Metamathematics I Fall 2001

Problem Set II Due Monday September 17

1) Let ϕ be an L-sentence and let A be an L-structure. Prove that if there is an assignment α such that (A, α) ⊧ ϕ, then (A, β) ⊧ ϕ for all assignments β. [Hint: You will probably want to prove this by induction on formulas. You will need to first formulate the right generalization to formulas.]

2)a) Let L be the language {+, 0} and consider the structure R with universe R (the real numbers) where + is interpreted as the usual addition and 0 as zero. Show that there is no formula ϕ(v, w) such that R ⊧ ϕ(a, b) if and only if a < b for all a, b ∈ R. (Hint: Find a and b and an automorphism of R such that a < b but F (a) > F (b)).

b)^{†} Let L be the language of rings {+, ·, 0, 1} and let R be R with the usual interpretation. Find a formula ϕ(v, w) such that a < b iff R ⊧ ϕ(a, b).

3) Let T be an L-theory. We say that T′ is an axiomatization of T if A ⊧ T if and only if A ⊧ T′ for any L-structure A. Suppose T′ is an axiomatization of T. Show that T ⊧ ϕ if and only if T′ ⊧ ϕ for all L-sentences ϕ.

4) If ϕ is a sentence, the spectrum of ϕ is the set of all natural numbers n such that there is a model of ϕ with exactly n elements.

i) Let L = {E} where E is a binary relation. Write down a sentence ϕ asserting that E is an equivalence relation and every equivalence class has exactly three elements. Show that the spectrum of ϕ is {n > 0: 3 divides n}.

ii) Let L = {P, Q, f} where P and Q are unary predicates and f is a binary function. Let ϕ be the conjuction of:

∃x∃y x ≠ y ∧ P (x) ∧ P (y)

∃x∃y x ≠ y ∧ Q(x) ∧ Q(y)
∀z∃x∃y P (x) ∧ Q(y) ∧ f(x, y) = z
∀x_{1}∀x_{2}∀y_{1}∀y_{2} [(P (x_{1}) ∧ P (x_{2}) ∧ Q(y_{1}) ∧ Q(y_{2}) ∧ f(x_{1}, y_{1}) = f(x_{2}, y_{2})) →
(x_{1} = x_{2} ∧ y_{1} = y_{2})]

Show that the spectrum of ϕ = {n > 3 : n is not prime}.^{1} iii) Find a sentence with the spectrum {n > 0 : n is a square }. iv) Find a sentence with the specturm {p^{n} : p prime n > 0}. v)^{∗∗} Find a sentence with spectrum {p : p is prime}.

Remarks • An interesting theorem from computer science says that X ⊆ N is the spectrum of a sentence if and only if X is recognizable in non-deterministic exponential time.

• Open Question: Suppose X is the spectrum of a sentence. Is N^{+} \ X also a spectrum?]

1 Note: Another sentence with the same spectrum is the sentence in the language of rings asserting that we have an integral domain with zero divisors