A while ago, I took a long break from set theory and came back recently. Now, I've been more able to understand concepts which used to be completely foreign to me, for example ultrapowers and mice in inner model theory.

I'm still having trouble understanding the "intuition" of forcing, and how one is supposed to come up with models by forcing. Even though I understand each individual concept, and I can read about the proofs, I can't really understand the whole concept as a unit.

I'm not really looking for a guide as a "beginner's forcing book," more just an explanation of how one might create a forcing notion to solve a certain problem. For example, how did Cohen know which poset to force with to create a model in which CH is false?

Sorry if this question sounds a bit ambiguous, I'm not the best at putting things into words. The point is, I don't quite understand how to come up with specific posets for certain problems.

  • $\begingroup$ In general coming up with the "right" forcing notion is a very difficult process. But the explicit example you mention is straightforward: Suppose you have a model of CH and want its negation. What do you do? You add a bunch of reals and then try to show that you haven't collapsed their size. That's the naiive approach (once you have the method of forcing), that's what Cohen did and it worked. In current day settings, the naiive approach almost never seems to work which is why we end up with these insanely complicated forcing notions. $\endgroup$ – Stefan Mesken Oct 7 '18 at 9:43
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    $\begingroup$ To add to what @Stefan said, there is a lot of subtlety in coming up with a forcing notion. Objects tend to get coded into the generic filter, whether you like or not. So you need to be very careful to approximate what you want to add, but not collapse the whole damn thing to $\omega$. There are very few "obvious" posets, Cohen subsets and Levy collapses (and Cohens can be seen as a particular case of Levy, as is appropriate from Jewish tradition). Other than this, I guess the way would be the ask "What do I want to approximate" and then modify to exclude things you want to avoid. $\endgroup$ – Asaf Karagila Oct 7 '18 at 12:26
  • $\begingroup$ @StefanMesken I don't quite understand how he knew that he was adding more reals with his poset. Sorry if I'm a bit naiive, I'm just having trouble understanding this. $\endgroup$ – Keith Millar Oct 7 '18 at 16:18
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    $\begingroup$ To add to what @ Asaf said, If $F$ is a non-empty ground-model family of dense subsets of a forcing notion $(P,\leq) $ then forcing with $(\cup F,\leq)$ produces the same forcing extensions. So you may dig into $P$ to see if you can find a useful $F$ or you may assemble $F$ from raw materials. Sometimes the members of $P$ are called promises, as they sometimes can be thought of as small (e.g. finite) approximations to a large (desired) object. $\endgroup$ – DanielWainfleet Oct 7 '18 at 16:19
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    $\begingroup$ @Keith In this, as is true in many forcing applications, you build a poset of approximations to the objects you want to add and then use a density argument to show that some version of which you considered all the approximations of does make it into the generic extension. To gain a better understanding of how one might go about constructing a poset of such approximations, I'd encourage you to carefully look at a bunch of examples (different kinds of reals, shooting a club, specialising trees, ...). (But, at least for now, avoid the more complicated forcing notions.) $\endgroup$ – Stefan Mesken Oct 7 '18 at 16:38

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