TITLE: Better understanding dividing and forking
QUESTION [14 upvotes]: Recently I learned about dividing and forking of formula / partial types:
We say that $φ(x,b)$ divides over $C$ (where $b$ in the monster and $C$ is small) if there exists an indiscernible sequence $\{b_i\}_{i\in\omega}$ over $C$ of elements with the type $\operatorname{tp}(b/C)$ such that $\{φ(x,b_i)\}$ is inconsistent.
We say that $φ(x,b)$ fork over $C$ if it implies finite disjunction of dividing formulas over $C$.

I am trying to get a better intuition about those definitions, especially about forking.
From my understanding, $φ(x,b)$ divides over $C$ if $φ(C,b)$ (i.e. the set $φ$ defines with $b$) is not "big" in the sense that for every $A\subseteq ω$, $|A|=k$ (for some fixed $k$) we have $\bigcap\limits_{i∈A} φ(C,b_i)=∅$
This view is reinforced when we see that the forking formulas are basically the ideal generated from the dividing formulas (I mainly worked in set theory with filters which are "defining big sets", so my intuition about their dual, the ideals, is "defining the small sets").
But this view doesn't feel that good to me, it is somewhat lacking: specifically, what is the role of the second parameter ($b,b_i$) here, why do we look at what happens to the defined subset when we change it. And why are we fixing $C$? Isn't the first parameter is usually the parameter we move around?
Further more, I feel like this view falls completely for forks: the set of forking formulas is not always a proper ideal, even over the empty set (e.g. in $(X,\mathcal P(X),∈)$ or circular order, in which $x=x$ forks over $∅$)
My understanding on forking/dividing partial types is basically the same as my understanding of forking.


So I am guessing that my question is: is there a standard way to think about dividing and forking formulas and partial types?
Also, I focused a lot on the study of set theory/formal logic, and kind of neglected abstract algebra for some time (which comes to bites me in the butt now...), so if there is a view that appeals to set theorists and relies less on algebraic intuition I would love to hear about it.

REPLY [6 votes]: Non-forking and invariance (a notion easier to grasp) coincide in a large class of theories.
Below I elaborate with a model $M$ instead of a set $C$ to avoid some technical issues (b.t.w., these technical issues are quite interesting, just not now).
Things are clearer when working with global types.
We say that $\mathscr{D}\subseteq\mathscr{U}$ is invariant over $M$ if $\mathscr{D}$ is (setwise) fixed under the action of Aut$(\mathscr{U}/M)$.
To a global type $p(x)$ and a formula $\varphi(x,y)$ we associate the set $\mathscr{D}=\{a\in\mathscr{U}:\varphi(x,a)\in p\}$.
The global type $p(x)$ is invariant if all sets $\mathscr{D}$ as above are invariant (as $\varphi(x,y)$ ranges over all formulas).

Theorem Assume $T$ is NIP (the definition of NIP is irrelevant - just a large class of theories that include all stable theories) then the following are equivalent for every global type $p(x)$

$p(x)$ does not fork over $M$;

$p(x)$ is invariant over $M$.



See Theorem 5.21 in P.Simon's book
For an easier comparison with non-dividing, note that a simple argument of compactness proves the following

Fact For every $T$ the following are equivalent for every global type $p(x)$

$p(x)$  is invariant over $M$;

$\varphi(x,b)\in p$ and every $b_1\equiv_M\dots\equiv_Mb_n\equiv_Mb$ there is a $c$ such that $\varphi(c,b_i)$ for every $i$.



It is evident that non-dividing is a weak analogue of 2.
What about non-NIP theories? What about non-global types? Can we say something meaningful about 2 and/or 4?
I wish I could answer YES. But there is a practical problem. The notions in 2 an 4 are semantic in nature. Though we (human being) can figure out semantic better, we mainly can work with syntactic notions. Dividing and forking offer a syntactic hand-grip.
Between a natural notion with few theorems and a unnatural one with many theorems mathematicians clearly choose the second (who would disagree).