Mathematical Logic
Answers of Ex.1, 20.12.2013.
Henri Hansen
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3. Let us assume Σ is an axiomatizable first-order theory that has a model M = (R, R, c),
i.e., where the domain is the set of real numbers.
(a) Is it true that there exists a countable model M 0 , such that M 0 is model of Σ?
Justify your answer!
Answer: Yes it is true. This follows from the downward Löwenheim-Skolem
theorem. In the proof that we had, we showed that every infinite model has an
elementary substructure for every cardinality that is at least as large as that of
the language. As our language is countable, this proves the property.
(b) Is it true that, in general, the model M ∗ = (N, R∗ , c∗ ), which is the substructure
of M , restricted to natural numbers, is an elementary substructure of Σ? Justify
your answer!
Answer: No, it is not an elementary substructure. For instance, the formula
∀x∃y(x + y = 0) is true in one but not in the other.
4. An ordered field, is a structure with functions + and · and two constanst 0 and 1, and
a total order relation ≤. An ordered field is archimedean iff for every x, y ≥ 0 there is
an n such that y ≤ x + · · · + x (with n occurrences of x). Use compactness to show that
any first-order theory of the real numbers has a non-archimedean model. What does
this tell you about the relationship between the archimedean property and first-order
logic?
Answer: Let Σ be a first-order theory such that (R, {+, ·, ≤}, {0, 1}) |= Σ Consider a
new constant symbol ω, and a family of formulas A = ∪i Ai ≡ 1+1+· · ·+1 < ω, where
the number of 1s is i for each i. Then consider the set of formula Σ ∪ A. Now, for every
finite subset A ⊆ A we have (R, {+, ·, ≤}, {0, 1, ω}) |= Σ ∪ A, when ω > 1 + 1 · · · + 1,
when the number of 1s is larger than the largest n such that An ∈ A. Compactness
then implies that Σ ∪ A has a model, say M . M |= Σ by definition, but it cannot be
archimedean; ω and 1 fail the requirement, because ω > 1 + · · · + 1 for any number of
1s.
5. Choose two of the following and explain what they are or mean. Use examples if
applicable.
(a) The tableaux method
Answer: Explanation of α, β, δ and γ rules. Explanation that a tableaux for a
formula closes if the formula is UNSAT, and is open otherwise. Example of some
of the rules as steps.
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Mathematical Logic
Answers of Ex.1, 20.12.2013.
Henri Hansen
(b) Ordered Binary Decision Diagrams
Answer: Explanation of what an OBDD is, that it represents a logical function
and therefore an equivalence class of formulas. Example of a formula and a BDD,
and an explanation how logical operators (at least one or two of them) have a
counterpart as operations on BDDs.
(c) Unification
Answer: Explanation that unification is a substitution that makes two (nonground) terms equal. MGU is a unifier θ such that any other unifier µ can be
expressed as µ = θρ. Unification is used in resolution, especially logic programming.
(d) Tarki’s theorem
Answer: Truth of arithmetic formulas is not an arithmetic property of formulas.
Some scetching of how this is done, i.e., by numbering formulas and showing
that even though syntactical correctness is arithmetically representable, truth of
syntactically correct formulas is not.
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