tp(ai , cj /A) =
.
 s(x, y) i > j
Exercise 3.19. Let A ⊂ M |= T and suppose (bi )i<κ is an indiscernible sequence over A.
(a) Prove there is a model M |= T containing A such that (bi )i<κ is indiscernible over M .
(b) Prove there is an increasing sequence of models (Mi )i<κ such that for all i < κ
(i) A(bj )j<i ⊆ Mi ;
(ii) (bj )i≤j<κ is indiscernible over Mi .
Exercise 3.20. Let R be a stable relation. If R is B-invariant for some B ⊂ M, prove there does not exist
a B-indiscernible sequence (ai , bi )i<ω such that R(ai , bj ) if and only if i ≤ j.
Exercise 3.21. Suppose ϕ(x) ∈ LM and α is an ordinal. Prove that the following are equivalent:
(i) RM(ϕ) ≥ α + 1;
(ii) there are (ψi (x))i<ω ⊆ LM such that ψi (M) ⊆ ϕ(M), RM(ψi ) ≥ α, and for all i 6= j, RM(ψi ∧ ψj ) < α;
(iii) there are (ψi (x))i<ω ⊆ LM and k ≥ 2 such that ψi (M) ⊆ ϕ(M), RM(ψi ) ≥ α, and for all I ⊆ ω, |I| = k,
\
ψi (M) = ∅.
i∈I
Exercise 3.22. Let T be a theory, M |= T , and A ⊆ M. Prove that there are at most degRM (T ) many
types p ∈ S1 (A) such that RM(p) = RM(T ).
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SELECTED LOGIC EXERCISES
Exercise 3.23. Suppose ψ ∈ LA for some A ⊆ M. If RM(ψ) = α and deg(ψ) is minimal among LA -formulas
with rank α, then there is q ∈ Sn (A) such that ψ ∈ q and RM(q) = α.
Exercise 3.24. Suppose T is ω-stable and ϕ is an LM -formula with RM(ϕ) = α. Show that for all ordinals
β < α there is an LM -formula ψ such that M |= ψ → ϕ and RM(ψ) = β.
Exercise 3.25. Suppose that T is a complete ω-stable theory. Show that there is an ordinal α < ω1 such
that for all A ⊆ M if p ∈ Sn (A), then RM(p) < α.
Exercise 3.26. Suppose that T is ω-stable, M, N |= T , and M ≺ N . If X ⊆ N k is definable in N , then
X ∩ M k is definable in M.
Exercise 3.27. Suppose T is an ω-stable theory and M |= T . Prove that any type-definable subset of M
is definable.
Exercise 3.28. Let M ≺ N . Prove that every type in S(M) realized in N is definable if and only if for all
definable X ⊆ N n , X ∩ M n is definable in M.
Exercise 3.29. Suppose G is an ω-stable group and G ≺ H. Prove that H ◦ ∩ G = G◦ .
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Exercise 3.30. Let M be a sufficiently saturated model of a complete theory T . Define the notion of
algebraic independence as the following ternary relation of subsets of M:
a
A^
| CB ⇔
acl(AC) ∩ acl(BC) = acl(C).
(a) Suppose A, B, C ⊂ M such that A ∩ B = ∅ = B ∩ C and C is algebraically closed. Prove that there is
some A0 ≡C A such that A ∩ B = ∅.
(b) Prove that algebraic independence satisfies full existence, i.e., for all A, B, C ⊂ M there is A0 ≡C A such
a
that A ^
| C B.
Exercise 3.31. For any A ⊆ M and a ∈ M, prove that tp(a/Aa) does not divide over A if and only if
a ∈ acl(A).
Exercise 3.32. Suppose A ⊆ B ⊂ M, p ∈ S(A) is algebraic, and q ∈ S(B) extends p. Prove that q does
not divide over A.
Exercise 3.33. The following are equivalent:
d
(i) a ^
| CB
(ii) for any C-indiscernible sequence I = (bi )i<ω , with b0 an enumeration of BC, there is some I 0 ≡BC I
such that I is aC-indiscernible.
d
a
Exercise 3.34. a ^
| C acl(BC) ⇒ a ^
| CB
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SELECTED LOGIC EXERCISES
d
d
Exercise 3.35. a ^
| CB ⇒ a^
| C acl(BC).
