Let $V=L$ denote the axiom of constructibility. Are there any interesting examples of set theoretic statements which are independent of $ZFC + V=L$? And how do we construct such independence proofs? The (apparent) difficulty is as follows: Let $\phi$ be independent of $ZFC + V=L$. We want models of $ZFC + V=L + \phi$ and $ZFC + V=L + \neg\phi$. An inner model doesn't work for either one of these since the only inner model of $ZFC + V = L$ is $L$ and whatever $ZFC$ can prove to hold in $L$ is a consequence of $ZFC + V=L$. Forcing models are of no use either, since all of them satisfy $V \neq L$.
Statements Independent of ZFC + V=L
lo.logicset-theory
Related Solutions
Let me address the part of your question seeking algebraic statements independent of ZFC+V=L.
The basic situation is that in set theory our tools are not so flexible for finding statements independent of ZFC+V=L, as opposed to finding statements independent of ZFC. The main reason for this is that one cannot directly use forcing to prove that a statement is independent of ZFC+V=L, because no nontrivial forcing extension can satisfy this theory. Simply put, forcing extensions never satisfy V=L. Thus, our main tool for proving independence over ZFC does not work at all for proving independence over ZFC+V=L.
This phenomenon has led some set theorists to view the theory ZFC+V=L as "nearly complete", settling essentially every set-theoretic question. But of course, the theory isn't really complete---it cannot be by the incompleteness theorem---it just feels so complete in comparison with the ubiquitous independence phenomenon for ZFC. Nevertheless, one can find several classes of statements independent of ZFC+V=L. Let me list a few of them.
There are of course the consistency statements, such as Con(ZFC), which if true, are independent of ZFC+V=L.
The existence of certain large cardinals, such as inaccessible, Mahlo, hyper-Mahlo, weakly compact, indescribable, or unfoldable cardinals, if consistent with ZFC, is independent of ZFC+V=L. The reason is that if there are such cardinals, then they exist also in L, and truncating the universe shows the consistency of the non-existence of such cardinals. Perhaps some of these large cardinals can be cast in purely algebraic terms, and I expect this may be the most promising kind of example for you.
The existence of transitive models of any given extension of ZFC, such as ZFC + `there is a supercompact cardinal', if consistent with ZFC, is independent of ZFC+V=L. The reason, as I explain in a recent paper on the axiom of constructibility, is that $V$ and $L$ have transitive models of the same theories. So if there is such a transitive model inside a model of ZFC, then there is such a transitive model inside a model of ZFC+V=L, and meanwhile, it is consistent with V=L that there are no transitive models of ZFC at all.
The number of ordinals $\alpha$ such that $L_\alpha\models\text{ZFC}$ is highly independent (provided that it is consistent to be large, which is essentially a mild large cardinal assumption). The reason is that if there are, say, $\omega^2+\omega\cdot 5+7$ many such ordinals $\alpha$, then by chopping off at the $\omega^2+\omega\cdot 4+26^{th}$ such ordinal, the number of such $\alpha$ drops accordingly.
Unfortunately, it seems that few of these kinds of independent statements have an essentially algebraic nature. I think the best examples will arise simply by stating the large cardinal axioms in algebraic-like terms, such as they are.
This principle is inconsistent: consider the formula $\theta(x)$ = "$x^+$ is the smallest infinite cardinal at which $\mathsf{CH}$ fails." The formula $\theta$ cannot hold on more than one infinite cardinal, let alone on all infinite cardinals, yet your principle (applied to $\phi:=\neg\theta$) would require this.
(I originally omitted the "$+$"-superscript; that's actually a mistake, since there are constraints on when $\mathsf{CH}$ can fail first. Looking at a successor simplifies things.)
Note that this argument is rather flexible: for example, it also shows that we can't "go from internally countable failures to global failures" by considering $\psi(x)=$ "$x^+$ is one of the first $\omega_1$-many failures of $\mathsf{CH}$."
