Axioms and Theorems

Axioms and Theorems

Axioms or postulates – proposition which are not yet proven but consider being self- evident or subjecting to necessary decision.

Theorems- statements which have been proven on the basis of previously established statements such as other theorems, and previously accepted statements such as axioms.

Brief Backgrounds of Axiom
The early Greeks developed a logico- deductive method whereby conclusions follow from premises.

Common notions by Euclid:

a. Things which are equal to the same thing are also equal to one another.
b. If equals be added to equals, the wholes are equal.
c. If equals be subtracted to equals, the remainders are equal.
d. Things which coincide with one another are equal to one another.
e. He whole is greater that part.

Axiomatic System
-any set of axioms from which some or all axioms can be used in conjunction to logically derived theorem.

Properties of an Axiomatic System:
1. consistent if it lacks contradiction
2. complete if for every statements, either itself or its negotiation of contradiction is derivable
3. independent if it is not a theorem that can be derived from other axioms in the system

Indicative conditional
Many theorems are of the form of an indicative conditional: if A, then B. In this case, A is called the hypothesis (antecedent) of the theorem and B the conclusion (consequent)

*Theorems are true precise in the sense that they posses proofs.