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?

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    $\begingroup$ A cynical viewpoint: It's "effective" in settling questions. But mathematicians would rather have open problems (gives them busy work). $\endgroup$ – MyNinthAccount May 19 '19 at 22:37
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    $\begingroup$ What does it mean to "adopt" $V=L$? Does it mean that all research into alternative theories (with large cardinals, determinacy, etc.) should be halted, and papers about such research no longer be published? Otherwise, if people are going to continue investigating consequence both of $V=L$ and its alternatives, how is that different from the present situation? $\endgroup$ – bof May 19 '19 at 22:41
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    $\begingroup$ Does anyone know of a journal rejecting a proof on the grounds that it assumed $V = L$? $\endgroup$ – Andrej Bauer May 19 '19 at 22:54
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    $\begingroup$ A perhaps less contentious way of phrasing the question would be to ask why $V=L$ does not have the same epistemological status in ordinary mathematical practice as the axiom of choice (see here and here for parallels between the two). $\endgroup$ – Gro-Tsen May 19 '19 at 23:04
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    $\begingroup$ @Andrej Many papers in general topology are like that. There are also quite a few (current) examples in real analysis, group theory, model theory, etc. $\endgroup$ – Andrés E. Caicedo May 20 '19 at 11:03

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.

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!)

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    $\begingroup$ @YCor Yes, I know - the point was that the speaker didn't (or more reasonably didn't think about it and temporarily forgot that new axioms can't rule out old theorems), and interpreted "every set is constructible" as ruling them out. The point I was making was that the naive justification for V=L is actually quite dangerous as an explanation - because the notion of constructibility here is actually subtle and technical, the slogan "every set is constructible" is really not useful. $\endgroup$ – Noah Schweber May 20 '19 at 4:27
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    $\begingroup$ @MonroeEskew First of all, note that - per the beginning - most of my answer is descriptive rather than prescriptive. I do separately happen to actually support these (per the end of the answer) but that's a separate point. Re: (1), I think you underestimate how complicated that is for someone with no logic background. I'm basing this off of numerous conversations I've had with such mathematicians. Re: (2), I honestly find that quite distasteful - I really think that one of the key criteria of a foundational theory is that it be maximally accessible. (cont'd) $\endgroup$ – Noah Schweber May 20 '19 at 6:42
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    $\begingroup$ (And to be honest, I do have sympathy for the criticism of ZFC that it's already too messy in terms of implementing arguments.) Basically, I think the threshold for understanding what a "community-valid proof" is should be as small as possible; I don't really see "lots of natural problems are undecidable" as an overriding concern, since that's always going to happen per Godel (and if we adopt a "sufficiently dynamic" approach to foundations, developing our base theory over time, there will be occasions where we have to walk everything back as we discover an inconsistency in old commitments; $\endgroup$ – Noah Schweber May 20 '19 at 6:50
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    $\begingroup$ @MonroeEskew Concerning "once you know what the ordinals are and what the satisfaction relation is": The problem is that a lot of mathematicians don't know and don't want to find out what the satisfaction relation is. As far as I know, that's why Gödel circumvented satisfaction in his book by defining $L$ in terms of what are now called the Gödel operations. Those seem to be just an ugly way of disguising satisfaction to not look like logic. $\endgroup$ – Andreas Blass May 20 '19 at 12:37
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    $\begingroup$ "Replacement is transfinite recursion which isn't really that alien." Can't we be a lot more down to earth than that? Replacement is set-builder notation. Non-set theorists use set-builder notation in two ways: $\{x\in S|P(x)\}$ and $\{f(x)|x\in S\}$. The axiom that guarantees the existence of a set of the first form is separation. The axiom that guarantees the existence of the second form is replacement. This seems like a much less alien justification than an appeal to only somewhat alien transfinite recursion. $\endgroup$ – ziggurism May 20 '19 at 13:00

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.

