Why not adopt the constructibility axiom $V=L$?

Question

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?

Answer 1

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.

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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.

I think that in order to be a good candidate axiom, a principle needs to be motivated by non-foundational ideas; and this requires more than just demonstrating its power outside of logic.

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 think that even if we go full Platonist, and even if we're fully convinced after deep technical work that a given principle is true of the universe of sets, that's not enough to add it as an axiom to ZFC.

(I'm not claiming this matches the actual history of ZFC at all, by the way!)

Answer 2

Let me adopt a somewhat contrarian view, and slightly overstate it for the sake of clarity:

The historical window for adopting new "foundational axioms" has closed. V = L, and any other axiom you might propose today, has missed the boat. It's too late.

I'd argue that the last time an axiom gained "foundational status" was when the axiom of choice made the cut, and that was roughly a hundred years ago. The general attitude today, I submit, is pluralistic. If you're interested in an axiom, then by all means, go ahead and investigate its consequences. But since we understand today that the status of axioms such as V = L is never going to be settled in the way that mathematicians are accustomed to "settling" open problems in mathematics, and since there is such a plethora of axioms to choose from, the tendency is to live and let live.

This state of affairs is reinforced by the fact that set-theoretic axioms such as V = L are not (or at least seem not to be) visibly relevant to the sorts of questions that most mathematicians are interested in. So there's less motivation for people to take a partisan stance on such questions, unless they've "drunk the foundation Kool-Aid" and start to care a lot about foundational questions. Among Kool-Aid drinkers you can have a debate (about whether V = L violates the "maximize" principle, or whether, as Shelah says, it seems to be a permanent resident lacking citizenship), but it's going to be regarded as a side-show by the general mathematical community, which by and large no longer sees the point in taking a strong stance on questions like this.

Answer 3

When I explain the cumulative hierarchy at the beginning of a set theory course, I point out that it involves two rather vague ideas: (1) forming arbitrary subsets of a given set (which is what we do at each successor step of the hierarchy), and (2) continuing forever. More technically, these two sorts of vagueness are represented by the notions of full power sets and arbitrary ordinals.

The constructible universe $L$ still has vagueness (2) built in, but (1) has been replaced with the much less vague formation of definable subsets. The intuitive (for me) content of $V=L$ is therefore that all vagueness of the form (1) can be reduced to form (2).

That sort of reduction strikes me as implausible for two reasons. First, there seems to be no natural connection between (1) and (2). Second, this sort of reduction can actually be disproved in some more restricted contexts (where we and ZFC can actually see what's going on). What I have in mind here are results like the theorem (of Sacks, if I remember correctly) that for almost all reals $r$, the Church-Kleene ordinal ${\omega_1}^{CK}$ relativized to $r$ is the same as the unrelativized one. (Here "almost all" can be taken with respect to either Lebesgue measure or Baire category.) In other words, much of the complexity (one might even say most of the complexity) in a real $r$ does not show up in the ordinals computable from $r$ (or even the ordinals hyperarithmetical in $r$, since those are the same ordinals). To me, that strongly suggests that ordinals capture only a small part of the complexity that can occur in arbitrary sets. And in that sense, arbitrary sets should not all (or even mostly) be constructible.

Answer 4

I think this question deserves a careful conceptual analysis, and I would like to raise two conceptual issues that I consider to be relevant. The first issue is about set theory only, without reference to the role it plays in the foundations of mathematics. The second issue is about the relation between set theory and mathematics. In my opinion, to say that $V=L$ is inconsistent with other axioms without further thoughts does not answer the question, for it just pushes forward the problem: one could also ask for a conceptual basis for preferring the other axioms. Now, the first issue:

1. Set theory is not supposed to go so much against its original direction of inquiry based on unlimited set-formation. The adoption of $V=L$ represents a radical rupture with the original concept set theory was supposed to be about.

The original direction of set theory was given by the usual naive set concept which is open-ended in the sense that the corresponding set-formation notion is unlimited, the naive operation of set of applies without limitation to any plurality of particular objects, no matter what. That open-endedness is present in Cantor's famous paragraph and the corresponding undisciplined gathering by which sets can thus be obtained in the naive conception leads to well-known paradoxical conclusions. For instance, based on unlimited set-formation one can say that a universe of sets can be seen as just a set in another universe, that whenever a plurality of sets is considered, there might be new sets outside, as nothing in the open-ended set concept prevents the application of set-formation to that plurality itself.

After the discovery of the now well-known paradoxes in the realm of unlimited set-formation, iterative set-formation has assumed the position of preferred conceptual basis for set theory. The iterative conceptual direction for set theory is a suitably organized in stages set-formation notion in which the production component is dominant and the organization level is reduced to a minimum, just enough to avoid the known paradoxes derived from unlimited set-formation. Iterative set-formation is unbalanced, but still points in the acceptable direction in which production is dominant. On the other hand, the organization component of the constructible variation of iterative set-formation is dominant with respect to its production component, which is responsible for its strength. The unbalanced constructible set-formation is capable of justifying the very strong constructibility axiom, deciding basically every question that is supposed to be decided. However, in spite of its virtues, this kind of unbalance is not acceptable mainly because set theory is not supposed to go against its original direction of inquiry based on unlimited set-formation.

