Hey!

Sorry for posting again so quickly, but I also have an issue for a second concept. This one is regarding applying the compactness theorem in first order logic, the framework is a Hilbert system:

L is a FOL (First order language) with R, where R is a single binary predicate symbol. Suppse A = ⟨V,E⟩ is a structure for this language domain V = |A|. Suppose also that E = RA, is the interpretation of the symbol R in A.

So ⟨V, E⟩ can be viewed as a directed graph; i.e., a (possibly infinite) set of vertices in V connected by edges in E.

Note that A Hamiltonian cycle in a graph is a finite sequence of vertices a_{1}, a_{2},. . . , a_{n}such that the following 3 conditions are met:

- a_{1}, a_{2},. . . , a_{n}are distinct,

- V = {a_{1},...,a_{n}}

- ⟨a_{1},a_{2}⟩ ∈ E, ⟨a_{2},a_{3}⟩ ∈ E,...⟨a_{n−1},a_{n}⟩ ∈ E, ⟨a_{n},a_{1}⟩ ∈ E.

Also note that if ⟨V,E⟩ has Hamiltonian cycle then V is finite.

How do you describe a sentence σ_{n}in the language L that has the property ⟨V,E⟩ |= σ_{n}if and only if ⟨V,E⟩ has a Hamiltonian cycle with n vertices. The question requires to give σ_{n}explicitly in the case that n = 4.

Could you provide a hint or suggestion as to how I can begin to go about this!

Many thanks!