I) Take an English-into-English dictionary (any other language will also do). Start with any word and note down any word occurring in its definition, as given in the dictionary. Take this new word and note down any word appearing in it until a vicious circle results. Prove that a vicious circle is unavoidable no matter which word one starts with , (Caution: the vicious circle may not always involve the original word).
For example, in geometry the word “point” is undefined. For example, in set theory, when we write or say : ; the element “a” ‘belongs to’ “set A” —- the word “belong to” is not defined.
So, in all branches of math or physics especially, there are such “atomic” or “undefined” terms that one starts with.
After such terms come the “axioms” — statements which are assumed to be true; that is, statements whose proof is not sought.
The following are the axioms based on which equations are solved in algebra:
- If to equals we add equals, we get equals.
- If from equals we take equals, the remainders are equal.
- If equals are multiplied by equals, the products are equal.
- If equals are divided by equals (not zero), the quotients are equal.