Axiom vs Theorem

An axiom is a statement that is considered to be true, based on logic; however, it cannot be proven or demonstrated because it is simply considered as self-evident. Basically, anything declared to be true and accepted, but does not have any proof or has some practical way of proving it, is an axiom. It is also sometimes referred to as a postulate, or an assumption.

An axiom’s basis for its truth is often disregarded. It simply is, and there is no need to deliberate any further. However, lots of axioms are still challenged by various minds, and only time will tell if they are crackpots or geniuses.

Axioms can be categorized as logical or non-logical. Logical axioms are universally accepted and valid statements, while non-logical axioms are usually logical expressions used in building mathematical theories.

It is much easier to distinguish an axiom in mathematics. An axiom is often a statement assumed to be true for the sake of expressing a logical sequence. They are the principal building blocks of proving statements. Axioms serve as the starting point of other mathematical statements. These statements, which are derived from axioms, are called theorems.

A theorem, by definition, is a statement proven based on axioms, other theorems, and some set of logical connectives. Theorems are often proven through rigorous mathematical and logical reasoning, and the process towards the proof will, of course, involve one or more axioms and other statements which are already accepted to be true.

Theorems are often expressed to be derived, and these derivations are considered to be the proof of the expression. The two components of the theorem’s proof are called the hypothesis and the conclusion. It should be noted that theorems are more often challenged than axioms, because they are subject to more interpretations, and various derivation methods.

It is not difficult to consider some theorems as axioms, since there are other statements that are intuitively assumed to be true. However, they are more appropriately considered as theorems, due to the fact that they can be derived via principles of deduction.

Summary:

1. An axiom is a statement that is assumed to be true without any proof, while a theory is subject to be proven before it is considered to be true or false.

2. An axiom is often self-evident, while a theory will often need other statements, such as other theories and axioms, to become valid.

3. Theorems are naturally challenged more than axioms.

4. Basically, theorems are derived from axioms and a set of logical connectives.

5. Axioms are the basic building blocks of logical or mathematical statements, as they serve as the starting points of theorems.

6. Axioms can be categorized as logical or non-logical.

7. The two components of the theorem’s proof are called the hypothesis and the conclusion.

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