Simplest algebraic definition of set and collection which distinguishes the two I'm Googling for concise algebraic definitions of what it means to be a set, and in addition what it means to be a collection which is not a set.  I'm looking at this in the context of the definition of a probability space which looks like this:
A probability space $(\Omega,{\mathscr F},P)$ consists of


*

*An arbitrary nonempty set $\Omega$

*A collection ${\mathscr F}$ of subsets of $\Omega$ which is also a $\sigma$-algebra on subsets of $\Omega$

*A probability measure $P: {\mathscr F} \rightarrow [0,1]$
So to the extent that this is formal, I am looking for a formal way of saying what it means for $\Omega$ to be a set and for ${\mathscr F}$ to be a collection which is not necessarily a set.
I can go to the Zermelo-Fraenkel Axioms, which gives me 9 axioms and one fundamental object (the empty set).  
Q1. Is that the easiest and simplest and most algebraic definition I can get?  I am looking for something like the definition of a ring or a field.
Q2. Assuming ZFC axioms are it for sets, how do I axiomatize the definition of a collection?  Is there a ZFC of collections?

You definition of a probability space is strange. Where did you find it?
I ask the question because according to Wikipedia definition of a probability space, the $\mathscr{F}$ you mention in your own definition is not mentioned to be a collection but a set as $\Omega$.
This is more coherent as according to $\mathtt{ZF}$, any subset of a set is also a set. While a collection is different in $\mathtt{ZF}$ from a set. See collections in Zermelo–Fraenkel set theory.
