2 Struck out "hot air"

Because of Goedel's Incompleteness Theorems, we know that we cannot describe a complete axiomatization of mathematics. Any proposed axiomatization $T$, if consistent, will be unable to prove the principle Con(T) asserting that $T$ itself is consistent, although we have reason to desire this principle once we have committed ourselves to $T$. Adding the consistency principle Con(T) simply puts off the question to Con(T+Con(T)), and so on, in a process that proceeds into the transfinite.

Thus, we come to know that there should be a transfinite tower of theories above our favorite theories, transcending them in consistency strength. The incompleteness theorems imply that there is a tower of theories above PA, above ZFC, each level transcending the consistency strength of the prior levels.

How fortunate and wonderful that we have also independently come upon such a tower of theories: the large cardinal hierarchy. Numerous large cardinal concepts arose very early in set theory, from the time of Cantor, before Goedel's theorems and before the notion of consistency strength was formulated. These large cardinal concepts arose from natural set-theoretic questions in infinite combinatorics: Can there be a regular limit cardinal? Can there be a countably-complete measure measuring all subsets of a set? Does every $\kappa$-complete filter on a set extend to a $\kappa$-complete ultrafilter? And so on.

Eventually, it was realized that these large cardinal notions separate into a very tall hierarchy, with the property that from the larger cardinals, one can prove the consistency of the smaller cardinals. For example, if $\kappa$ is the least Mahlo cardinal, then the universe $H_\kappa$ is a model of ZFC + there is a stationary proper class of inaccessible cardinals + there are no Mahlo cardinals. If $\delta$ is the least measurable cardinal, then $H_\delta$ satisfies ZFC + there are a proper class of Ramsey cardinals, but no measurable cardinal.

Thus, the large cardinal hierarchy provides exactly the tower of theories, whose levels transcend consistency strength, that we knew should exist. And it does so in a way that is mathematically robust and interesting, with its foundations arising, not in some syntactic diagonalization, but in mathematically fulfilling and meaningful questions in infinite combinatorics.

The case of Vopenka's principle is just like this. VP is a large cardinal axiom at the higher end of the large cardinal hierarchy, implying the consistency of the existence of supercompact cardinals, say, which are far stronger than strong cardinals, which imply entire towers of measurable cardinals, which imply numerous Ramsey cardinals and so on down the line.

Illustrating the essential large cardinal nature, the VP axiom is elegantly stated: for every proper class sequence $\langle M_\alpha | \alpha\in\text{ORD}\rangle$ of first order structures, there is a pair of ordinals $\alpha\lt\beta$ for which $M_\alpha$ embeds elementarily into $M_\beta$. (It is equivalently stated in terms just of graphs, if you like.) It's simple and clear---beautiful! And the consequences are far-reaching and often profound, as you have observed in category theory, in the way that VP implies that the set-theoretic universe is regular and organized.

These are the reasons you should be attracted to Vopenka's principle. It is an elegant combinatorial principle, with far-reaching consequences that interest you, which has not yet been refuted.

In contrast, I find the philosophical heuristics that seek to justify the large cardinal axioms, on the grounds of reflection or some other means, to be so much hot air ultimately unsatisfying. These arguments are not mathematically sound, and cannot be made to be, by the Incompleteness Theorems. Philosophically, they seem much more like rationalizations after the fact. For example, even at the much lower (and therefore seemingly easier-to-justify) level of inaccessible cardinals, one sometimes hears an appeal to reflection type views, that since we have no definable unbounded map from a set into the ordinals, that there should be a level $V_\kappa$ of the universe also with this feature, and that such a level would be inaccessible cardinal. Of course, the conclusion outstrips the argument, with the conclusion seeming to justify at most $V_\kappa\models$ZFC, which is a weaker notion, and the meta-reflection principle appealed to amounts anyway to a large cardinal principle of its own.

Ultimately, we must recognize the uncertain nature of all our mathematical enterprise. As our hypotheses rise higher in the large cardinal hierarchy, we must become less sure of consistency---perhaps they will be shown to be inconsistent. This issue arises even at the lowest levels of our mathematical axiomatizations, for we may find at any time (as mentioned in a recent MO question) that even PA is inconsistent. As Woodin says, we all have in our minds the image of a railway line, lined by a sequence of telegraph poles, proceeding into infinity; but when the physicists tell us that the universe is finite, we realize that this picture is pure imagination. Perhaps it is simply inconsistent? So skepticism about consistency has nothing especially to do with the infinite.

Meanwhile, the large cardinal axioms are fascinating and have fascinating consequences. Let's seek out the boundary of consistency, with an attitude tempered by the realization that we may find inconsistency.

In summary, we cannot ever be sure that our axioms are consistent, and we know that above the mathematical theory about which we may be sure, there is a tall tower of theories whose levels transcend consistency. Among them are fascinating theories that are elegantly stated with far-reaching consequences, and which we have not yet refuted. So let's study them! Let's find the boundary between consistency and inconsistency!

