Short Course on Fundamentals of Mathematics. Part I

This is the first part of the shortest course which has various goals. Firstly, to confront somehow the growing fragmentation of mathematics. Also to create a feeling that mathematics is a very simple science, if one starts to study it not with very formal and complicated general theories but with simplest examples. Any mathematical mind has some picture of mathematics as the (Egyptian) pyramid, but it should not be brought in from outside, but to appear step-by-step from many intuitively evident assertions.