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The property of being true in almost all finite, initial segments of the standard model of arithmetic is ∑20 -complete. Thus, it admits a kind of a truth definition. We define such an arithmetical predicate. Then, we define its modal logic SL and prove a completeness theorem with respect to finite models semantics. The proof that SL is the modal logic of the approximate truth definition for finite arithmetical models is based on an extension of SL by a fixed-point construction. [ABSTRACT FROM AUTHOR]

The paper presents the current state of knowledge in the field of logical investigations of finite arithmetics. This is an attempt to summarize the ideas and results in this area. Some new results are presented – these are mainly generalizations of the earlier results related to properties of sl-theories and some nontrivial cases of FM-representability theorem. [ABSTRACT FROM AUTHOR]