2. $V=L$ reinforces the already dominant arithmetical/combinatorial character of set theory over the geometrical/dynamical component of the mathematical thought.

The pythagorean view embodied by $V=L$ and according to which everything is completely determined by the ordinals is not very faithful to the geometrical component of the mathematical thought (since the discovery of the incomensurability of the diagonal of the square). Indeed, Jensen opposes it to the newtonian view according to which the continuum admits no simple arithmetical/combinatorial reduction. We always had the arithmetical thought in which things are supposed to be counted and the geometrical thought in which things are supposed to be measured, not counted. In set theory, thanks to the (very combinatorial-like) axiom of choice, everything is supposed to be counted and not every part of the continuum can be measured. We have the anti-geometrical Banach-Tarski paradox, a consequence of this asymmetrically combinatorial framework. Since we have been exposed to the arithmetization of simple geometrical notions such as that of limit for more than a century, the arithmetical reduction of geometry was partially naturalized, but it is not very natural. It is quite cumbersome to do, for example, geometry of bundles and connections in set theory, as one must keep track of a lot of annoying identifications. $V=L$ emphasizes this asymmetry, it is just too much arithmetic-friendly and geometric-unfriendly. The asymmetry is already present in usual set theory anyway, and maybe topos theory must be considered more neutral with respect to the arithmetical and geometrical components of mathematical thought, but that is another story.

Answer 5

Although the axiom of constructibility is often resisted by set theorists with the view that it is restrictive, nevertheless there are a variety of ways in which the axiom is compatible with strength in set theory, which I survey and explain in my article:

Hamkins, Joel David, A multiverse perspective on the axiom of constructibility DOI:10.1142/9789814571043_0002, Chong, Chitat (ed.) et al., Infinity and truth. Based on talks given at the workshop, Singapore, July 25--29, 2011. Hackensack, NJ: World Scientific (ISBN 978-981-4571-03-6/hbk; 978-981-4571-05-0/ebook). Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore 25, 25-45 (2014). ZBL1321.03061. Blog post.

Abstract. I argue that the commonly held $V\neq L$ via maximize position, which rejects the axiom of constructibility $V=L$ on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of $V=L$ inside larger set-theoretic universes.

From my blog post:

To support the main philosophical thesis of the article, I survey a series of mathemtical results,  which reveal various senses in which the axiom of constructibility $V=L$ is compatible with strength in set theory, particularly if one has in mind the possibility of moving from one universe of set theory to a much larger one.  Among them are the following, which I prove or sketch in the article:

Observation. The constructible universe $L$ and $V$ agree on the consistency of any constructible theory. They have models of the same constructible theories.

Theorem. The constructible universe $L$ and $V$ have transitive models of exactly the same constructible theories in the language of set theory.

Corollary. (Levy-Shoenfield absoluteness theorem)  In particular, $L$ and $V$ satisfy the same $\Sigma_1$ sentences, with parameters hereditarily countable in $L$. Indeed, $L_{\omega_1^L}$ and $V$ satisfy the same such sentences.

Theorem. Every countable transitive set is a countable transitive set in the well-founded part of an $\omega$-model of V=L.

Theorem. If there are arbitrarily large $\lambda<\omega_1^L$ with $L_\lambda\models\text{ZFC}$, then every countable transitive set $M$ is a countable transitive set inside a structure $M^+$  that is a pointwise-definable model of ZFC + V=L, and $M^+$ is well founded as high in the countable ordinals as desired.

Theorem. (Barwise)  Every countable model of  ZF has an end-extension to a model of ZFC + V=L.

Theorem. (Hamkins, see here)  Every countable model of set theory $\langle M,{\in^M}\rangle$, including every transitive model, is isomorphic to a submodel of its own constructible universe $\langle L^M,{\in^M}\rangle$. In other words,  there is an embedding $j:M\to L^M$, which is elementary for quantifier-free assertions.

Another way to say this is that every countable model of set theory is a submodel of a model isomorphic to $L^M$. If we lived inside $M$, then by adding new sets and elements, our universe could be transformed into a copy of the constructible universe $L^M$.

