# The Model of Solovay

The objective of measure theory is to make rigorous the intuitive notions of length, surface area, volume, and measurement of quantity. A mathematical model of all of these can be found in the concept of measure on a set , which in its simplest form is a function such that the following hold:

•  Monotonicity:
• Normalization: and
• Atomlessness:

On the one hand, this notion replicates a number of facts about volume in the physical world. For instance, the first condition can be rephrased as “Bigger objects have bigger volume” and the second one translates to “The combined volume of a collection of disjoint objects is the sum of their respective volumes”. On the other hand, we can interpret a measure as an order-preserving map between partially ordered sets. Here the set is ordered by inclusion, and is ordered by . From this perspective, (i) just means that is order-preserving, (ii) restricts the value of on certain joins, and (iii) and (iv) are normalization conditions. In analogy of the representation theory of groups, the structure of as a poset is compared trough to the poset structure of . This interrelates the two structures and allows to analyze the structure of either one.With this definition in mind, there are a few questions that arise naturally. From the practical point of view, me might ask:

• Does euclidean space admit a measure that is translation invariant and hence reflects our experience of volume?

From the set theoretic point of view, one might ask:

• Which sets admit a measure? In particular, since admits a measure exactly when does, which cardinals admit measures? This question is known as the Measure Problem.

As we will show in this semester project, the answer to the first question is no under . This has peculiar consequences such as the possibility to decompose a solid ball into a small number of pieces, rearranging them, and ending up with two exact copies of the initial ball. This mathematical possibility of copying a ball without adding any mass is known as the Banach-Tarski paradox, and would be impossible if a measure existed on all of , for then mass would have to be conserved.

From here, there are essentially two possible directions to proceed. Either one can weaken the requirement that the measure be translation invariant, in which case there may exist a solution, or one can weaken , which is the route we are taking. Robert M. Solovay showed in 1970  that there is a model of set theory in which every set of reals is Lebesgue measurable and where a such a weakening of , the axiom of dependent choice , still holds.

The main goal of this thesis is to construct this model and to discuss how the structure of the real line relates to strong assumptions, such as further axioms of set theory or large cardinal hypotheses.In the first chapter, we introduce essential concepts and tools from descriptive set theory, such as a representation of the reals used in set theory, the Borel sets and alongside Lebesgue measurability two other regularity properties of the reals, the property of Baire and the perfect set property. The second chapter constructs the Model of Solovay and proves that any subset of the real line in it is Lebesgue measurable, has the Property of Baire and the perfect set property. The proof requires the existence of an inaccessible cardinal to define the Lévy collapse, which is a forcing notion that allows to overwrite any bad behaviour the reals might display. With this a possible solution to the measure problem is presented if is not assumed.In the last chapter we present the work of Shelah and Specker, illuminating when the assumption of the inaccessible cardinal is really needed, and in which cases it can be avoided. Finally, we will compare different alternatives to and present their impact on the regularity of the reals.We generally follow the books by Jech and Kanamori.

The rest of this semester project can be found here.

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# Shelah Product Tree Forcing

This forcing is used to prove the independence of the number of Ramsey ultrafilters from . We will give a brief overview of the results discussed in Halbeisen’s book.  To be precise:

Theorem: Let be any cardinal. Then there is a model of  with exactly Ramsey ultrafilters.

The idea behind the proof is to start with a lot of Ramsey ultrafilters, say many, for instance by starting in a model where holds. Then we have that implies , which in turn can be  used to construct RUF. Next, we choose of them to preserve, and destroy all of the others. This can be done with an iteration of some forcing notion whose job it is to destroy exactly one RUF. One then has to make sure of course that:

• The  iterated forcing notions don’t interfere with each other.
• No new RUF are added at limit stages.

This forcing notion which will be responsible for eliminating a RUF at a time is Shelah’s product tree forcing. First some preliminary definitions.

We define for that extends if:

We say a subset is a (product) tree if for all and such that we have , i.e. is closed under taking initial segments.  A tree is called perfect, if it splits indefinitely, i.e. for each element, we can find two different extensions.  Product trees are a little more complicated than the usual trees encountered in Sacks or Miller forcing, since at level , they potentially split into branches. Let’s formalize that idea.

is called the restriction of up to level .  If we have an element and some , then we can concatenate them to get an element . Let be given. Then:

Now we are ready to define the -th meta level

Next, we say a binary tree is a -tree, if it is a full binary tree when restricted to its first levels. We can then look at how “branching” the levels of are by examining:

So in other words: means that every node on the -th meta level branches into at least nodes on the next meta-level. We are now finally in good shape to define the Shelah tree forcing. Consider a fixed P-family . The conditions of the Shelah tree forcing restricted to are then given by perfect product trees, for which for every . The order is given by reverse inclusion. Together we denote them by .

Let us now take a look at the properties of this forcing notion.

Proposition is proper and -bounding.

Lemma: Suppose is a Ramsey ultrafilter in the ground model and let be a P-point in the ground model. Then if:

We must have . In other words, the Shelah tree forcing destroys only RUF with are below in the Rudin-Keisler order of ultrafilters.

Lemma: Let be a P-point and the corresponding Shelah tree forcing. If we have an -iteration of other Shelah tree forcings  for some P-points , then they do not intere with each other in the following sense:

Let be -generic over and be -generic over , then does not generate an ultrafilter in . In other words,  killed the ultrafilter , and no -iterations of Shelah forcing can revive it.

