Say we have DC-λ where λ is some inaccessible cardinal. Is that enough to develop all of ordinary mathematics? If not, is there a strengthening that is but that nevertheless does not assume full choice high up in the universe of sets?



How much of the axiom of choice do you need in mathematics?
Your hypothesis is in a sense stronger than just assuming ZFC outright.


Namely, if we have $\lambda$-DC for some inaccessible cardinal $\lambda$, and ZF in the background, then in particular, we will have the full axiom of choice inside the universe $V\_\lambda$, consisting of all sets of rank less than $\lambda$, since $\lambda$-DC implies choice for all families of size less than $\lambda$. So $V\_\lambda$ will be a model of full ZFC.


Therefore, by simply jumping inside this universe $V\_\lambda$, we recover full ZFC for all the purposes of "ordinary mathematics." If all ordinary mathematics would take place inside such a universe, then the answer would be yes.


But let me note that it is somewhat subtle to define what one means by inaccessible cardinal in the absence of the axiom of choice, since the usual definition would be that $\lambda$ is an uncountable regular strong limit, but being a strong limit should mean that if $\kappa<\lambda$ then $P(\kappa)<\lambda$ as well, and so in particular, $P(\kappa)$ is well-orderable. But in this case it follows by definition that if $\lambda$ is inaccessible, then the axiom of choice holds in $V\_\lambda$ just as a consequence of inaccessibility.


In other words, we get ZFC in $V\_\lambda$ even without your $\lambda$-DC assumption. In this sense, the power of your hypothesis for ordinary mathematics consists of your having an inaccessible cardinal in the first place, rather than in your having $\lambda$-DC.


**Controversial counterpoint.** Meanwhile, let me also say that my view also is that we would make a fundamental disservice to mathematics and to ourselves by attempting to limit our mathematical conceptions to those ideas that have proved productive in the past, limiting ourselves to the ideas used in "ordinary" realms of mathematics. Set theory has discovered a vast new tranfinite realm of mathematical reality and fundamental principles that govern it, such as the axiom of choice but also large cardinals and many new strong principles with transformative global effects. In my view, the fact that those principles do not apply as much to the older "ordinary" questions do not show the impotence of the new ideas, as much as they show the impotence of the old ideas in capturing the vast new lands before us.


