We say that a first order theory T is locally finite if every finite part of T has a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theory T a locally finite theory FIN(T) which is syntactically (in a sense) isomorphic to T.
Our construction draws upon the main idea of Paris and Harrington  (I have been influenced by some unpublished notes of Silver  on this subject) and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. (Our proof is syntactic, and it is simpler than the proofs of ,  and . This reminds me of the simple syntactic proofs of several variants of the Craig-Lyndon interpolation theorem, which seem more natural than the semantic proofs.)
The first mathematically strong locally finite theory, called FIN, was defined in  (see also ). Now we get much stronger ones, e.g. FIN(ZF).
From a physicalistic point of view the theorems of ZF and their FIN(ZF)-counterparts may have the same meaning. Therefore FIN(ZF) is a solution of Hilbert's second problem. It eliminates ideal (infinite) objects from the proofs of properties of concrete (finite) objects.
In  we will demonstrate that one can develop a direct finitistic intuition that FIN(ZF) is locally finite. We will also prove a variant of Gödel's second incompleteness theorem for the theory FIN and for all its primitively recursively axiomatizable consistent extensions.
The results of this paper were announced in .