I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more applicable in one case more than the other? I never hear of axioms in geometry or postulates in set theory. Are axioms more formal and postulates used more informally?
The terms "postulates" and "axioms" can be used interchangeably: just different words referring to the basic assumptions - the "building blocks" taken as given (assumptions about what we take to be true), which together with primitive definitions, form the foundation upon which theorems are proven and theories are built.
The choice to use one particular term rather than the other is largely a function of the historical development of a given branch of math. E.g., geometry has roots in ancient Greece, where "postulate" was the word used by the Pythagoreans, et.al.
So it's largely a matter of history, and context, and the word favored by the mathematicians that introduced or made explicit their "axioms" or "postulates." "Postulate" was once favored over "Axiom", with the development of analytic philosophy, particularly logical positivism, the term "axiom" became the favored term, and its prevalence has persisted since. Perhaps I should be corrected: As Peter Smith points out, it is "more likely" that the "uptake" of the term "axiom" can be attributed to "mathematicians like Hilbert (who talks of axioms of geometry), Zermelo (who talks of axioms of set theory), etc.".
Neither term is more formal than the other. I personally prefer "postulate" over "axiom", since a "postulate" transparently conveys (or connotes - as in connotation) that what we are calling a postulate is "postulated" as a "supposition", from which we agree to work in building theorems or a theory. In contrast, to me, the connotation of an "axiom" is that of a "law" of some sort, which MUST be followed or MUST be true, though it is no stronger than, or different from, a postulate. But again, this is simply a personal observation and preference, and the term "axiom" seems to have more "uptake" at this point in history.
Same thing, different name. Much like how the axiom of choice is equivalent to Zorn's lemma and to Zermelo's theorem (also known as the Well-ordering principle).
These are just words indicating our basic assumptions of what is true in the universe.
To the edit, I think that axioms just got a better foothold as a term. The Peano Axioms are also known as Peano Postulates; and I have a book written by Tarski (Cardinal Algebras, 1949) in which he begins with Postulates rather than axioms.
Postulate and axiom are used interchangeably in modern mathematics and they mean a statement that is assumed to be true within a specified domain. Every deductive mathematical system (such as Euclidean Geometry) normally will have statements that are self-evident (or assumed to be true) and don’t need proofs. Such statements are called axioms and always form the basis of that deductive system. Then there come theorems which are statements with proof (using axioms or other theorems). It is worth noting that, Euclid in his book “Elements” mentioned about 23 definitions, 5 postulates and 5 axioms; and it is not sure why he differentiated between postulates and axioms. However, these two terms can be interpreted as having the same meaning.
Primary source: Introduction to Mathematical Proofs, Second Edition: A Transition to ...By Charles Roberts
For consideration, both describe declarations that join primitives in sweet reason. The choice of word depends on attitude. Postulate is a more affixative term, axiom is more transitive. In history, the use of axiom gained favor by mathematicians challenging the permanence of posted speculation (e.g. finite observations about infinity)...thus opening the door, and paving the way, for experiments with relativity.