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    $\begingroup$ I think that the axiom of choice made the cut prior to replacement and regularity. Both of which are arguably foundational axioms just as much as they are set theoretic axioms. $\endgroup$ – Asaf Karagila May 20 '19 at 13:59
  • $\begingroup$ I have to say, I still don't understand why. The Whitehead problem was considered important. If history had turned out differently and it had been announced that Shelah had solved it by using deep structural facts about sets, rather than that Shelah showed it to be independent of the axioms, then maybe people would have said, "Hmm so those set theorists figured out something useful after all!" It seems that the decision not to promote V=L was made by set theorists during the time when foundations were very unclear. $\endgroup$ – Monroe Eskew May 20 '19 at 15:24
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    $\begingroup$ @MonroeEskew "It seems that the decision not to promote V=L was made by set theorists during the time when foundations were very unclear." I think there's a separate, and very interesting, question about the history of attempts on the logicians' side to argue for new axioms. But that's different from acceptance by the mathematical community. $\endgroup$ – Noah Schweber May 22 '19 at 1:39
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    $\begingroup$ @AsafKaragila : Regularity is sort of irrelevant to mathematics, but replacement is an interesting case. It makes me think that an axiom is much more likely to gain foundational status if mathematicians use it "unconsciously"; i.e., assuming it tacitly without fully realizing that they're invoking an "extra" assumption. If that is true, then V=L is at a disadvantage because it's not the sort of thing that anyone is going to use unconsciously. $\endgroup$ – Timothy Chow May 26 '19 at 16:18
  • $\begingroup$ Is an axiom true because the majority of mathematicians believe it? What makes an axiom true? $\endgroup$ – Jesse Elliott Jul 11 '19 at 5:59

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.

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    $\begingroup$ Is the restriction to countable models a technical convenience, a necessity, or something else? $\endgroup$ – Andrej Bauer May 20 '19 at 14:35
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    $\begingroup$ If you are referring to Barwise's theorem or mine, then yes, one needs the restriction to countable models in order to prove the theorems. $\endgroup$ – Joel David Hamkins May 20 '19 at 14:38
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    $\begingroup$ "Needs" as in "it is known to be false for uncountable models"? I am asking because I am not sure how to understand the significance of the theorems for the question at hand (should we adopt $V = L$?). $\endgroup$ – Andrej Bauer May 20 '19 at 15:19
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    $\begingroup$ My argument concerning the significance of the results for $V=L$ is that they undermine the "restrictive" aspect of $V=L$, which is the main mark against it. The results explain how we can have large cardinals in one universe, and $V=L$ in a larger universe, on and off again as the ordinals arrive. But yes, these facts are proved in ZFC only for countable models, which is viewed as a "toy" multiverse. We have in principle no way to prove things like this for the full actual multiverse, except in analogy with the toy multiverse. $\endgroup$ – Joel David Hamkins May 20 '19 at 15:29
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    $\begingroup$ Meanwhile, yes, the theorems are known to fail (consistently) for uncountable models. For example, in my paper jdh.hamkins.org/…, we show that there are incomparable $\omega_1$-like models of set theory, which violate the comparability property of the theorem I had stated for countable models. $\endgroup$ – Joel David Hamkins May 20 '19 at 15:32

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.

  1. $V=L$ 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.

