Homework IV, Math 736, Model-Theory.

Elements a_{0}, …, a_{k−1} of an abelian group are called additively independent if

∑_{i<k} n_{i}a_{i} ≠ 0

for all integers n_{0} …, n_{k−1}, not all of which are 0.

Problem 7. Let Q be the abelian group of rational numbers. Show that there is an abelian group G such that:

(∗) G ≡ Q, and

(†) G contains n additively independent elements for every n in ω.

Then show that any two such countable groups are isomorphic.

Now let L be an arbitrary signature. If M, N ∈ 𝔐𝔬𝔡(L) and M ⊆ N, let us write

M ≼_{1} N

if the inclusion of M in N preserves universal formulas of L.

Problem 8. Prove that the following are equivalent:

(∗) M ≼_{1} N

(†) there is R in 𝔐𝔬𝔡(L) such that M ≼ R and N ⊆ R.

Suppose {M_{n} : n ∈ ω} is a subset of 𝔐𝔬𝔡(L) forming a chain, that is, M_{n} ⊆ M_{n+1} for all n in ω. Then the union of this chain is defined to be the structure N, where:

(∗) N = ⋃_{n∈ω} M_{n}, and

(†) for all basic formulas ϕ, if a is a tuple from M_{n}, and M_{n} ⊧ ϕ(a), then N ⊧ ϕ(a).

(You should verify that N is well-defined, but you need not submit the verification.) The chain is called elementary if M_{n} ≼ M_{n+1} for all n.

Problem 9. Show that the union of an elementary chain is an elementary extension of each structure in the chain.

Recall that a theory T of L is called model-complete if M ≼ N whenever M ⊆ N and both structures are models of T. A formally weaker notion is 1-model-completeness: T is 1-model-complete if M ≼_{1} N whenever M ⊆ N and both structures are models of T.

Problem 10. Prove that 1-model-completeness and model-completeness coincide.