Theorem vs. Postulate — What's the Difference?

A rule or law, especially one expressed by an equation or formula.

A statement that is taken to be true, to serve as a premise or starting point for further reasoning.

A statement or proposition that is proven by argument.

A fundamental principle or assumption that is accepted without proof.

A theoretical proposition that follows logically from established facts.

To assume the existence or truth of something as a basis of discussion.

A general proposition in mathematics that is demonstrably true or is assumed to be so.

A foundational element of a logical or mathematical system that is not questioned.

In logic, a formula that can be derived from the axioms of a formal system.

Suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief

In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system.

(in ecclesiastical law) nominate or elect (someone) to an ecclesiastical office subject to the sanction of a higher authority

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

A thing suggested or assumed as true as the basis for reasoning, discussion, or belief

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

To assume or assert the truth, reality, or necessity of, especially as a basis of an argument

(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.

To propose as a hypothesis or explanation

A mathematical statement that is expected to be true

To assume as a premise or axiom; take for granted.

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

(Archaic) To make claim for; demand.

(transitive) To formulate into a theorem.

Something assumed without proof as being self-evident or generally accepted, especially when used as a basis for an argument

That which is considered and established as a principle; hence, sometimes, a rule.

A fundamental element; a basic principle.

A statement of a principle to be demonstrated.

(Mathematics) An axiom.

To formulate into a theorem.

(Archaic) A requirement; a prerequisite.

A proposition deducible from basic postulates

Something assumed without proof as being self-evident or generally accepted, especially when used as a basis for an argument. Sometimes distinguished from axioms as being relevant to a particular science or context, rather than universally true, and following from other axioms rather than being an absolute assumption.

An idea accepted as a demonstrable truth

A fundamental element; a basic principle.

(logic) An axiom.

A requirement; a prerequisite.

Postulated.

To assume as a truthful or accurate premise or axiom, especially as a basis of an argument.

To appoint or request one's appointment to an ecclesiastical office.

To request, demand or claim for oneself.

Something demanded or asserted; especially, a position or supposition assumed without proof, or one which is considered as self-evident; a truth to which assent may be demanded or challenged, without argument or evidence.

The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.

Postulated.

To beg, or assume without proof; as, to postulate conclusions.

To take without express consent; to assume.

To invite earnestly; to solicit.

(logic) a proposition that is accepted as true in order to provide a basis for logical reasoning

Maintain or assert;

Take as a given; assume as a postulate or axiom;

Require as useful, just, or proper;

To assume or suggest something as a basis for reasoning or argument.

A theorem is a statement in mathematics or logic that has been proven based on previously established statements, axioms, or postulates.

A postulate is a statement that is assumed to be true without proof and is used as a foundation for further reasoning.

While commonly used in geometry, postulates are also fundamental in other areas of mathematics and logic.

No, not all mathematical statements are theorems; some are postulates, axioms, or conjectures.

No, a theorem cannot become a postulate because theorems require proof and postulates do not.

Axioms and postulates are similar in that they are both accepted without proof, but axioms are generally more broadly applicable across different branches of mathematics.

Theorems are important because they expand our understanding of mathematical relationships and logical structures.

Postulates serve as the starting assumptions from which theorems are derived; they shape the scope and nature of theorems that can be proven.

A postulate can be rejected if it leads to contradictions or does not align with the logical structure of the mathematical system.

Mathematicians accept statements as postulates if they are seemingly self-evident and if they provide a useful foundation for building a logical framework.

If a theorem is disproven, it means there was an error in the proof, and it must be re-evaluated or discarded.

Postulates are not proven as they are the foundational assumptions upon which proofs are built.

A theorem is proven through a logical sequence of statements that follow from accepted axioms, postulates, and previously proven theorems.

Many theorems have practical applications in fields such as physics, engineering, computer science, and economics.

A mathematical system can have any number of postulates, but they must be consistent and not lead to contradictions within the system.