In this post I will give a few examples of forcing notions and explore their combinatorial properties. Most of this is directly taken from “combinatorial set theory with a gentle introduction to forcing” by Halbeisen and condensed down a bit.
Ultrafilter Forcing :
What makes this one useful, is that a -generic filter over is also a Ramsey Ultrafilter in the extension, different from all ultrafilters in the ground model. Hence, when forcing with this notion, one adds at least one Ramsey Ultrafilter to the ground model. There are a few things one needs to check to be able to conclude:
- RUF’s are characterised by the fact that for each coloring
there is a such that is constant, i.e. for each coloring, contains a monochromatic set. We will conclude that the new filter has such a set for each coloring in the ground model. However one also needs to make sure that no new colorings get added, so that indeed the property holds true for all colorings in . For this one checks that is -closed, hence preserves all cardinals , and in particular all real numbers . Since the reals are in one-to one correspondence with the two colorings of , there are no new colorings appearing in .
- is in fact an ultrafilter which is different from every ultrafilter in the ground model. For the first fact consider the open dense set:
being -generic means it intersects this set, and hence either or . For the second fact let be an ultrafilter in the ground model and consider the following open dense set:
Cohen Forcing :
This forcing notion can be used to show that is consistent with . Suppose was a model of . Then forcing with adds distinct new reals to the model, the so called Cohen reals, and preserves all cardinals. In fact if is -generic over , we get that is a function . This can be concluded by looking at the open dense set:
which shows that is defined for all . Since counts exactly the number of reals, if we choose we get:
Hence we see that the continuum hypothesis no longer holds. However there are a few things one needs to check given a -generic filter over :
- The added reals are indeed are all distinct and different from all existing reals. First, for , we define the new reals by:
Then you look at:
which exactly tells you that no two ‘s are the same, and consider a similar open dense set to show that they also differ from all reals in the ground model.
- The forcing doesn’t collapse any cardinals, since otherwise might shrink down to again and we wouldn’t have proved anything. This is shown using the fact that is by a consequence of the -system lemma, and hence preserves all infinite cardinals.
Collapsing Forcing :
The conditions are given by partial functions with countable domain, and the order is the inclusion. As the name suggests, this forcing can be used to collapse a cardinal to . In particular for we have that for a -generic filter over :
One needs to show a couple of things for that:
- No new reals get added. This can be assured by verifying the -closedness of .
- adds a surjective function , so that
Sacks Forcing :
Here the conditions are given by:
and . A tree is a subset that is closed under taking initial segments, i.e. if , then for all . We also say that extends , written , if is an initial segment of . A perfect tree, is a tree that splits indefinitely, i.e. . Sacks forcing allows us to build a model [G], such that there is no “intermediate model” between and [G], meaning if is a model of then either or . Forcing with perfect trees was used by Gerald Enoch Sacks to produce a real a with minimal degree of constructibility.
Sacks Forcing adds neither splitting nor unbounded reals
Miller Forcing :
Here the conditions are given by:
and . Here superperfect means that for :
is infinite for all .
It adds unbounded reals but no splitting reals.
Mathias Forcing :
where is a fixed Ramsey family. The conditions are given by:
and the preorder is given by:
Mathias forcing can be used to construct a pseudointersection of an ultrafilter.
Restricted Laver Forcing :
Here is a fixed ultrafilter. Now a Laver tree is a tree if it has a special element , called the stem, such that for any we have , and for every with , we have that is infinite. A Laver tree restricted to is a Laver tree such that . The conditions of Laver forcing are exactly the Laver trees restricted to , with reverse inclusion.
This forcing is -centred and adds a real which is almost homogenous for all colorings in the ground model.
Silver Forcing :
Here is a fixed P-family. Define:
and for :
Then . For write and just call it Silver forcing. Silver reals are given by:
In general, Silver filters cannot be reconstructed from Silver reals, since multiple Silver filters yield the same real. However, if is an ultrafilter we can always do the reconstruction.
Shelah’s Tree forcing : See my other post here.