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Rodrigo Freire
Rodrigo Freire
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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 betweenwith respect to the arithmetical and geometrical components of mathematical thought, but that is another story.

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 between the arithmetical and geometrical components of mathematical thought, but that is another story.

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.

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Rodrigo Freire
Rodrigo Freire
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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 between the arithmetical and geometrical components of mathematical thought, but that is another story.