From the introduction to the article:

Let me briefly summarize the position I am defending in this article, which I shall describe more fully section in 4. On the upwardly extensible concept of set, one holds that any given concept of set or set-theoretic universe may always be extended to a much better one, with more sets and larger ordinals. Perhaps the original universe even becomes a mere countable set in the extended universe. The "class of all ordinals," on this view, makes sense only relative to a particular set-theoretic universe, for there is no expectation that these extensions cohere or converge. This multiverse perspective resonates with or even follows from a higher-order version of the maximize principle, where we maximize not merely which sets exist, but also which set-theoretic universes exist. Specifically, it would be limiting for one set-theoretic universe to have all the ordinals, when we can imagine another universe looking upon it as countable. Maximize thereby leads us to expect that every set-theoretic universe should not only have extensions, but extremely rich extensions, satisfying extremely strong theories, with a full range of large cardinals. Meanwhile, I shall argue, the mathematical results of section 3 lead naturally to the additional conclusion that every set-theoretic universe should also have extensions satisfying $V=L$. In particular, even if we have very strong large cardinal axioms in our current set-theoretic universe $V$, there is a much larger universe $V^+$ in which the former universe $V$ is a countable transitive set and the axiom of constructibility holds. This perspective, by accommodating both large cardinals and $V=L$ in the multiverse, appears to dissolve the principal thrust of the $V\neq L$ via maximize argument. The idea that $V=L$ is permanently incompatible with large cardinals evaporates when we can have large cardinals and reattain $V=L$ in a larger domain. In this way, $V=L$ no longer seems restrictive, and the upward extensible concept of set reveals how large cardinals and other strong theories, as well as $V=L$, may all be pervasive as one moves up in the multiverse.

The final section of the article includes a philosophical defense of the upwardly extensible concept of set, a view by which the axiom of constructibility $V=L$ becomes a kind of on-again off-again set-theoretic switch, as the ordinals grow.

Answer 6

Keith Devlin wrote a book called The Axiom of Constructibility --- A Guide for the Mathematician which espouses just this point of view.

My impression is that most working set theorists regard $V = L$ as "boring", perhaps for precisely the reason mentioned in the question, that it seems to settle nearly all ordinary mathematical questions. [Edit: this statement is not controversial. Problems arising in mainstream math which are independent of ZFC are already rare. I do not know any examples of problems which legitimately arose in mainstream math that are known to be independent of $V = L$.] Whereas large cardinal axioms have consequences, such as the existence of $0^\#$, which are of great interest to set theorists (though only peripheral interest outside the field). The existence of $0^\#$ is itself a pretty strong argument against $V = L$ because once one admits that $L$ exists one is hard pressed to deny $0^\#$.

In the opposite direction, when Chuck Akemann and I showed that a counterexample to Naimark's problem can be constructed using diamond, I gave some talks for general audiences which included a brief discussion of diamond and $V = L$. After several of these talks people asked me why we don't adopt $V = L$ as an axiom. So a good part of the reason there hasn't been a strong movement for adopting $V = L$ is probably because the ordinary mathematicians to whom it would make its strongest appeal are only barely aware of it.

A personal comment: in my case, I went through a phase where I thought $V = L$ made a lot of sense on philosophical grounds. Namely, there should not be "unobservable" sets. The picture that most set theorists seem to have, of a well-defined universe of sets "out there" somewhere, in which all questions have definite answers, still seems to me naive, and frankly I think I have a better understanding of what sets are than those who take this picture seriously. Abstract objects are only linguistic devices and have no actual content. But when I really thought through these issues I came away with something much stronger than $V = L$, namely that all sets are constructible and all sets are countable.

Answer 7

Since this question is more philosophical than mathematical, let me wax poetic for a moment. Consider the following thought experiment. You build a machine that will flip a quarter, once every minute, for the rest of eternity. When heads is flipped the machine writes to an infinite tape the number 1. When tails is flipped it writes 0. Thus, you get an infinite sequence of 0's and 1's, which can be thought of as a real number.

You could get a different real number by turning on the machine a little later. Or using a different quarter. Or using a different mechanism than a coin flip. etc...

Are each of these real numbers in $L$? Is there any reason to suppose that all real numbers that we intuitively believe might "exist" are also the result of the operations which define the constructible universe?

Answer 8

I believe the main problem with $V=L$ is how it restricts reflection. It has been discovered that $V$ obeys several reflection properties, the most famous theorem. Intuitively this makes sense; if Set Theory is considered the supereme foundation of mathematics, then the universe $V$ of Set Theory is the absolute infinite. As such, it stands to reason that for any $M$ there exists some $M_0\supseteq M$ that is in a sense, a substructure of $V$ (This is in fact what the reflection theorem states).

Let's take this a step further. What if we suppose that there exists another $M$ "embbedable" into $V$; that you can construct a set that is identical to a certain construction of $V$. Moroever, we must assume this embedding is non-trivial, to avoid simply repeating the Reflection Theorem. Assuming something similar, we can derive the existence of a measurable cardinal, which $V=L$ sates does not exist. And here we get to the root of the problem; $V=L$ postulates, in a sense, the finitude of mathematics; that every object has an elemntary characterization in terms of parameters and formulas, which just doesn't seem right. In all the vast vistas of mathmetics, everything is just a simple collection of paramtered formulas, with handful of connective and two quantifiers?