  • $\begingroup$ I don't buy your point (2) yet, although I think it's potentially quite interesting. I don't see how V=L goes further in the "antigeometric" direction. In particular, you describe the "geometrical component of mathematical thought" as being opposed to a "simply (combinatorially/arithmetically) definable" universe, but the example you give doesn't seem to fit this picture to me: it's not a situation where we abandon a commitment to definability, but rather where we prefer one notion of definability (= "geometrically constructible") to another. (cont'd) $\endgroup$ – Noah Schweber May 20 '19 at 3:06
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    $\begingroup$ In fact, I generally consider "geometrically-flavored" mathematics (indeed, most math) to be stricter - e.g. there's rarely a role for sets more complicated than analytic and ZFC already is enough to give a decent picture of the analytic sets. My own "hot take" for V=L is that it minimizes the extent to which set theory goes beyond what's needed - it's not that the geometrical viewpoint encourages us to think beyond the constructible, but rather that set theory encourages us to think beyond the geometrically meaningful. (And then large cardinals ruin this hot take, but that's a deep fact.) $\endgroup$ – Noah Schweber May 20 '19 at 3:08
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    $\begingroup$ I have not characterized the geometrical component of mathematics, but I think it is fair to say that V=L goes further in the antigeometric direction. First, the continuum becomes too arithmetical under V=L, as it is too close to the natural numbers. On the contrary, the geometric continuum seems to be something very distant, inaccessible from the natural numbers and incomensurable with it. Second, V=L seems to impose further dificulties to the measure problem by excluding measurables. Third point, I think Suslin lines are very antigeometric. But I agree that these may not seem very strong. $\endgroup$ – Rodrigo Freire May 20 '19 at 4:21
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    $\begingroup$ In short: Deciding things about unintended objects will not improve mathematics. The constructible real numbers seems not to be the intended real numbers, so proving statements about this object is not the same thing as solving a classical mathematical problem. $\endgroup$ – Rodrigo Freire May 20 '19 at 9:48
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    $\begingroup$ I will try to give an example of the sort of incorrect behavior I have in mind: to admit a low complexity definable well- ordering is not the sort of thing one expects of a continuum object. Admiting nonmeasurable parts is another, but this is in ZFC already (which is not neutral with respect to geometry and combinatorics according to what I understand). $\endgroup$ – Rodrigo Freire May 20 '19 at 12:44

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.

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  • $\begingroup$ +1 What would you think about the powerset of the ordinals capturing 'most'/all of the complexity that occurs when forming arbitrary subsets and continuing forever? Some people identify $\mathcal{P}(\omega)$ with $\mathbb{R}$ so this seems to be the case for the reals, and I would be tempted to identify $\mathcal{P}(O_n)$ with the surreals which (it has been suggested (by you)) may encode the entire universe taken along with their birthday structure. $\endgroup$ – Alec Rhea May 21 '19 at 2:53
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    $\begingroup$ @AlecRhea In the framework of ZFC, the power "set" of the ordinals captures all sets, i.e., every set $x$ can be coded as a set of ordinals as follows. Using AC, fix a bijection from the transitive closure of $x$, TC$(x)$, to an ordinal $\alpha$, and use this bijection to transfer the membership relation $\in\upharpoonright$TC$(x)$ to a binary relation $E$ on $\alpha$. Using a pairing function on ordinals, code $E$ as a set of ordinals. You can reconstruct $x$ because TC$(x)$ is the Mostowski transitive collapse of $(\alpha,E)$. $\endgroup$ – Andreas Blass May 21 '19 at 12:26
  • $\begingroup$ Very cool but slightly over my head when you begin to discuss Mostowski collapse, thank you though -- is the same fact true in the absence of choice? It looks like Mostowski's lemma holds without choice (ncatlab.org/nlab/show/Mostowski%27s+collapsing+lemma), but I don't know if the above argument carries through. $\endgroup$ – Alec Rhea May 22 '19 at 23:04
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    $\begingroup$ @AlecRhea My argument definitely requires choice, because I need to biject TC$(x)$ to an ordinal. I'm pretty sure it's possible for two different transitive models of ZF (without choice) to have the same sets of ordinals. $\endgroup$ – Andreas Blass May 23 '19 at 0:20
  • $\begingroup$ Interesting, thanks again Andreas. $\endgroup$ – Alec Rhea May 24 '19 at 6:10

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.

