In this post, we present the proofs of the following two theorems, using Baire’s categoricity theorem.
Theorem (Fraisse, 1954). Let 
be a countable class of relational structures over a finite relational signature, which is closed under induced substructures and under amalgamation. Then there is a homogeneous structure 
with age .
We note that this is not the most general version of the theorem.
Theorem (Completeness; Godel, 1929). Let 
be a theory such that 
. Then has a countable model.
The standard proofs of these theorems involve some sort of constructions which proceed in stages, in each stage satisfying some requests which appeared in the previous stages. The proofs presented below rely on the following.
Theorem (Categoricity; Baire, 1899). Let 
be a complete metric space, and let 
be a family of open and dense subsets of . Then 
is dense in .
The first proof of Godel’s completeness theorem using Baire’s categoricity theorem appeared in Mathematics of Metamathematics (1963) by Helena Rasiowa (my great-great-phd-grand-mother) and Roman Sikorski. I don’t know who was the first to prove Fraisse’s result using Baire’s categoricity; definitely it is not new in this post (see e.g. W. Kubiś, Fraisse sequences: category-theoretic approach to universal homogeneous structures). Other applications of Baire’s categoricity theorem in logic include the Omitting Type’s theorem and Forcing.
denote the Rado graph and 
its automorphism group. We show that any action of 
– the Rado graph with a constant – in 
does not interpret (without parameters) in the homogeneous poset 
nor in the random graph 
or 
does not, this proves that there is no nontrivial continuous action of 
.
-categorical structure has a unique (up to isomorphism) model-complete core. We pose the question whether the model-complete core of a definable structure (say, over the equality atoms) is again definable.
, i.e., 
is definable using the parameter 
. A definable relational structure is a relational structure 
where 
and each relation 
is a definable set. Up to isomorphism, definable structures are the same as structures which interpret in 
, the pure countable set.
and 
are isomorphic, No otherwise.
. A straight semigroup is an equivariant semigroup whose universe is a straight set. We raise the following: