When reading or writing mathematical papers, you’ll often encounter terms like theorem, lemma, corollary, and more. At first glance, they might seem interchangeable — after all, they’re all statements that are true, right?

But in mathematics, these terms aren’t just labels. They help structure complex ideas, clarify the importance of results, and guide the reader through a logical argument. Understanding the role of each type of statement makes it easier to follow proofs, communicate your own results effectively, and appreciate how mathematical reasoning is built.

In this post, we’ll walk through the most common types of mathematical statements — what they mean, how they’re used, and why each one matters.

Basic assumptions::

Axiom / Postulate:
A basic assumption accepted without proof.

Not yet proven statements:

Conjecture:
A statement believed to be true but not yet proven.

Proven statements:

Theorem:
A fundamental and significant mathematical statement that has been rigorously proven.

Proposition:
A true and provable statement, usually less central than a theorem but still meaningful.

Lemma:
A helper result used to simplify or support the proof of a theorem.

Corollary:
A result that follows directly from a theorem, usually without requiring a separate proof.

Example:

Lemma:
Any connected acyclic graph with $n$ vertices has at most $n-1$ edges.
Proposition:
If a graph is connected and has $n-1$ edges, then it is a tree.
Theorem:
Every tree with $n$ vertices has exactly $n-1$ edges.
Corollary:
Any tree with more than one vertex has at least two leaves.

Usually used in informal contexts:

Claim:
A statement intended to be proven later within a larger proof.

Remark:
A non-central observation that offers insight or intuition.

Note:
An explanatory comment for context or clarification.

Property:
An interesting and useful feature of a mathematical object.