There are (possibly) two questions here:

Why should we believe that large cardinal axioms are consistent?

Why, *if* we believe that large cardinal axioms are consistent, should we believe that they are true?

Here are some reasons for believing that large cardinal axioms are consistent (or at least, that small large cardinal axioms that have been studied for a long time are consistent.)

First, there is the empirical fact no one has published a proof of a contradiction from the assumption $\mathsf{ZFC} + {}$"there is an inaccessible cardinal" (for example) despite a long period of study in which many theorems have been proved from this assumption. Although some large cardinal hypotheses (such as Reinhardt cardinals) have turned out to be inconsistent, this was discovered relatively quickly, in the period during which most people were still skeptical of them.

Second, there is "fine structure" which gives canonical models for the smaller large cardinal axioms (so far, up to Woodin cardinals and a bit further.) It seems reasonable to expect that a systematic study of the structure of the models of a theory would eventually reveal the inconsistency of the theory if it were inconsistent, and this has not happened yet.

For question 2, let us now assume (informally, for the sake of non-mathematical argument) that large cardinal axioms are consistent. Why should we then believe that they are true?
Most people find it natural to believe the assumptions that they use in their day-to-day work, so this question is closely related to the question of which axioms people should *use*. Of course, the answer will depend on what types of theorems they want to prove. In most areas of mathematical research, $\mathsf{ZFC}$ seems to be sufficient in a practical sense and there does not seem (to me) to be a compelling argument that people working in these areas should use, or believe, any kind of axiom beyond $\mathsf{ZFC}$.

So perhaps the question should be revised to "why should mathematicians who want to prove theorems beyond $\mathsf{ZFC}$ use large cardinal axioms, instead of alternatives such as $V=L$?" A practical answer is that doing so allows us to prove lots of interesting theorems. Suppose that I assume $\mathsf{ZFC} + {}$"there is a measurable cardinal" and you assume "$\mathsf{ZFC} + V=L$." Then for every theorem that you prove, I could have proved (if I were clever enough) a corresponding theorem of the form $L \models \ldots.$ On the other hand, I may have the opportunity to discover an interesting theorem about measurable cardinals that you do not have the opportunity to discover (unless you investigate countable transitive models with measurable cardinals, which seems like an unnatural thing to do if you believe that $V=L$, even though it is presumably formally consistent for you to assume the existence of such models.)

This last point is summarized by the slogan "maximize interpretive power." Many of the points I made above are better made in the following paper. I think that what I wrote here leans toward Steel's viewpoint, but I do not claim to have rendered it faithfully.

Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope & Steel, John R. (2000). Does mathematics need new axioms? *Bulletin of Symbolic Logic* 6 (4):401-446.

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