Motivation behind Generic filters in forcing.

Question

So I was reading about forcing in wikipedia to try to get an intuitive idea about forcing in set theory. There is this paragraph in it:

A subtle point of forcing is that, if $X$ is taken to be an arbitrary "missing subset" of some set in $M$, then the $M[X]$ constructed "within $M$" may not even be a model. This is because $X$ may encode "special" information about $M$ that is invisible within $M$ (e.g. the countability of $M$), and thus prove the existence of sets that are "too complex for $M$ to describe".

Then there is another line about how forcing avoids this problem:

Forcing avoids such problems by requiring the newly introduced set $X$ to be a generic set relative to $M$. Some statements are "forced" to hold for any generic $X$. For example, a generic $X$ is "forced" to be infinite. Furthermore, any property (describable in $M$) of a generic set is "forced" to hold under some forcing condition. The concept of "forcing" can be defined within $M$, and it gives $M$ enough reasoning power to prove that $M[X]$ is indeed a model that satisfies the desired properties.

Here is the definition of a "generic filter"

For $G$ to be "generic relative to $M$" means:
$\bullet$ If $D \in M$ is a "dense" subset of $\mathbb{P}$ $\space$(that is, for each $p \in \mathbb{P}$ , there exists a $q \in D$ such that $q \leq p$), then $G\cap D \neq \emptyset $.

Why is the definition of generic filter designed like this? (intersecting with dense subsets of $\mathbb{P}$ in $M$). What does it do to avoid the problem mentioned in the first paragraph?

Note: I don't have any background in set theory except I read some stuff from Enderton's set theory textbooks.

Answer 1

I think the modern abstract approach to forcing obscures the underlying motivation. Cohen's original approach (as in his monograph Set Theory and the Continuum Hypothesis) is much more concrete and shows why one might think of defining forcing as we do.

I'll briefly describe a hybrid version that shows why the dense open set requirement comes up in the modern approach.

You want to have that, for every sentence $\varphi$ in the language of forcing, there's some condition $p\in G$ which decides $\varphi.$ This is how we ensure that every first-order fact about $M[G]$ depends only on some individual condition being in $G$ and so can be talked about in $M$ (rather than depending on all of $G,$ or on more of $G$ than $M$ can know about).

An equivalent way of phrasing this requirement is that for every $\varphi$ in the language of forcing, $$G\cap\{p\mid p\Vdash \varphi \text{ or }p\Vdash\neg\varphi\}\neq\emptyset.$$

(By the way, I'm oversimplifying a bit here. You need to do all this with an auxiliary notion of "strong forcing," which is defined via a somewhat elaborate transfinite induction, and then you can define the normal notion of forcing, sometimes called "weak forcing," using that. But that doesn't affect the principle here, so I won't delve into those details.)

Note that for every sentence $\varphi$ in the language of forcing, the set $\{p\mid p\Vdash \varphi \text{ or }p\Vdash\neg\varphi\}$ is a dense open set of conditions.

So if you require that $G$ meet every dense open set of conditions in $M,$ it will meet all the sets $\{p\mid p\Vdash \varphi \text{ or }p\Vdash\neg\varphi\},$ which are the dense open sets that we really care about.

But the dense open set requirement is more elegant, in that it can be described simply just based on the partial ordering, algebraically, without having to develop the forcing relation first.

So that's why we look at dense open sets in defining genericity.

Answer 2

One way to motivate generic filters is that generic filters are hidden points (from $V$) that exist almost everywhere. Let us recall the Baire category theorem:

Baire category theorem. Let $X$ be a locally compact regular space, then every countable intersection of open dense sets of $X$ is non-empty.

Also, we can view ultrafilters as points in a topological manner (see the convergence of a filter.) Similarly, we can view forcings as topological spaces: one way of doing it is imposing the Scott topology over a forcing poset (a topology generated by sets of the form $\{q\mid q\le p\}$.) Or you may consider the Stone dual of the forcing poset (given by the set of all ultrafilters over the poset.) In either way, we may understand a forcing poset as a basis for a topological space.

Under the Scott topology, a dense set (a la partial order) $D\subseteq \mathbb{P}$ corresponds to a topologically open dense set over the space. Thus a filter (or a point) $G$ being a generic can mean that it belongs to every open dense set in the ground model $M$, so the generic point is "generic" in every open dense condition in the ground model.

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But I do not think the above explanation justifies why it avoids the issues you mentioned. We can prove that $M[G]$ is a model of $\mathsf{ZFC}$ if $M$ is, but it seems not to follow from the above informal explanation. We should rather view it happening by chance. There are other model constructions (like, classical realizability) that are not forcing but also give a new model of $\mathsf{ZFC}$ from another.

