[Note: the numbering A1..A9 below follows the numbering on Wikipedia instead of the book]

ZFC axioms of set theory (the axioms by Zermelo, Fraenkel, plus the axiom of Choice).

For details see Wikipedia "Zermelo-Fraenkel set theory". Note that the descriptions
there are quite formal (They need to be, because the goal is to reduce the rest of
math to these axioms. So to avoid circular reasoning, you have to state the axioms
without using things you know from the rest of math!)

Since the formal description may be difficult to read at this point, I
typed a more informal description here, and added explanations:


A1. This Axiom says that two sets are the same if their elements are the same.
    [You can think of this axiom as defining what a set is.]

A2. Axiom of regularity says that
	x can't be an element of x,
	x can't be an element of an element of x,
	x can't be an element of an element of an element of x,
	etc.

A3. Axiom schema of specification. Says that if A is a set, and P is some
    property, then A3 says that the elements of A that satisfy property P
    also form a set.  We denote that set as { x in A | P(x) }.

A4. Axiom of pairing. Says that if x,y are sets then {x,y} is also a set.

A5. Axiom of union. Says that if we have a set, whose elements are again sets,
    then the union of those sets is again a set.

A6. Axiom of replacement. Suppose phi(x,y) is a statement such that for every x
    there is precisely one y for which phi(x,y) is true.  Lets denote that y as f(x).

    [Remark: f looks a lot like a function, but we may not yet call f a function.
     In math you may only call f a function after specifying the domain AND the co-domain.]

    Axiom A6 says that if we apply f to all elements of some set A,
    so take f(x) for every x in A, then A6 says that all those f(x) form
    a set. We denote that set as {f(x) | x in A}.

A7. Axiom of infinity. Says that there exists an infinite set.

    [It does not matter which infinite set, as long as we have at least one
     infinite set then the other axioms allow us to construct our
     favorite infinite set, which is N = {0,1,2,3,....}.]

    [You may wonder, why do we need an axiom that tells us that an infinite
     set exists, when we've already known about {0,1,2,3,....} for a long
     time?  The issue here is:  How do you define the dots .... in
     the notation {0,1,2,3,....}?  The only way we can do that is if
     we first accept the unprovable statement that there exists at least one
     infinite set].

    [So wait, any time we need an unproven statement, we can simply
     call it an axiom?
     No, because other mathematicians won't accept additional axioms.
     The current axioms of set theory have been thoroughly put to the test
     for at least a century. It is outside of the scope of this course to detail
     the ways in which the axioms have been investigated, but the upshot
     is that mathematicians are very confident that the standard axioms (called ZFC),
     combined with the rules of logic, do not lead to errors.
     Mathematicians are unlikely to accept more axioms; we do not need more
     axioms, and we are confident about the ones we have.]

A8.  Axiom of power set.  If A is a set then P(A) is a set.

A9.  Axiom of Choice (AC)

     This axiom is equivalent to several other statements in the book (that we won't cover):
	1) Zorn's lemma
	2) The well-ordering theorem
	3) For any cardinals d,e we have d <= e or e <= d.
     If you replaced A9 by one of these three statements, then ZFC set theory stays the same.

     The axiom of choice says that if A is a set, whose elements are non-empty sets,
     then one can pick an element from each of these non-empty sets.
     This sounds harmless, however, if A is an infinite set, then we have to choose
     one element from infinitely many sets. How do you make infinitely many choices?
     Even though we may not be able to do this ourselves, the Axiom of Choice says
     that there does exist a function that makes all these choices. In case you think that
     this is fishy, that is not unreasonable, but consider this:
     One can prove (using only A1-A8) that if A1-A9 is contradictory, then A1-A8
     must also be contradictory. So if you want to prove that something in A1-A9
     is wrong, then you also have to prove that something in A1-A8 is wrong
     (remember from the comments under A7 that the axioms have been put to the
     test in many ways).


There you have it, a full list of all statements that mathematicians accept without
proof. Any statement other than A1-A9 will only be accepted by other mathematicians
if it has a proof that only uses previously accepted statements plus the rules of logic.
Proved statements are called theorems, and mathematicians trust them even when they
are counter-intuitive (like the existence of infinite sets with different cardinalities!).
Conversely, other than A1-A9, mathematicians don't accept statements that don't have a
proof, no matter how plausible they may sound.


Final question: what about other axioms? (Euclid's axioms, Peano's axioms, field axioms)
The objects that they deal with (points and lines, numbers, etc) can be defined in terms
of set theory. Those other axioms can be proved from the axioms of set theory.  So at the
end of the day, the only statements in math that are accepted without proof are the axioms
of set theory.