Exercise 3.36. Let Ab ⊂ M and ϕ(x, y) ∈ LA . For any k ∈ Z+ , prove that the following are equivalent.
(i) There is (bi )i<ω such that bi ≡A b for all i < ω and {ϕ(x, bi ) : i < ω} is k-inconsistent.
(ii) There is (bi )i<ω , indiscernible over A in tp(b/A), such that {ϕ(x, bi ) : i < ω} is k-inconsistent.
(iii) There is (bi )i<ω , indiscernible over A in tp(b/A), such that {ϕ(x, bi ) : i < ω} is inconsistent.
Exercise 3.37. Recall that a partial type forks over a set A if it proves a finite disjunction of formulas
that divide over A. Let A ⊆ B and π(x, b) be a (consistent) partial type over B. Prove the following are
equivalent:
(i) π(x, b) forks over A;
(ii) there is some C ⊇ B such that every extension of π(x, b) to a complete type over C divides over A.
Exercise 3.38. Suppose M |= T and A ⊂ M such that M is |A|+ -saturated. Prove that if p ∈ Sn (M ) forks
over A then p divides over A.
Exercise 3.39. Let M be κ-saturated L-structure, and A ⊂ M. A type p ∈ S(M) is A-invariant if for all
ϕ(x, y) ∈ L and b, b0 ∈ M, if b ≡A b0 then ϕ(x, b) ∈ p if and only if ϕ(x, b0 ) ∈ p. A type p ∈ S(M) is finitely
satisfiable in A if for all ϕ(x, b) ∈ p there is some a ∈ A such that M |= ϕ(a, b). A type is invariant (resp.
finitely satisfiable) if it is A-invariant (resp. finitely satisfiable over A) for some A ⊂ M.
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(a) Suppose p ∈ S(M) and A ⊂ M. Prove that if p is A-definable or finitely satisfiable in A, then p is
A-invariant.
(b) Suppose p ∈ S(M), A ⊂ M and A ⊆ M ≺ M with M |A|+ -saturated. Prove that if p is A-invariant then
p is the unique A-invariant type in S(M) extending p|M .
(c) Suppose p ∈ S(M) and A ⊂ M with p A-invariant. Prove that if p is definable then it is A-definable.
(d) Suppose p ∈ S(M) and A ⊂ M with p A-invariant. Prove that if p is finitely satisfiable then it is finitely
satisfiable in any model M containing A.
Exercise 3.40. Let M ≺ N be κ-saturated L-structures. If p ∈ S(M) is invariant over A ⊂ M then, by
definition, for all ϕ(x, y) ∈ L there is a collection dp ϕ of y-types over A such that for all b ∈ M,
ϕ(x, b) ∈ p ⇔
tp(b/A) ∈ dp ϕ.
Suppose M ⊆ C ⊆ N . Define p|(C/A) by
ϕ(x, c) ∈ p|(C/A) ⇔
tp(c/A) ∈ dp ϕ,
for c ∈ C. Note that p|(C/A) is A-invariant by definition.
For p definable over A, we can take dp ϕ = {q(y) ∈ S(A) : dϕ(y) ∈ q(y)} and, in this case, p(C/A) is the
definitional extension of p to C over A.
(a) Suppose p ∈ S(M) is A-invariant for some A ⊂ M and M ⊆ C ⊆ N . Prove that p|(C/A) is a consistent,
complete type containing p.
(b) Suppose p ∈ S(M) is A, A0 ⊂ M such that p is invariant over A and A0 . Prove that if M ⊆ C ⊆ N
then p|(C/A) = p|(C/A0 ). Therefore, for an invariant type p ∈ S(M) and M ⊆ C ⊆ N , we may define
p|C to be p|(C/A) where p is A-invariant. For p definable, p|C is the definitional extension of p to C.
(c) Suppose p ∈ S(M) is invariant and M ⊆ C ⊆ N .
(i) Prove that if p is A-invariant for some A ⊂ M then p|C is A-invariant.
