Can all laws be derived from axioms?

Question

I don't really understand the purpose of an axiom if some laws cannot be derived from them. For example, how is one supposed to prove De Morgan's laws with only the axioms of probability?

Answer 1

You probably can, and it's extremely rare that you have to. De Morgan's laws probably wouldn't be the worst (and they don't really involve probability), but one of the points of theorems is that we can use the ones we already have proven to prove new ones, without going all the way back to axioms. Because if we use theorem $A$ in the proof of theorem $B$, we know that given a proof of $A$ using only axioms, we could substitute that into our proof of $B$ and have a proof of $B$ using only axioms.

(I skipped over some details in an attempt to make it clearer).

Answer 2

In fact, you will hardly find any contemporary mathematical work providing an explicit, self-contained axiomatic system; it's just too good to be possible in pratice. One can only hope that they state clearly all those postulates that are too technical to be considered alredy known. In Probability Theory they may take (naive) Set Theory for granted, and thus consider De Morgan's laws as something easy to prove from what you ‘already know’.

There is a special branch of mathematics, called Mathematical Logic, which is actually dedicated to investigating how modern mathematical theories can be reduced to fully-rigorous axiomatic systems (they are trying ‘formalize’ mathematics). To take a glimps of how complex this task is, have a look at this site:

http://us.metamath.org/mpegif/mmset.html

(Of course, what they do in Metamath is an extreme instance of what professional mathematical logicians are concerened with.)

Answer 3

In theory, every theorem in some practical formal system is by definition a sentence that can be deduced in finitely many steps by strictly following the allowed inference rules. Inference rules are more general than axioms, and commonly are of the form "If you can deduce ... then you can deduce ...". Axioms can be easily captured by the special case of inference rules where the condition is vacuous (you need not be able to deduce anything), in other words by inference rules of the form "You can always deduce ...".

Concerning the axioms of probability, they are not stated in a vacuum, but within an over-arching formal system. Let's take a simpler example of the axiomatization of groups:

A group is an ordered triple $(G,∘,e)$ where $G$ is a collection and $∘$ is a binary operation on $G$ and $e \in G$ such that:

• $\forall x,y \in G\ ( x∘y \in G )$.

• $\forall x,y,z \in G\ ( (x∘y)∘z = x∘(y∘z) )$.

• $\forall x \in G\ ( x∘e = x = e∘x )$.

• $\forall x \in G\ \exists y \in G\ ( x∘y = e = y∘x )$.

Notice that these axioms are more-or-less first-order sentences applied to $(G,∘,e)$. But we certainly cannot deduce a lot about groups using only the axioms, such as the fact that every element in a finite group has a finite order that divides the size of the group. There is no easy explanation of this, but intuitively it is because if you restrict yourself to working within the group axiomatization you would only be able to deduce things from the perspective of a general group, and the language you have is not even able to refer to multiple elements, iterated operations or collections of groups, not to say reason about them.

This is why group theory (and every other branch of mathematics) is investigated 'outside' the group axiomatization, in the foundational system which allows you to set up axiomatizations in the first place. The most common foundational system in modern mathematics is usually said to be ZFC set theory, but actually it is ZFC plus a fair amount of syntactic sugar to help with actual usage. The language of pure ZFC only has a single binary predicate symbol "$\in$", making it very cumbersome to express anything of interest, but if we add on-the-fly definitorial expansion then it very drastically condenses proofs and the resulting system ZFC* is a reasonably usable foundational system.

In real mathematical practice mathematicians do not even use ZFC*, but rather most mathematics is done in a fairly informal way, with the main criterion of correctness being that any sufficiently trained reader is capable of convincing himself/herself that the argument can be translated to a formal proof in ZFC*. (Actually some mathematicians do not even know the formal specifications of ZFC, but they know what they are allowed to do that set theorists tell them is translatable to ZFC.)