TITLE: Why not adopt the constructibility axiom $V=L$?
QUESTION [32 upvotes]: 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?

REPLY [1 votes]: 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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