(ii) Prove that if q ∈ S(C) extends p and is invariant over some A ⊂ M, then q = p|C.
(iii) Prove that if p is finitely satisfiable in A ⊂ M then p|C is finitely satisfiable in A.
(iv) Prove that if p is definable over A ⊂ M then p|C is definable over A by the same defining schema.
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SELECTED LOGIC EXERCISES
Exercise 3.41. Let M |= T and p ∈ S(M ).
(a) If M ⊆ A, prove that p has an heir q ∈ S(A).
(b) If M ⊆ A, prove that p has a coheir q ∈ S(A). Furthermore, prove that q does not split over M .
(c) Suppose M ⊆ N |= T , N is |M |+ -saturated, and q ∈ S(N ). If q does not split over M then prove q does
not divide over M .
Exercise 3.42. Let p ∈ Sn (M).
(a) Prove that if p is definable and N M, then the definitional extension of p to N (see Exercise 3.40) is
an heir of p.
(b) Prove that p is definable if and only if it has a unique heir to any N M.
Exercise 3.43. Let M |= T and Ac ⊂ M. Prove that tp(c/A) is an heir of tp(c/M ) if and only if tp(a/M c)
is a coheir of tp(a/M ) for all a ∈ A. Likewise, prove tp(c/A) is a coheir of tp(c/M ) if and only if tp(a/M c)
is an heir of tp(a/M ) for all a ∈ A.
◦
Exercise 3.44. Let A, b ⊂ M |= T and suppose ^
| is a ternary relation satisfying automorphism invariance,
finite character, extension, and monotonicity. Prove that for any infinite cardinal κ, there is a sequence (bi )i<κ
◦
of realizations of tp(b/A) such that (bi )i<κ is indiscernible and ^
| -independent over A.
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◦
Exercise 3.45. Suppose ^
| is a ternary relation satisfying automorphism invariance, local character, and
monotonicity. If A ⊆ M and (bi )i<ω is indiscernible over A, prove there is a model M containing A such
◦
that (bi )i<ω is indiscernible and ^
| -independent over M .
4. Set Theory
Exercise 4.1. Let α be an infinite ordinal. Prove that αα = 2α .
Exercise 4.2.
(a) Suppose A is a set and B ⊆ A such that |A| ≤ |B|. Prove that |A| = |B|.
(b) Suppose A and B are sets such that |A| ≤ |B| and |B| ≤ |A|. Prove that |A| = |B|.
Exercise 4.3. Suppose (κi )i∈I and (λi )i∈I are collections of cardinals such that κi < λi for all i ∈ I. Prove
that
X
i∈I
κi <
Y
λi .
i∈I
Exercise 4.4. For α an ordinal, define cof(α) to be the least ordinal β such that there is some unbounded
map f : β −→ α.
(a) For all ordinals α, cof(α) ≤ |α|.
(b) Suppose there is an unbounded map from α to β. Prove that there is an increasing unbounded map
from α to β.
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SELECTED LOGIC EXERCISES
(c) For all ordinals α, cof(cof(α)) = cof(α).
(d) For all ordinals α, cof(α) is a cardinal.
Exercise 4.5. Let κ be an infinite cardinal.
(a) Suppose κ is a successor cardinal. Prove that cof(κ) = κ.
(b) Suppose κ is a limit cardinal. Prove that there is a cofinal map f : cof(κ) −→ κ such that f (α) is a
cardinal for all α < cof(κ).
(c) Prove that κcof(κ) > κ.
(d) Prove that if λ ≥ 2 then κ < cof(λκ ).
Exercise 4.6.
(a) Prove that for all ordinals α, α ≤ ℵα .
(b) Prove that if α > 0 then ℵα is a limit cardinal if and only if α is a limit ordinal.
(c) Prove that
cof(ℵα ) =


ℵα
 cof(α)
if α = 0 or α is a successor ordinal
.
if α > 0 is a limit ordinal
(d) Prove that if κ > ℵ0 is a regular limit cardinal then κ = ℵκ .
Exercise 4.7. Suppose α is a limit ordinal with countable cofinality. Prove that 2ℵ0 6= ℵα .