Return to Question

For the existence of LCAs in general, "There is an explanation along the lines of, we developed a theoretical framework that can be modified and strengthened in somewhat obvious ways, and this results in a coherent system of hypotheses. We looked at the limits of this framework, and tried to find the exact breaking point" (Eskew). One might ask: Are there any other theoretical frameworks extending ZFC that preclude the existence of nearly all large cardinals? One might be $V = L$, or even the axiom of restriction, but most set theorists dismiss them and instead seek core models that accommodate nearly all large cardinals. (In a vein related to LCAs, there are also various forcing axioms and axioms of determinacy.) One is hard-pressed to find an alternative coherent system of hypotheses that "complete" the axioms of ZFC that preclude the large cardinals besides those, such as the axiom of restriction, that preclude them by fiat.

For the existence of LCAs in general, "There is an explanation along the lines of, we developed a theoretical framework that can be modified and strengthened in somewhat obvious ways, and this results in a coherent system of hypotheses. We looked at the limits of this framework, and tried to find the exact breaking point" (Eskew). One might ask: Are there any other theoretical frameworks extending ZFC that preclude the existence of nearly all large cardinals? One might be $V = L$, or even the axiom of restriction, but most set theorists dismiss them, as they rule out large large cardinals and inaccessible cardinals, respectively, and instead seek core models that accommodate nearly all large cardinals. (In a vein related to LCAs, there are also various forcing axioms and axioms of determinacy.) One is hard-pressed to find an alternative coherent system of hypotheses that "complete" the axioms of ZFC that preclude the large cardinals besides those, such as the axiom of restriction, that preclude them by fiat.

FINAL EDIT: Sorry to answer my own question, but I wanted to summarizes the answers of Monroe Eskew and Joel David Hamkins and some of the comments that were provided.

The main argument by those who insist on the truth of LCAs is that their truth is the only explanation of their many consequences, including their consistency (Eskew). In other words, "it is the existence [of] the large cardinals and not merely their consistency [that] presents a coherent picture of mathematical reality, which gives rise to all the other consequences of large cardinals of which we are familiar. The view is that we believe in the consistency assertions because we think that the stronger large cardinals exist. Without that existence, we wouldn't have any reason to believe in even the much weaker consistency assertions" (Hamkins). Steel, in particular, objects to theories merely asserting consistency results as "the intrumentalist dodge." However, we often want to compare consistencies of theories without asserting the theories themselves, so it is not clear that the instrumentalist dodge should be dismissed as "unnatural" as Steel claims it to be. See further objections to Steel's claim in Hamkins' papers, "The set-theoretic multiverse" and "Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength." For example, in the former paper, Hamkins notes that "The believer in large cardinals is usually happy to consider those cardinals and other set-theoretic properties inside a transitive model, and surely understanding how a particular set-theoretic concept behaves inside a transitive model of set theory is nearly the same as understanding how it behaves in $V$."

In favor of the existence of large cardinals, note that LCAs that are not too far from 𝖹𝖥𝖢 (like inaccessibles) are accepted by many category theorists in a form of Grothendieck universes; however stronger LCAs require better justification (Eskew). A problem, however, is where, if anywhere, to draw the line: the stronger the LCA, the more likely it is to be inconsistent, and therefore the stronger LCAs are less likely to be accepted by the general mathematical community. However, evidence for "the mutual consistency of the large cardinals" is "their linear ordering in the hierarchy" (Eskew).

For the existence of LCAs in general, "There is an explanation along the lines of, we developed a theoretical framework that can be modified and strengthened in somewhat obvious ways, and this results in a coherent system of hypotheses. We looked at the limits of this framework, and tried to find the exact breaking point" (Eskew). One might ask: Are there any other theoretical frameworks extending ZFC that preclude the existence of nearly all large cardinals? One might be $V = L$, or even the axiom of restriction, but most set theorists dismiss them and instead seek core models that accommodate nearly all large cardinals. (In a vein related to LCAs, there are also various forcing axioms and axioms of determinacy.) One is hard-pressed to find an alternative coherent system of hypotheses that "complete" the axioms of ZFC that preclude the large cardinals besides those, such as the axiom of restriction, that preclude them by fiat.

Large cardinal hypotheses and related hypotheses like projective determinacy are well-known to be gauges of the consistency strength of various theories. What reasons are there to believe in their truth, rather than merely in their consistency, or at most their truth in some transitive model?

Various authors have presented and argued for the more conservative position in the literature. On the other hand, one argument presented for the truth of projective determinacy is the following theorem of Woodin.

Theorem (Woodin). The following are equivalent:
(1) Projective Determinacy (schematically rendered).
(2) For every $n < \omega$, there is a fine-structural, countably iterable inner model $M$ such that $M \models$ “There are $n$ Woodin cardinals”.

However, one can still quarantine projective determinacy and its equivalent above to be true in some model (or transitive model) of ZFC, since models can be talked about within models.

If one quarantines large cardinal axioms to models, then their existence is just a matter of what your background set theory is (as the existence of models and the existence of transitive models depends only on that), rather than a matter of what large cardinals one decides "exist" in some absolute sense. Moreover, this still allows for the use of large cardinals axioms as a gauge for the consistency strengths of other theories and for each other. Thus, my question more precisely is, what reasons are there to believe in the the "truth" of large cardinal axioms, rather than its being a matter of whether or not they are true in some model, or in some transitive model, relative to the background theory one is using?

Large cardinal hypotheses and related hypotheses like projective determinacy are well-known to be gauges of the consistency strength of various theories. What reasons are there to believe in their truth, rather than merely in their consistency, or at most their truth in some transitive model?

Various authors have presented and argued for the more conservative position in the literature. On the other hand, one argument presented for the truth of projective determinacy is the following theorem of Woodin.

Theorem (Woodin). The following are equivalent:
(1) Projective Determinacy (schematically rendered).
(2) For every $n < \omega$, there is a fine-structural, countably iterable inner model $M$ such that $M \models$ “There are $n$ Woodin cardinals”.

However, one can still quarantine projective determinacy and its equivalent above to be true in some model (or transitive model) of ZFC, since models can be talked about within models.

If one quarantines large cardinal axioms to models, then their existence is just a matter of what your background set theory is (as the existence of models and the existence of transitive models depends only on that), rather than a matter of what large cardinals one decides "exist" in some absolute sense. Moreover, this still allows for the use of large cardinals axioms as a gauge for the consistency strengths of other theories and for each other. Thus, my question more precisely is, what reasons are there to believe in the the "truth" of large cardinal axioms, rather than its being a matter of whether or not they are true in some model, or in some transitive model, relative to the background theory one is using?

Edits (additions given some comments):

From Chow--Related posts: Philosophical arguments in defense (or against) large cardinals What "forces" us to accept large cardinal axioms? Philosophical arguments in defense (or against) large cardinals

From Hamkins: As Hamkins points out in his answer, my question is related to the question of whether or not the "instrumentalist dodge" or variants thereof are truly enough to "dodge" large cardinals or not.