Differences between theorems, definitions, axioms, lemmas, corollaries, propositions, statements, hypothesis, definition, conjecture, claim and paradox

1. Statement

A sentence which has objective and logical meaning.

2. Proposition

a statement which is offered up for investigation as to its truth or falsehood.

3. Axiom

Is used throughout the whole of mathematics to mean a statement which is accepted as true for that particular branch.

4. Theorem

Is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch.

Note: In a mathematical paper, the term theorem is often reserved for the most important results.

5. Lemma

A minor result whose sole purpose is to help in proving a theorem.

Note: lemma is seen more as a stepping-stone than a theorem in itself (and frequently takes a lot more work to prove than the theorem to which it leads).
Some lemmas are famous enough to be named after the mathematician who proved them (for example: Abel’s Lemma and Urysohn’s Lemma), but they are still categorised as second-class citizens in the aristocracy of mathematics.)

6. Corollary

A proof which is a direct result, or a direct application, of another proof.

Note: It can be considered as being a proof for free on the back of a proof which has been paid for with blood, sweat and tears. Also, a result in which the (usually short) proof relies heavily on a given theorem

7. Hypothesis

an assumption made.

8. Definition

A precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.

9. Conjecture

A statement that is unproved, but is believed to be true

10. Claim

An assertion that is then proved. It is often used like an informal lemma.

11. Paradox

A statement that can be shown, using a given set of axioms and definitions, to be both true and false.

Note: paradoxes are often used to show the inconsistencies in a flawed theory. The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules

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