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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.

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    $\begingroup$ Good luck! This is partially sarcasm, partially a true encouragement. Beware of answers that try to convince you that this notion is easy/natural. In fact, for 20 years people tried to get rid of it. Indeed, in most important cases nonforking reduces to an much more intuitive notions. But the general notion has resurrect in the 90's, I guess because of the (re)discover of simple theories. I will post a more practical contribution as a answer below. $\endgroup$ Apr 28 at 18:36
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    $\begingroup$ @PrimoPetri Easy? No. Natural? Yes! $\endgroup$ Apr 28 at 18:45
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    $\begingroup$ @PrimoPetri I hope you do add your own answer, by the way, especially if you want to elaborate on the reductions to more intuitive notions in important cases. This is important, and I left it out of my answer. $\endgroup$ Apr 28 at 19:45
  • $\begingroup$ @PrimoPetri oof, thanks. I will be waiting for the answer $\endgroup$
    – ℋolo
    Apr 28 at 20:02

2 Answers 2

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Some years ago I wrote an answer about the motivation for the definitions of dividing and forking. You may find it helpful: https://math.stackexchange.com/a/1613033/7062

But I'll add to what I wrote there, addressing some specific points in your question.

You wrote:

From my understanding, $\varphi(x,b)$ divides over $C$ if $\varphi(C,b)$ (i.e. the set $\varphi$ defines with $b$) is not "big".

That's exactly the right idea, except that we shouldn't think about the set $\varphi(C,b)$, but rather the set $\varphi(\mathscr{U},b)$, where $\mathscr{U}\models T$ is the monster model. Often $\varphi(C,b)$ might be empty (as small as possible!) even though $\varphi(x,b)$ does not divide.

In fact, to start getting intuition for these notions, it's probably best to think about the case $C = \varnothing$.

If I give you a definable set in $\mathscr{U}$, say defined by the formula $\varphi(x,b)$, I can also find "copies" of this definable set that "look the same". What makes a copy "look the same"? Well, it should be defined by the same formula, using parameters that "look the same": $\varphi(x,b')$, where $\mathrm{tp}(b) = \mathrm{tp}(b')$. By the strong homogeneity of the monster model, $\mathrm{tp}(b) = \mathrm{tp}(b')$ if and only if there is an automorphism $\sigma\in \mathrm{Aut}(\mathscr{U})$ such that $\sigma(b) = b'$, and in that case $\sigma(\varphi(\mathscr{U},b)) = \varphi(\mathscr{U},b')$.

Intuitively, we say a formula divides (over $\varnothing$), if it has a lot of copies with very little overlap. In the answer linked above, I tried to give some intuition for why this is a good notion of "smallness".

Now it's often useful to draw finer distinctions for when two tuples "look the same" than looking at their type over $\varnothing$. We could instead fix a set $C$ in the background and insist that a "copy" of $\varphi(\mathcal{U},b)$ should be defined by $\varphi(x,b')$, where $\mathrm{tp}(b/C) = \mathrm{tp}(b'/C)$. Equivalently, these definable sets are conjugate by an automorphism of $\mathscr{U}$ which fixes $C$ pointwise. This is the role of "over $C$" in the definition of "dividing over $C$".

Whenever you think about dividing or forking over $C$, you could just as well think about dividing or forking over $\varnothing$, in the expanded language where we name every element of $C$ by a constant.

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").

That's exactly right: once we have a basic notion of "smallness" (dividing), it's natural to want the small sets to form an ideal in the Boolean algebra of definable sets. The forking formulas are exactly the ones which are "small" in the sense that they are in the ideal generated by the dividing formulas.

From a technical viewpoint, the fact that the forking formulas form an ideal lets us take any non-forking partial type (that is, any filter in the Boolean algebra of definable sets which does not meet the forking ideal) and extend it to a non-forking complete type (an ultrafilter which does not meet the forking ideal). In practice, when $C\subseteq B\subseteq A$, this is used to take a type $p(x)\in S_x(B)$ which does not fork over $C$ and extend it "generically" to a type in $S_x(A)$ which does not fork over $C$. In other words if $p(x)$ doesn't fork over $C$, we can extend it to any larger set of parameters without being "forced" into any definable sets which are "small" from the point of view of $C$.

I feel like this view falls completely for forks: the set of forking formulas is not always a proper ideal.