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    $\begingroup$ "[$V=L$] seems to settle nearly all ordinary mathematical questions." It settles the Riemann Hypothesis? and $P=NP$? and Navier-Stokes? Sign me up! $\endgroup$ – Gerry Myerson May 20 '19 at 12:51
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    $\begingroup$ @Gerry: Perhaps we need to qualify that with "interesting", because who even cares about RH or PvNP? :-) $\endgroup$ – Asaf Karagila May 20 '19 at 13:12
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    $\begingroup$ Abstract objects are only linguistic devices and have no actual content. I'm not sure this phrase has any meaning whatsoever. It is just some sort of religious proclamation and I find the religion declaring that mathematical notions are linguistics constructs no better than the one declaring that the visible universe is just a mathematical object, while, if we talk about personal preferences, the latter holds much more appeal to me. I would settle on much milder phrase: "mathematical objects can be viewed as linguistic constructs and that viewpoint produces some interesting considerations" $\endgroup$ – fedja May 20 '19 at 14:35
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    $\begingroup$ @GerryMyerson: okay, I understand the confusion now. I should have said "decide", not "settle". Fair enough. $\endgroup$ – Nik Weaver May 20 '19 at 23:05
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    $\begingroup$ @PaceNielsen, Nik has a list of papers here: mathoverflow.net/a/298759/44143. I recommend “Analysis in $J_2$” through “Is set theory indispensable?”, even though this is all “predicativism given the natural numbers”, which takes their canonical status as a motivational starting point. $\endgroup$ – Matt F. May 21 '19 at 11:19

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?

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  • $\begingroup$ At least one possible (strong) interpretation of the physical Church-Turing thesis would be that, yes, not only is the real number in $L$, but it is in fact Church-Turing computable. ⁂ A variant of the question might be to ask whether a physical dimensionless constant (say, the fine structure constant) belongs to $L$. I have no opinion on this, or whether it even makes sense to ask that question (but I find it interesting nonetheless). $\endgroup$ – Gro-Tsen May 20 '19 at 10:07
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    $\begingroup$ But you are certainly right to emphasize that one intuitive reading of the philosophical content (or at least, implication) of $V=L$ is that “true randomness does not exist (even conceptually)”, which is something that, presented in this form, I suspect most working mathematicians would not agree with. $\endgroup$ – Gro-Tsen May 20 '19 at 10:09
  • $\begingroup$ @Gro-Tsen Let's take it another direction! Fix an effective list of all arithmetic statements. Consider the sequence whose $i$th digit is $0$ if the $i$th arithmetic statement is false (in the standard model), otherwise it is $1$. This is not computable by a Turing machine. Godel's incompleteness theorem tells us that most(all?) theories we care about are ambivalent about what many of the actual digits are. It isn't random in the sense of indescribable, but it is certainly random in the sense that we are at a loss to explicitly write it down. Should we believe $L$ captures such constructions? $\endgroup$ – Pace Nielsen May 20 '19 at 12:01
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    $\begingroup$ @PaceNielsen You have described a constructible set, one that lives in $L_{\omega+\omega}$. $\endgroup$ – Monroe Eskew May 20 '19 at 12:24
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    $\begingroup$ Not necessarily, but $L$ isn't unique in having this problem. (If $M$ is a model of enough basic stuff, the set of numbers that code sentences true in $M$, is not definable in $M$. But it could live there as an undefinable element.) $\endgroup$ – Monroe Eskew May 20 '19 at 12:46

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?

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  • $\begingroup$ This seems to be an argument against $V=\mathrm{HOD}$. How do you distinguisn between both situations? $\endgroup$ – Andrés E. Caicedo Jun 8 '19 at 23:36
  • $\begingroup$ It is actually an argument against $V=HOD$, and I dislike both axioms. But it is also an argument against $V=L$. One axiom states that every set can be defined hereditarily via a formula with ordinal parameters ($V=HOD$), and another axiom states every set can be constructed via parameters in set-many steps ($V=L$). $\endgroup$ – Master Jun 8 '19 at 23:50
  • $\begingroup$ Wouldn't this be an argument against the idea that the theorems of geometry, for example, can be fully expressed in the language of set theory? $\endgroup$ – Monroe Eskew Jun 9 '19 at 22:52
  • $\begingroup$ I don't see your logic. I am saying that claiming everything is definable is inaccurate, not that claiming geometry is definable is inaccurate. $\endgroup$ – Master Jun 9 '19 at 22:57
  • $\begingroup$ And you're claiming it's inaccurate because the "vast vistas of mathematics" don't look like they should be reducible to simple logical components. You're giving an intuitive/aesthetic argument. Do you see how this intuition could be stretched to just be completely anti-reductionist? $\endgroup$ – Monroe Eskew Jun 10 '19 at 1:52

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