The concept of infinity and the foundations of mathematics

Abstract：We could not use the concept of Actual Infinity randomly; There are contradictions in current theories of real number and ZFC. The facts of not having the end of some division calculating and some evolution calculating and the fact of not having enough accuracy to measure continuous quantity must be respected. For all the terms of point, straight line, parallel line, real number, equality, function, derivative and so on, we must put forward three different terms of ideal, approximate and omnipotent approximation. The mathematics theory without error, is obtained by taking limit of the error bounds tending to 0 in approximation situation in practice. The acceptability of real number system depends on practice.

Keyword：the axiom of infinity; the omnipotent approximation real number; the omnipotent approximate function; noneuclidean geometry; the axis of real numbers; the set of real numbers; the laws of the unity of opposites;