  # Axiom vs. Theorem: What's the Difference? Edited by Huma Saeed || By Sawaira Riaz || Updated on November 16, 2023
An axiom is a self-evident truth or starting principle, while a theorem is a proposition proven based on axioms or other theorems. ## Key Differences

An axiom is a statement or principle that is accepted without proof due to its inherent truth. It's foundational in nature. A theorem, conversely, is a statement or proposition in mathematics or logic that requires proof using axioms or previously proven theorems.
Axioms serve as the building blocks in mathematics or logical systems. They lay the groundwork from which other statements are derived. Theorems, on the other hand, are conclusions drawn from these axioms or from previously established theorems.
An axiom is typically universally accepted without controversy, given its self-evident nature. Theorems, however, can be more contentious and may undergo scrutiny and rigorous testing before achieving widespread acceptance.
Axioms don't require validation or justification within a system, they simply are. Theorems, by contrast, necessitate proof, and their validity hinges upon the correctness of this proof.
Over time, some axioms may evolve into theorems when they're proven using more fundamental principles. Similarly, some theorems, when deeply ingrained into a system, might be treated with the reverence of axioms.

## Comparison Chart

### Nature

Self-evident truth or starting principle
Proposition requiring proof

### Proof Requirement

Does not require proof
Requires proof based on axioms or other theorems

### Role

Foundational statement
Derived statement or conclusion

### Acceptance

Universally accepted due to inherent truth
Accepted after rigorous proof

### Evolution

Can evolve into theorems when proven by more basic axioms
When deeply accepted, can sometimes be seen as axiomatic

## Axiom and Theorem Definitions

#### Axiom

An axiom does not need validation within its system.
The axiom that a whole is greater than any of its parts is taken as self-evident.

#### Theorem

A theorem is a conclusion or proposition in math or logic.
According to the theorem, the angles of a triangle always sum up to 180 degrees.

#### Axiom

An axiom serves as a foundational statement in logical or mathematical systems.
The axiom of equality states that if two things are equal to the same thing, they are equal to each other.

#### Theorem

A theorem is a statement proven based on axioms or other theorems.
Pythagoras' theorem relates the sides of a right triangle.

#### Axiom

An axiom stands as a starting point for reasoning.
Based on the axiom of identity, an object is identical to itself.

#### Theorem

A theorem requires validation through rigorous proof.
The fundamental theorem of calculus connects differentiation and integration.

#### Axiom

An axiom is a self-evident principle accepted without proof.
In geometry, the notion that two parallel lines never meet is considered an axiom.

#### Theorem

A theorem undergoes scrutiny before acceptance.
After Euler's theorem was presented, it became a cornerstone in number theory.

#### Axiom

An axiom is a universally accepted truth.
It's an axiom in logic that something cannot both be and not be at the same time.

#### Theorem

A theorem can be foundational, but always derives from axioms or other theorems.
The binomial theorem provides a method to expand powers of binomials.

#### Axiom

A self-evident or universally recognized truth; a maxim
“It is an economic axiom as old as the hills that goods and services can be paid for only with goods and services” (Albert Jay Nock).

#### Theorem

An idea that has been demonstrated as true or is assumed to be so demonstrable.

#### Axiom

An established rule, principle, or law.

#### Theorem

(Mathematics) A proposition that has been or is to be proved on the basis of explicit assumptions.

#### Axiom

A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate.

#### Theorem

(mathematics) A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called lemmas.

#### Axiom

(philosophy) A seemingly self-evident or necessary truth which is based on assumption; a principle or proposition which cannot actually be proved or disproved.

#### Theorem

A mathematical statement that is expected to be true
Fermat's Last Theorem'' was known thus long before it was proved in the 1990s.

#### Axiom

A fundamental assumption that serves as a basis for deduction of theorems; a postulate (sometimes distinguished from postulates as being universally applicable, whereas postulates are particular to a certain science or context).

#### Theorem

(logic) A syntactically correct expression that is deducible from the given axioms of a deductive system.

#### Axiom

An established principle in some artistic practice or science that is universally received.
The axioms of political economy cannot be considered absolute truths.

#### Theorem

(transitive) To formulate into a theorem.

#### Axiom

A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted; as, "The whole is greater than a part;" "A thing can not, at the same time, be and not be."

#### Theorem

That which is considered and established as a principle; hence, sometimes, a rule.
Not theories, but theorems ( ), the intelligible products of contemplation, intellectual objects in the mind, and of and for the mind exclusively.
By the theorems,Which your polite and terser gallants practice,I re-refine the court, and civilizeTheir barbarous natures.

#### Axiom

An established principle in some art or science, which, though not a necessary truth, is universally received; as, the axioms of political economy.

#### Theorem

A statement of a principle to be demonstrated.

#### Axiom

A saying that widely accepted on its own merits

#### Theorem

To formulate into a theorem.

#### Axiom

(logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident

#### Theorem

A proposition deducible from basic postulates

#### Theorem

An idea accepted as a demonstrable truth

## FAQs

#### What is the primary distinction between an axiom and a theorem?

An axiom is accepted without proof, while a theorem requires proof based on axioms or other theorems.

#### Can a theorem ever become an axiom?

While deeply accepted theorems might be treated as axiomatic, technically, a theorem always has a proof and an axiom doesn't.

#### Do axioms exist outside of mathematics?

Yes, axioms are also present in logic and other disciplines as foundational truths.

#### Why are axioms essential in mathematics?

Axioms serve as the foundational principles from which other statements, including theorems, are derived.

#### How is a theorem validated?

A theorem is validated through rigorous proof using axioms or other previously proven theorems.

#### Can one derive multiple theorems from a single axiom?

Yes, multiple theorems can be derived from a single axiom, depending on the system.

#### Are all theorems universally accepted?

No, theorems require rigorous proof, and until accepted, can be a topic of debate among scholars.

#### Do axioms change over time?

While axioms are foundational, they can evolve or be replaced when more fundamental principles are discovered.

#### Is a theorem's proof always mathematical?

While often mathematical, theorems in logic or other fields might use different forms of rigorous proof.

#### Are axioms always true?

Axioms are considered self-evidently true within the system they belong to, but they can vary between systems.

#### Is every theorem derived from axioms?

Directly or indirectly, every theorem is derived from axioms or from other theorems that trace back to axioms.

#### Can theorems be proven wrong?

If a flaw is found in the proof or the underlying axioms, a theorem can be disputed or refuted.

#### Can something be both an axiom and a theorem?

While some principles might be treated as both in different systems, by definition, an axiom is unproven while a theorem has proof.

#### Can one prove an axiom?

By definition, axioms are taken as self-evident and do not require proof within their system.

#### Why are theorems important in mathematics?

Theorems expand upon foundational axioms, providing depth, richness, and broader understanding to a mathematical system.

#### Are axioms universal across all mathematical branches?

No, axioms can vary depending on the specific branch or system of mathematics.

#### What's an example of a famous theorem?

Pythagoras' theorem in geometry is a well-known example.

#### How many axioms are there in a system?

The number of axioms varies with the system; some systems might have few axioms, while others might have many.

#### How does one decide on the axioms for a system?

Axioms are chosen based on their self-evident truth and their utility in building the desired mathematical or logical system.

#### Can a theorem be based on multiple axioms?

Yes, a theorem can be derived using reasoning from multiple axioms.  