On Modal Arithmetic of Indexed Natural Numbers: Possible Worlds Made of Numbers

The purpose of the article is to construct an arithmetic of indexed natural numbers which would be a generalized version of Peano's arithmetic. It is assumed within that theory that there may be several numbers one, several numbers two, several numbers three, etc. Moreover several different operations of addition and multiplication can be defined for these numbers. Such formalism makes it possible to construct new kinds of numbers; semi-whole numbers, semi-rational numbers and semi-real numbers. The construction is inspired by Kant's philosophy.