Gödelian incompleteness seems to ruin the idea of mathematics offering absolute certainty and objectivity. But Gödel‘s proof gives examples of independent statements that are often remarked as having a character that is too metamathematical. Other methods, such as Cohen’s forcing, are able to produce examples of independent statements that look more “ordinary.” However, the axiom $V=L$, when added to ZFC, settles “nearly all” mathematical questions. Furthermore, it can be motivated by constructivist philosophy. Here is Gödel (1938) introducing his theorem on the relative consistency of AC and GCH with ZF:
This model, roughly speaking, consists of all "mathematically constructible" sets, where the term "constructible" is to be understood in the semiintuitionistic sense which excludes impredicative procedures. This means "constructible" sets are defined to be those sets which can be obtained by Russell's ramified hierarchy of types, if extended to include transfinite orders. The extension to transfinite orders has the consequence that the model satisfies the impredicative axioms of set theory, because an axiom of reducibility can be proved for sufficiently high orders. Furthermore, the proposition "Every set is constructible" (which I abbreviate by "A") can be proved to be consistent with the axioms of [ZF], because A turns out to be true for the model consisting of the constructible sets…. The proposition A added as a new axiom seems to give a natural completion of the axioms of set theory, in so far as it determines the vague notion of an arbitrary infinite set in a definite way.
We note that Gödel later rejected this philosophical viewpoint. We also note that later developments on the structure of $L$, especially those due to Jensen, gave a wealth of powerful combinatorial principles that follow from the axiom $V=L$.
Question: Given the effectiveness of the axiom $V=L$ at settling mathematical questions, and the fact that it can be motivated by constructivist views that are still widely held today, why hasn’t there been historically a stronger push to adopt it as a foundational axiom for mathematics?
Best Answer
Let me add to the existing answers a point which may seem "vulgar" at first, but I think is actually important:
V=L is complicated.
And whether or not this ought to be a reason to not raise it to ZFC-esque status - I think it is, and see below - I think it's clear that it is in practice going to be an issue with any such attempt (and V=L isn't alone in this ...).
The ZFC axioms may be difficult to work with, but they're ultimately not that hard to understand. Union, Powerset, Pairing, and Extensionality are obvious; Separation is just restricted comprehension; Replacement is "transfinite recursion," which isn't really that alien; and Choice both has "obvious enough" forms and is sufficiently famous that general mathematicians are at least familiar with it in the abstract. Foundation poses a bit of an issue, but not because it's complicated, but rather because it often seems pointless; and that's fine since it really plays no essential role in general mathematics since we can "implement" everything within the hereditarily well-founded sets.
V=L, by contrast, is genuinely complicated. The slogan "only things you can construct exist" is snappy enough, but hides a ton of subtlety and is easy to use incorrectly: for example, can you construct the sets required for the Banach-Tarski paradox? (I've actually seen it claimed, by a competent non-set-theorist, that V=L prevents Banach-Tarski - by virtue of implying that every set of reals is Borel.)
And there are even deeper issues. For one thing, it's not even clear that V=L is actually first-order expressible! Similarly, the manner in which it resolves concrete questions actually requires some direct handling of logic. It takes a very little time to get a basic competency with ZFC; it takes some serious effort to achieve the same for V=L.
This winds up pushing back against "common readability" - the idea that mathematicians should be generally able to read a paper in their own field without having to be competent in an unrelated field. Of course even ignoring logic this frequently fails, but giving V=L the same status as the ZFC axioms would essentially endorse its fairly global failure.
In a bit more detail:
At the end of the day this gets to a question about what the purpose of foundations of mathematics is. I take a very "profane" approach: the point is to facilitate mathematics. Adding axioms in order to solve questions is fundamentally a cheat, especially when those candidate axioms are fundamentally technical.
Of course one can argue about the extent to which the original ZFC axioms satisfy this point, and in my opinion it would be incredibly dishonest to pretend that ZFC is the "a priori correct" foundational theory rather than historically contingent, but going forwards I think that the above is important. I see no way in which V=L meets this criterion.
Note that this criterion also pushes against (large) large cardinals, forcing axioms, etc. And indeed I take the "profane" view of foundations to what I believe is a rather unpopular extreme:
(I'm not claiming this matches the actual history of ZFC at all, by the way!)