Let K be an infinite field, and L/K be a finite algebraic extension. Show that L is interpretable in K. Is K is interpretable in L? Exercise 4 (one types in Q) Determine ...
Exercise 1 (number of definable sets) Let M be an L-structure. Compare the cardinality of the set of definable subsets of M with the set of all subsets of M . Give a few examples. Exercise 2 (definable maps) Let M be an L-structure and f : M n −→ M m a map. We say that f is definable if its graph is a definable subset of M m+n . 1. Let f : M n −→ M m and f : M m −→ M ` . If f and g are definable, is g ◦ f definable? If g ◦ f is definable, are f and g definable? 2. Let f : M n −→ M . If f is definable, is the image of f definable? If the image of f is definable, is f definable? 3. If f : M n −→ M is a definable bijection, is f −1 definable? Exercise 3 (on interpretability) 1. Let L1 , L2 , L3 be three languages, Mi an Li -structure for each i ∈ {1, 2, 3}. M3 is interpretable in M2 and If M2 is interpretable in M1 , show that M3 is interpretable in M1 . 2. Show that if M is an infinite L-structure, then any finite structure (in a finite language) is interpretable in M . 3. Let K be an infinite field, and L/K be a finite algebraic extension. Show that L is interpretable in K. Is K is interpretable in L? Exercise 4 (one types in Q) Determine all the 1-types over Q in the ordered set (Q,
Determine the atomic Lmon-formulas and their interpretations in a model of Σ1. 2. Let G, H be two models of Σ2. Let ¯a in G and ¯b in H be two n-tuples such that ...
If Ï is an atomic formula, it is of the form r(t1, ..., tm) for a relation symbol r and terms t1 ... By induction hypothesis, Ï is logically equivalent to a prenex formula Ï .
Exercise 1 (algebraically closed fields). 1. Let K be an infinite field and F a subfield. Show that if K/F is algebraic and F infinite, then K and F have the same ...
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Let A(a1,...,an) be a sentential formula in sentential variables a1,...,an. ... Exercise 2 (a few formal proofs) Let Ï1,...,Ïn,Ï and Ï be formulas, Î a set of formulas.
Model theory. 9. Axiomatisable classes. Exercise 1 (universal and existential axiomatisations) Let C be an axiomatisable class of L-structures. 1. Recall an ...
A subset A of Mn is definable if there is a formula Ï(¯x) such that ... by a quantifier-free formula, Ï(A) is also definable by a quantifier-free formula. 2.
Exercise 2 (on the isomorphism relation) Let L be a language, (M,LM ), (N,LN ), (S, LS) ... Show that Ï is a ring isomorphism from R1 to R2 if and only if Ï is an.
three kinds of elements: function symbols f, relation symbols r and constant ..... Î also satisfies the formula Ï, we say that Ï is a semantic consequence of Î and ...
Formulas, satisfaction. Exercise 1 (interpreting terms) Let M be an L-structure and t(x1,...,xn) an L-term. We write tM for the function from Mn to M that maps ¯a to ...
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3. Formal proofs (correction). Exercise 1. Note that f2. A = fA holds for every sentential formula A, since 02 = 0 and 12 = 1. Claim 1.1 For any sentential formulas A ...
The ordinal α + β is isomorphic to the disjoint union αâ β (defined to be α à {0} ... If β = γ + 1 and fγ : α + γ ââ αâ γ is an isomorphism sets, we define fβ : α + β ...
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Recall briefly what are the atomic Lring-formulas and their interpretations in a model of Σ. 2. Let ¯a in M and ¯b in N be two n-tuples such that for any atomic ...
Show that if α is an ordinal number obtained by finitely many applications of ordinal operations. (addition, multiplication and exponentiation) to Ï or natural ...
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Tp(¯a)Qq(¯a) for all 0 ⤠i ⤠n + m. By (2), one must also have. Qi(¯b) · T(¯b) · Sn(¯b) = â p+q=i. Tp(¯b)Qq(¯b) for all 0 ⤠i ⤠n + m, hence the decomposition ...