What is the difference between a conjecture and an axiom?

Answer 1

First off, I'll explain what axioms are since they are of fundamental importance to abstract algebra and math in general. In mathematical logic, an axiom is an underivable, unprovable statement that is accepted to be truth. Axioms are, therefore, statements which form the mathematical basis from which all other theorems can be derived. The most well-known modern examples of mathematical axioms are those of the axiomatic set theory known as ZFC.

The Zermelo-Fraenkel set theory with the axiom of choice added to it, abbreviated ZFC, is the axiomatic set theory which provides our basis for math. There are nine axioms in the theory:

1) Two sets are equal if and only if they have the exact same members.

2) A set can't be an element of itself.

3) If there is a property that is characteristic of the elements of a set, a subset of that set exists containing the elements that satisfy the property.

4) A set exists containing exactly all of the members of two given sets.

5) A set exists whose elements are the members of the members (the union) of a given set.

6) The image of any function on a set is also a set.

7) There is a set that contains all of the natural numbers.

8) Every set has a power set; i.e. the set of all possible subsets of the original set.

9) Every set can be well-ordered such that every subset of the set has a "least" element under the ordering.

All nit-picking aside, these nine axioms are mathematically unprovable and therefore must be assumed true for mathematics to work.

A conjecture, as opposed to an axiom, is an unproved (not unprovable) statement that is also generally accepted to be true. The subtle difference between the two terms is basically that an axiom has been proven to be unprovable, whereas a conjecture hasn't.

See the related link for more information about ZFC.