
Gamma
if we have gamma_1 be the set of V_< sentences that hold in every finite model of
T_LO .Let gamma_2 be the set of sentences saying that there exist at least n elements for each n=1,2,3,... Let T_LOF denote the closure of gamma_1 (union) gamma_2 . Show that T_FLO is a theory.
where T_LO denote the theory of linear orders

What are the definitions of closure and theory? Also, since there are not many model theorists here, it would help if you could indicate the context, i.e., the topic or chapter this problem is coming from and any theorems that you suspect may be relevant. It may also be useful to know the level of the course (undergraduate, introductory, graduate for logic majors, etc.).

theory: For any Vstructure M, the theory of M, denoted Th(M), is
the set of all Vsentences ϕ such that M = ϕ.
closure:the closure of gamma is the set of sentences {ϕ gamma = ϕ.
This = means gamma models ϕ. where gamma is a set of sentences.
Yes, it is undergraduate course. the chapter is chapter 5 in ( A first course in logic ) book. and It is Q 5.6 (a)
the chapter is about first order theories