With these properties, it is possible to prove that for instance the existence of just RUF is consistent with . The proof of these properties is rather complicated, and one still needs to prove things such that

• a RUF will generate a RUF in an extension, if the forcing notion is proper, -bounding, and preserves it as a P-point.
• non isomorphic RUF stay non isomorphic in -bounding forcing extensions.
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# Some Properties of Forcing Notions

Suppose is a forcing notion in some model of , and let be a -generic filter over . Then there are a few properties a can have that are especially useful.

Adding reals: A lot of forcing notions add reals to the ground model in some form or another. Prominent examples would be Cohen reals for Cohen forcing, Sacks reals for Sacks forcing, etc. However forcing with doesn’t only add a single -real, as reals definable trough that new real also get added. Hence it would be lovely to have a few properties that describe general reals added by forcing notions. Examples of such properties are:

1. Adding dominating reals:  In every forcing extension ,  there is a real that dominates .
2. Adding unbounded reals: In every forcing extension ,  there is a real that is not bounded by any real in the base model.
3. Adding splitting reals:  In every forcing extension ,  there is a real that splits any real in the base model, i.e.

It turns out that if adds a dominating real, it also adds both unbounded reals (this is easy) as well as splitting reals (this is not). However none of these inclusions are reversible as shown by the following table, exposing which forcing notions add which types of reals:

 Notion Dominating reals Unbounded reals Splitting reals Mathias ✔ ✔ ✔ Cohen ✘ ✔ ✔ Sacks ✘ ✘ ✘ Silver-like ✘ ✘ ✔ Miller ✘ ✔ ✘

Chain conditions: Let be a regular cardinal. We say that satisfies the -chain condition, if every maximal antichain in  has size strictly less than . The case is also called the countable chain condition or ccc for short. This condition is useful since we have the following

Proposition: If satisfies the  -chain condition, then all the cardinals are preserved in any forcing extension with .

This essentially follows from the fact that if we have a surjective  function for some cardinals in the extension, we also have a -name for it. Then for the set :

is an antichain in , and hence of cardinality strictly smaller than . Now this set consists of all the conditions above which the value of will eventually be forced to be some value . Since , we know that number the possible choices for must also be , since every such choice has to be forced by some condition in the filter, by the fundamental theorem of forcing. By the defineability of forcing, we can define another surjective function in the base model:

where the parameter is used to sweep all the possible choices  of . Hence for this implies that

So that also in the base model. If then if we would have by regularity of :

A contradiction to the surjectivity of . Hence in both cases, cardinals are preserved.

Closedness Properties:  Let be any cardinal. Then is called -closed , if for any ascending sequence of conditions there is a condition such that  for all . For , this is called -closedness. This property complements the -chain condition, in that the following hols:

Proposition: If satisfies the  -closedness condition, then all the cardinals are preserved in any forcing extension with .

This follows from the fact, that if is a function in with , then is already in the ground-model. Hence no cardinals less than can be collapsed. Now why is this the case? has a -name in . Start with any condition . Define the condition by:

for some . This is possible, since the conditions that define are open dense in . Again by open density find such that for some with:

Continue in this manner to obtain an ascending sequence . Now at the limit stage, utilize the -closedness to find for any . Continue this process further until we have a condition such that for all

So above each , there is a which essentially says that is definable in , hence no matter which generic filter we choose, there will be some , and hence is already a function in

Properness: A generalization of the . Remember that assures that no  uncountable cardinals get collapsed. Also gets preserved  under finite support iterations. However this is often too much to ask from a forcing notion, hence the notion of properness, for which the following hold:

Proposition 1: If is proper, then it does not collapse .

Proposition 2: If is a countable support iteration of , and for each   we have , then is proper.

Proposition 3:  If is a countable support iteration of proper forcing notions which satisfy one of the following properties

• Laver Property
• Preservation of P-points
• -boundedness

Then  is proper and has the respective property.

So now to the definition of properness. For a regular limit cardinal let:

These are all the sets which are hereditarily of cardinality . It turns out is a model of minus the power set axiom. For the definition of properness, we need large enough. Assume in the following, this will suffice.  Let be an elementary submodel of , i.e. any statement about elements of is true in  iff the corresponding statement (where all the quantifiers have to be relativised) is true in

Let . We say that is -generic if it has the following property: If and . So if the model says that a subset is open dense, should intersect it, similarly to genericity over . Next, a condition , which does not need to be in , is -generic if

where is the canonical -name for the -generic filter in .

Finally, a forcing notion is called proper, if for all countable elementary submodels of which contain , and every condition , there is a -generic condition .

Laver Property:  A property that makes sure no Cohen reals are added while forcing. Let be the family of all function , such that . The family is restricting the growth of its member to  exponential growth.  A forcing notion is said to have the Laver property, if for any function   in the ground model and any  -name of a function , such that we have:

Another way of looking at the Laver property is that if satisfies the Laver property, the models and cannot be too wildly different, since each bounded funtion in the extended model can at least be approximated in the ground model.

Preserving P-points: Let be a P-point in . Then we say that preserves , if generates an ultrafilter in any extension. Equivalently:

Note, that we don’t require the ultrafilter generated in the extension to be a P-point. This is where the following statement comes in:

Proposition: Let be a proper forcing notion and be a P-point in the ground model . If the filter generated in is an ultrafilter, then it is a P-point.

So nice enough forcing notions maintain P-points in this stronger sense.

Furthermore, the countable support iterations of P-point preserving  proper forcing notions also preserves P-points.

-bounding:  A forcing notion is called -bounding  if there are no unbounded reals in any forcing extension.

-boundedness of proper forcing notions is preserved under countable support iterations.

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# Some Forcing Notions

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:

Here, being -generic implies, that for every ultrafilter in the ground model, there is a set in that is not contained in , hence in particular .

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.

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