Answer 3

To answer your question, I think it's worth asking why we want the ground model $M$ to be able to describe the forcing relation in the first place. One justification for this is that we as mathematicians cannot perform infinite tasks without assistance from the assumed existence of a model in the first place. It's similar to the slogan that "programmers borrow randomness from the universe" to describe the inability to write an algorithm that yields a truly random output and the solution to sample things like time or temperature or wind patterns to use as inputs, as these can function as some natural form of randomness that we can steal from the world around us. As a mathematician, for instance, I cannot build a wellordering of the real line; I rely on the fact that such a wellordering already exists in a model of $ZFC$, as a consequence of the axioms that hold in that model. Similarly, I cannot build a model of $ZF(C)$ from scratch. This is why we deal in terms of relative consistency: given that a model $M$ of $ZF(C)$ exists, how can I use this to build a model $N$ of $ZF(C)$ with the properties that I would like to show to be consistent. In the context of forcing, this gives some intuitive justification for wanting $M$ to be able to describe the forcing relation, since we are using it to construct the new model.

With this motivation in place, we can begin to address your question. The ordinals of a model $M$ serve to describe all possible index-lengths of processes that $M$ can carry out; by a process, I mean one with a well-defined "next" step in the process at each point, so the indexing must be wellordered. This is one way to justify the constructible universe $L^M\subseteq M$: $L^M_\alpha$ is the class of sets that $M$ can build using defining formulae over some transfinite process of length $\alpha$, and $L^M=\bigcup_{\alpha\in OR^M}L^M_\alpha$ is the collection of such constructions over all such process lengths that $M$ is aware of (with these process lengths given by $OR^M$). When dealing with forcing over a poset $\mathbb{P}$, the $\mathbb{P}-$names give a way to describe all possible constructions that $M$ can describe (hence, indexed over the ordinals from $M$ as in the case for $L^M$) with reference to an arbitrary new set $G\subseteq\mathbb{P}$. The $\mathbb{P}-$names are built by $M$ without knowing yet what $G$ will be, but with a clear idea for how to determine what each one becomes once $G$ is given.

Let's pause and take a different approach: we can think of $\mathbb{P}$ as a collection of pieces of information $p\in\mathbb{P}$. Our goal is to sew the information together into a coherent new set of information from $\mathbb{P}$ that $M$ does not yet know about, namely $G\subseteq\mathbb{P}$. Critically, we need two things: first, that the new set of information from $\mathbb{P}$ is coherent. This is why we require $G$ to be a filter of information; if $p$ is a piece of information in $G$, then $G$ contains all weaker information than $p$, and any two pieces of information $p$ and $q$ from $G$ must be compatible. The second thing we need relates to the generic condition about meeting dense subsets. If a new subset $X\subseteq\mathbb{P}$ is considered, $M$ in general has no way of knowing what arbitrary route through $\mathbb{P}$ was taken to build $X$. As finite mathematicians who borrow infinite processes from $M$, we cannot ourselves build some new $X$ to put in, and now neither can $M$, so how are we supposed to gather knowledge about the new structure $M[X]$? This dilemma is captured by your first quote: if $X$ encodes some information that makes it so $M$ cannot describe it without extra outside help, we're at a dead end. The solution is genericity. $M$ knows about each partial fragment $p$ of $G$ in its construction and this simple condition on meeting dense subsets gives us a mechanism to predict just enough about how $G$ is built past a point $p$ to make claims about what statements must be true of $M[G]$ given that $p\in G$. Although we still cannot know ultimately which points $p\in\mathbb{P}$ are added to build $G$, we can make predictions as to what statements must be true given that a point $p$ has been included in $G$, i.e. what sentences $\phi$ that are forced to be true if $p$ is a piece of information included in your construction of $G$.

How does the generic condition on dense subsets let us predict enough of the construction of $G$ beyond a point $p\in\mathbb{P}$ to meet this goal? A dense subset $D$ can be thought of as representing some property (for instance, $D$ could be the set of points that claim that the range of a function we are building includes the value $7$). If $D$ is dense in $\mathbb{P}$, at any partial point $p$ in the construction of $G$, it is still possible to extend and include a point from $D$ since there is some $d\in D$ with $d\leq p$. The dense sets represent properties that it is never too late to throw into your construction, and genericity states that if it is never too late to include such a point from $D$, then it is guaranteed that this happens at some point in the construction. This is an extremely strong predictive tool for what happens beyond a point $p$ in the construction of $G$! In everyday terms, at any given point in the day, it is possible for me to wash my dishes, yet I may still not complete the task. If this were a generic process, the possibility of washing my dishes at any point in the day would guarantee that the task be completed at some point before midnight. With this predictive strength for what must be included in $G$ beyond a point $p$ in its construction, we can make conclusions about certain types of pieces of information from $\mathbb{P}$ that must and must not be added to $G$, in a way that is sufficient to conclude which sentences $\phi$ must be true in $M[G]$ based solely on the fact that $p\in G$. Thus, $M$ knows about each possible point $p$ in the construction of $G$, and can use this generic condition to make predictions of this form, which allows $M$ to fully define the forcing relation and make claims about what would necessarily be the case if it came to know about some new generic filter $G\subseteq\mathbb{P}$ with any particular $p\in G$.