Forcing and PDE's Anybody knows where can I read about some relations between PDE's and set theory? Something like "There exists a PDE such that the existence of a solution for it cannot be determined in ZFC" or "Given a PDE, the existence of a solution is absolute". I'm just looking for some references.
Thank you! 
 A: Shoenfield absoluteness shows that you're not going to get any simple examples of this. Roughly speaking, there is a hierarchy of mathematical propositions in terms of their quantifier complexity, and Shoenfield absoluteness states that sentences of a simple enough form cannot have their truth values altered by forcing. This level is $\Pi^1_2$: statements of the form "For every real $r$, there is a real $s$ such that [stuff]", where [stuff] involves only quantification over natural numbers (or equivalently rationals). The negation of a $\Pi^1_2$ sentence is $\Sigma^1_2$ and is also absolute.
Let's look at a very special case: the statement "$E$ has a solution defined on all of $\mathbb{R}$," where $E$ is some differential equation in $f, f', f'', . . . $. Well, continuous functions are coded by real numbers; so the outer quantifier is $\Sigma^1_1$, and we need to examine how complicated the statement "$r$ codes a solution to $E$" is.
With a bit of work, it's not hard to show that this is $\Pi^1_1$ (in fact, I believe this is overkill), so the original statement is $\Sigma^1_2$; hence its truth value cannot be altered by forcing. Similarly, saying that there is at most one solution to $E$ is $\Pi^1_2$: "For all $r, s$, either one of $r$ or $s$ does not code a solution to $E$, or $r=s$." So again, this statement's truth value cannot be altered by forcing.
(This shouldn't be too surprising, given the philosophy that the solution space carved out by a differential equation is a reasonably tame object; this suggests that the quantifier complexity of the relevant statements isn't too great.)
Now, more complicated statements such as "any differential equation of such-and-such a form has a unique solution" are on the face of it more susceptible to set-theoretic techniques, but even here we run into problems: most nice classes of differential equations have low-quantifier-complexity descriptions, and I believe most of the relevant statements can still be expressed in a $\Pi^1_2$ or $\Sigma^1_2$ manner.
Finally, even given a sufficiently complex statement, note that large cardinals provide extensions of Shoenfield absoluteness all the way up the projective hierarchy. So to get a statement which provably can be altered by forcing, we would need something like the Continuum Hypothesis which is fundamentally a statement about sets of reals. But such statements are quite rare outside of set theory.

This doesn't rule out the possibility of still using forcing in the context of PDEs: there are examples of theorems proved using forcing, as follows - force to produce a more-easily-analyzed model of set theory in which you show $\varphi$ holds, then argue via appropriate absoluteness that $\varphi$ must have been true to begin with. These arguments are (in my opinion) spectacular, but also few and far between. Currently, I do not know of any instances in differential equations.