1

Because of Goedel's Incompleteness Theorems, we know that we cannot describe a complete axiomatization of mathematics. Any proposed axiomatization $T$, if consistent, will be unable to prove the principle Con(T) asserting that $T$ itself is consistent, although we have reason to desire this principle once we have committed ourselves to $T$. Adding the consistency principle Con(T) simply puts off the question to Con(T+Con(T)), and so on, in a process that proceeds into the transfinite.

Thus, we come to know that there should be a transfinite tower of theories above our favorite theories, transcending them in consistency strength. The incompleteness theorems imply that there is a tower of theories above PA, above ZFC, each level transcending the consistency strength of the prior levels.

How fortunate and wonderful that we have also independently come upon such a tower of theories: the large cardinal hierarchy. Numerous large cardinal concepts arose very early in set theory, from the time of Cantor, before Goedel's theorems and before the notion of consistency strength was formulated. These large cardinal concepts arose from natural set-theoretic questions in infinite combinatorics: Can there be a regular limit cardinal? Can there be a countably-complete measure measuring all subsets of a set? Does every $\kappa$-complete filter on a set extend to a $\kappa$-complete ultrafilter? And so on.

Eventually, it was realized that these large cardinal notions separate into a very tall hierarchy, with the property that from the larger cardinals, one can prove the consistency of the smaller cardinals. For example, if $\kappa$ is the least Mahlo cardinal, then the universe $H_\kappa$ is a model of ZFC + there is a stationary proper class of inaccessible cardinals + there are no Mahlo cardinals. If $\delta$ is the least measurable cardinal, then $H_\delta$ satisfies ZFC + there are a proper class of Ramsey cardinals, but no measurable cardinal.

Thus, the large cardinal hierarchy provides exactly the tower of theories, whose levels transcend consistency strength, that we knew should exist. And it does so in a way that is mathematically robust and interesting, with its foundations arising, not in some syntactic diagonalization, but in mathematically fulfilling and meaningful questions in infinite combinatorics.

The case of Vopenka's principle is just like this. VP is a large cardinal axiom at the higher end of the large cardinal hierarchy, implying the consistency of the existence of supercompact cardinals, say, which are far stronger than strong cardinals, which imply entire towers of measurable cardinals, which imply numerous Ramsey cardinals and so on down the line.

Illustrating the essential large cardinal nature, the VP axiom is elegantly stated: for every proper class sequence $\langle M_\alpha | \alpha\in\text{ORD}\rangle$ of first order structures, there is a pair of ordinals $\alpha\lt\beta$ for which $M_\alpha$ embeds elementarily into $M_\beta$. (It is equivalently stated in terms just of graphs, if you like.) It's simple and clear---beautiful! And the consequences are far-reaching and often profound, as you have observed in category theory, in the way that VP implies that the set-theoretic universe is regular and organized.

These are the reasons you should be attracted to Vopenka's principle. It is an elegant combinatorial principle, with far-reaching consequences that interest you, which has not yet been refuted.

In contrast, I find the philosophical heuristics that seek to justify the large cardinal axioms, on the grounds of reflection or some other means, to be so much hot air. These arguments are not mathematically sound, and cannot be made to be, by the Incompleteness Theorems. Philosophically, they seem much more like rationalizations after the fact. For example, even at the much lower (and therefore seemingly easier-to-justify) level of inaccessible cardinals, one sometimes hears an appeal to reflection type views, that since we have no definable unbounded map from a set into the ordinals, that there should be a level $V_\kappa$ of the universe also with this feature, and that such a level would be inaccessible cardinal. Of course, the conclusion outstrips the argument, with the conclusion seeming to justify at most $V_\kappa\models$ZFC, which is a weaker notion, and the meta-reflection principle appealed to amounts anyway to a large cardinal principle of its own.

Ultimately, we must recognize the uncertain nature of all our mathematical enterprise. As our hypotheses rise higher in the large cardinal hierarchy, we must become less sure of consistency---perhaps they will be shown to be inconsistent. This issue arises even at the lowest levels of our mathematical axiomatizations, for we may find at any time (as mentioned in a recent MO question) that even PA is inconsistent. As Woodin says, we all have in our minds the image of a railway line, lined by a sequence of telegraph poles, proceeding into infinity; but when the physicists tell us that the universe is finite, we realize that this picture is pure imagination. Perhaps it is simply inconsistent? So skepticism about consistency has nothing especially to do with the infinite.

Meanwhile, the large cardinal axioms are fascinating and have fascinating consequences. Let's seek out the boundary of consistency, with an attitude tempered by the realization that we may find inconsistency.

In summary, we cannot ever be sure that our axioms are consistent, and we know that above the mathematical theory about which we may be sure, there is a tall tower of theories whose levels transcend consistency. Among them are fascinating theories that are elegantly stated with far-reaching consequences, and which we have not yet refuted. So let's study them! Let's find the boundary between consistency and inconsistency!