Best Answer
There are numerous examples of such statements. Let me organize some of them into several categories.
First, there is the hierarchy of large cardinal axioms that are relatively consistent with V=L. See the list of large cardinals. All of the following statements are provably independent of ZFC+V=L, assuming the consistency of the relevant large cardinal axiom.
There is an inaccessible cardinal.
There is a Mahlo cardinal.
There is a weakly compact cardinal.
There is an indescribable cardinal.
and so on, for all the large cardinals that happen to be relatively consistent with V=L.
These are all independent of ZFC+V=L, assuming the large cardinal is consistent with ZFC, because if we have such a large cardinal in V, then in each of these cases (and many more), the large cardinal retains its large cardinal property in L, so we get consistency with V=L. Conversely, it is consistent with V=L that there are no large cardinals, since we might chop the universe off at the least inaccessible cardinal.
Second, even for those large cardinal properties that are not consistent with V=L, we can still make the consistency statement, which is an arithmetic statement having the same truth value in V as in L.
Con(ZFC)
Con(ZFC+'there is an inaccessible cardinal')
Con(ZFC+'there is a Mahlo cardinal')
Con(ZFC+'there is a measurable cardinal')
Con(ZFC+'there is a supercompact cardinal').
and so on, for any large cardinal property. Con(ZFC+large cardinal property).
These are all independent of ZFC+V=L, assuming the large cardinal is consistent with ZFC, since on the one hand, if W is a model of ZFC+Con(ZFC+phi), then LW is a model of ZFC+V=L+Con(ZFC+phi), as Con(ZFC+phi) is an arithmetic statement. And on the other hand, by the Incompleteness theorem, there must be models of ZFC+¬Con(ZFC+phi), and the L of such a model will have ZFC+V=L+¬Con(ZFC+phi).
Third, there is an interesting trick related to the theorem of Mathias that Dorais mentioned in his answer. For any statement phi, the assertion that there is a countable well-founded model of ZFC+phi is a Sigma12 statement, and hence absolute between V and L. And the existence of a countable well-founded model of a theory is equivalent by the Lowenheim-Skolem theorem to the existence of a well-founded model of the theory. Thus, the truth of each of the following statements is the same in V as in L.
There is a well-founded set model of ZFC. This is equivalent to the assertion: there is an ordinal α such that Lα is a model of ZFC.
There is a well-founded set model of ZFC with ¬CH. (This is also equivalent to the previous statement.)
There is a well-founded set model of ZFC with Martin's Axiom.
and so on. For all the statements known to be forceable, you can ask for a well-founded set model of the theory.
There is a well-founded set model of ZFC with an inaccessible cardinal.
There is a well-founded set model of ZFC with a measurable cardinal.
There is a well-founded set model of ZFC with a supercompact cardinal.
and the same for any large cardinal notion.
These are all independent of ZFC+V=L, since they are independent of ZFC, and their truth is the same in V as in L. I find it quite remarkable that there can be a model of V=L that has a transitive model of ZFC+'there is a supercompact cardinal'. The basic lesson is that the L of a model with enormous large cardinals has very different properties and kinds of objects in it than a model of V=L arising elsewhere. And I believe that this gets to the heart of your question.
Since all these statements are studied very much in set theory, and are very interesting, and are independent of ZFC+V=L, I find them to be positive instances of what was requested.
However, how does this relate to Shelah's view in Dorais's excellent answer? He seems there to dismiss the entire class of consistency strength statements as combinatorics in disguise. What does he mean exactly? Since we set theorists are very interested in these statements, I don't think that he means to dismiss them as silly tricks with the Incompleteness theorem. Perhaps he means something like: to the extent that we believe that a large cardinal property LC is consistent, then we don't really want to consider the theory ZFC+V=L, but rather, the theory ZFC+V=L+Con(LC). That is, we aren't so interested in models having the wrong arithmetic theory, so we insist that Con(LC) if we are comitted to that. And none of the examples I have given exhibit independence from that corresponding theory.