Well, it's a common theme in mathematics that attempts to distinguish between large and small sets don't work in all situations. For example, from the point of view of Baire category, we introduce a basic notion of "smallness", namely nowhere denseness, and then we want the "small" sets to form a $\sigma$-ideal, so we define meager sets to be countable unions of nowhere dense sets. This works great for some spaces (those with the property of Baire), but in other spaces we find that the entire space is meager, so this particular large/small distinction is useless.

It's quite an amazing phenomenon in model theory that the analogous situation is so rare: there are only a few natural examples where the formula $x = x$ forks, so that the forking ideal is the entire Boolean algebra of definable sets. If we restrict attention to simple theories, we can prove that this never happens, which is one reason why the theory of forking and dividing is so well-suited to the context of simple theories.

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    $\begingroup$ (Characteristically) great exposition -- I wish I could +1 a second time. As a quick question, do you have a favorite survey on the applications of forking? $\endgroup$ Apr 28 at 20:00
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    $\begingroup$ @HallaSurvivor I don't know of a survey that's focused on the topic of applications of forking. But forking is so central to stable and simple theories that any survey of stability or simplicity theory is basically all about forking! From this point of view, I like Bradd Hart's article Stability theory and it's variants. $\endgroup$ Apr 28 at 20:09
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    $\begingroup$ @ℋolo (1) "non forking is a filter" is not quite right: the complement of an ideal is not usually a filter, and the conjunction of two non-forking formulas can fork. You should think of forking as like "measure 0" and non-forking as like "positive measure". (2) What I meant to express is that a type is a filter, and a non-forking type is a filter that doesn't meet the forking ideal. (3) I don't understand your second comment. Dividing/forking are not meant to capture "smallness in $C$". $C$ is background parameters that increase the expressivity of our language, we never look "in $C$". $\endgroup$ Apr 28 at 20:24
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    $\begingroup$ @ℋolo Ah good, I'm glad we're on the same page about that now. Continuing: (4) Simple theories are a class of theories containing the stable theories (e.g. vector spaces and algebraically closed fields) and also examples like the random graph. They can be characterized in by several good properties of forking. For example, a theory is simple if and only if non-forking independence is symmetric. They are defined and discussed in Hart's survery I linked to above. See also the "map of the universe". $\endgroup$ Apr 28 at 23:53
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    $\begingroup$ @ℋolo (5) By "generic", I meant nothing more than what I wrote: not containing any forking formula. There is an analogy here with the notion of generic in forcing. What properties does a generic (Cohen) real have? Well, it's not contained in any meager set. What properties does a random real have? Well, it's not contained in any measure $0$ set. What properties does a realization of a generic extension of a type over $C$ have? Well, it doesn't satisfy any formula which forks over $C$. $\endgroup$ Apr 29 at 0:00
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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 $p(x)$

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

  2. $p(x)$ 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 $p(x)$

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

  2. $\varphi(x,b)\in p$ 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).

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  • $\begingroup$ If I understand correctly the idea behind (2) is that as long as I don't touch $M$, any "shuffling" of $\mathscr{U}$ will not making inconsistencies ($σ(φ(x,a))∈p$), so $p$ is big(/maximal) in the sense that it doesn't have any "wiggle room" as long as we don't touch it's "base" ($M$) $\endgroup$
    – ℋolo
    Apr 28 at 23:03
  • $\begingroup$ And about (3), can't we force the $c$ to be such that $p(c)$? (We take $σ_i\in\operatorname{Aut}(\mathscr{U}/M)$ that fixes each parameter in $φ$ and sends $b_{i-1}$ to $b_i$ (with $b_0=b$)), so does it means that in NIP (for global types over models) being forking and dividing is the same? $\endgroup$
    – ℋolo
    Apr 28 at 23:06
  • $\begingroup$ > can't we force the $c$ to be such that $p(c)$?< No, recall that $p(x)$ is global. Therefore $p(c)$ only if $p(x)\leftrightarrow x=c$ $\endgroup$ Apr 28 at 23:09
  • $\begingroup$ Also note that I am only discussing global types (for simplicity). Hence I am avoiding the issue forking vs. dividing. $\endgroup$ Apr 28 at 23:12
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    $\begingroup$ I wish I'd know. But note that the reverse question is even more interesting: are there interesting classes of theories where invariance has a useful syntactic description? (By means of nonforking or else.) $\endgroup$ Apr 29 at 9:03

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