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Is the importance of developing forking machinery in the way we set it up, or is it in the fact that it allows us to come up with a notion of independence via the properties of non-forking? I'm currently reading Baldwin's Fundamentals of Stability Theory and would like to know if I need to have a very deep understanding of how to set up the machinery or whether it is enough to have a rudimentary understanding so that I can come back to the details in a second reading.

Also, are there any expository papers detailing the use of non-forking from an at least somewhat historical viewpoint? I'm thinking of something in the spirit of Baldwin's first chapter.

Thank you.

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For stable theories, it's really enough to know that the forking machinery exists and what properties it has. However, there are an awful lot of properties being used all the time (in stable contexts), which may or may not be obvious depending on which characterisations you are familiar with. And there is an awful lot of properties which it fails to have in general. This is why learning about the basics first tends to make sense, and this is also why it's good to know plenty of examples, both trivial and complicated, mathematically meaningful and artificial.

I think it's worth pointing out that Baldwin's first book Fundamentals of Forking is a bit dated. Even when it was still up to date and I used it for getting into the field via self-learning (1994), it was a bit hard to understand without the help of additional views such as those in Pillay's first book An Introduction to Forking and Lascar's Stability in Model Theory. But it was, and still is, useful as a reference to the classical material, as a pointer to the literature, and as a source of a myriad of little instructive examples and helpful illustrations. It's great for inspiration.

One thing that has changed in the meantime is that the focus of model theory has moved on beyond stable theories in the classical context. Some people are now working on generalisations of classical stability theory to certain generalisations of first-order theories. I guess (but I am not an expert for this) that for these Baldwin's first book is still relevant in that the notions used for the generalisations are often more complicated generalisations of the classical notions, which you should therefore understand first. But the majority of the community has moved towards unstable theories, while staying in the classical complete first-order context.

And this is where it gets interesting. For many years I wondered at Shelah's many notions related to independence in stable theories, and especially the prominence of dividing/forking. Why did he prefer these unintuitive definitions to the Paris approach using heirs and coheirs over a model? Was it just that he didn't think of the latter? When Byunghan Kim's thesis came out, the answer became clear: There is a world of stability theory outside stability, at least in simple theories. And there forking/dividing is exactly what we need. (In fact, it's defined so that if there is some notion of independence with certain nice properties, then non-dividing must be it.) Moreover, in simple theories the weaker assumptions result in generally less distraction, so that it's much easier to find the 'correct', i.e. short and natural, proofs for results:

First understanding forking in simple theories makes it easier, not harder than starting with stable theories!

(Currently something similar seems to be going on with NIP=dependent theories and with NTP2, a common generalisation of simplicity and NIP. The theory of NIP is difficult in part because forking is not symmetric there and so it's just not clear how to generalise results from the stable context. But part of the problem is again distractions by a general assumption that is too strong. By working in the more general NTP2 context, one can get rid of these distractions and find the proper, natural machinery that one can then also use in NIP theories. Of course this is on-going research, so I am not recommending to start with publications on NIP or even NTP2 right now.)

Now to get back to your question. Baldwin's first book is the kind of book where a rudimentary understanding might well be sufficient (depending on how you define that, and depending on what you want to learn from the book). This book is really all about the big picture, and for obtaining that it may still be the best source available. But once you start looking at the details, you will need additional books or research papers, some of which are actually highly readable. And as I said, for understanding the underlying basics of forking it's perhaps best to start with simple theories. (E.g. Enrique Casanovas' Simple Theories and Hyperimaginaries.) Just don't think too much about hyperimaginaries, since ordinary imaginaries are all you'll need in the stable case.

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There are people in this forum who are better qualified to address what is important or not. But I'll have a go.

At least in stable theories there are multiple ways of defining forking independence. E.g. from dividing, using definability or heir-coheir extensions. Actually the definition from dividing is applied very rarely. Instead one uses the properties of forking independence all the time. Besides we have a theorem asserting that if there is an independence relation possessing some properties, then the theory must be stable and the independence relation must be the forking-independence. Based on these, my opinion is that the properties of forking-independence are much more important than the exact definition from dividing.

On the other hand, in non stable theories (e.g. in simple theories) one still uses forking independence. However the other notions are no longer equivalent (signalling that the definition from dividing is the right one). The properties of forking independence are still very important though.

A useful reference for this is "Forking and Multiplicity in First Order Theories" by John Baldwin (available on his website) which has a bit of history as well.

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  • $\begingroup$ Thank you. This is very close to what I was looking for. I will take a look at the paper. $\endgroup$
    – user75685
    Jul 5 '15 at 19:14

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