Mathematical Induction Theorem

The Mathematical Induction Theorem is intertwined with the logical re-engineering within ZFC Set theory of our intuitively-apprehended natural number system, along with all its useful properties, so critical for Mathematics and all Sciences. Its consequence, Recursion Theorem, not only implies the useful categoricity of natural numbers but also the possibility of machine computation, and, thereby, pointing to the development of Artificial Intelligence. The Theorem is also a powerful method of logical inference unique to Mathematics with many versions, e.g., the Principle of Mathematical Induction and the Well-Ordering Theorem. It enables formalising the counting process, and offers different ways of counting finite as well as countable sets that are helpful in proving useful identities, inequalities, and propositions, like the Binomial Theorem, Arithmetic Mean-Geometric Mean Inequality, and the Pigeonhole Principle.