Mathematics based on incremental learning—Excluded middle and inductive inference

Learning theoretic aspects of mathematics and logic have been studied by many authors. They study how mathematical and logical objects are algorithmically “learned” (inferred) from finite data. Although they study mathematical objects, the objective of the studies is learning. In this paper, a mathematics whose foundation itself is learning theoretic will be introduced. It is called Limit-Computable Mathematics. It was originally introduced as a means for “Proof Animation”, which is expected to make interactive formal proof development easier. Although the original objective was not learning theoretic at all, learning theory is indispensable for our research. It suggests that logic and learning theory are related in a still unknown but deep new way.