البرهان في صف الرياضيات

Abstract

The importance of Mathematical proof made mathematics an exact and logic
science and led to the necessity of the axioms system. However, when there are
cases equivalent to one of the axioms, one is considered an axiom and the others
are related to as theories or of equivalent attributes. Mathematics science is based
on independent axioms, namely on fundamental cases that cannot be proven or
derived from other cases. In this article, the induction axiom as well as four
equivalent cases are addressed. Each of the four cases are equivalent to the
induction axiom. If we consider one of them to be undisputed the others cannot
be accepted but theories. When one case (induction) is considered an axiom, the
other four are related to as theories.