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The other day, I was reading the preface of Kanamori's The Higher Infinite and noticed that he says large cardinals provide a useful 'measuring stick' for consistency. That raised the question of whether large cardinals are motivated simply by searching for their consistency strengths or if there are other motivations. If large cardinals have appeal outside of consistency strength, like mathematical beauty or interesting properties, are there any examples that someone with a moderate knowledge of set theory could understand?

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    $\begingroup$ Measurable cardinals, as their name suggests, were originally motivated by questions arising from measure theory. It is not even obvious from their definition that they have to be large; that was established "after the fact" by Ulam. See A brief introduction to measurable cardinals by Edgar Bering IV for an elementary account. $\endgroup$
    – Timothy Chow
    Commented Jul 14, 2022 at 12:55
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    $\begingroup$ It seems somewhat disingenious to have asked math.stackexchange.com/questions/4486274/…, not interacted with the answer, and not to mention it here. $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 14, 2022 at 13:27
  • $\begingroup$ I personally use large cardinals to make music. $\endgroup$
    – Joseph Van Name
    Commented Jul 14, 2022 at 15:53
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    $\begingroup$ @AsafKaragila Is it common practice to interact with answers on MSE and MO? I thought your answer was really good, but also fairly comprehensive, and therefore didn't feel the need to comment asking for more information. I sincerely apologize if my actions seem to be in bad faith, I was just hoping for other perspectives by posting here. $\endgroup$
    – littleman
    Commented Jul 14, 2022 at 17:34
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    $\begingroup$ @littleman Although there can be exceptions, cross-posting is generally discouraged. You should not post the same question to two SE sites "just hoping for other perspectives." If you post on math.SE and the answer is not satisfactory, then posting here on MO may be okay, but in that case, you should say what you found to be unsatisfactory about the first answer. And when you post on MO, you should link to the math.SE question so that people know you are cross-posting, and won't duplicate what has already been said. $\endgroup$
    – Timothy Chow
    Commented Jul 15, 2022 at 1:36

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Asaf Karagila already wrote an excellent answer at math.SE, but here is a simple point that may be helpful. In any area of math, the natural course of research leads one to ask questions, pose conjectures (and prove them if possible), and impose extra structure on the objects of study if that seems to be necessary for them to have desirable properties. Set theory is not any different in this regard, except that in set theory, we can often detect that some of our "conjectures" are hopeless, in the sense that Gödel's incompleteness theorems tell us that we cannot possibly prove them in the conventional mathematical sense. So you can think of large cardinal axioms as "natural conjectures," except that we don't call them conjectures because we know we can't prove them. Alternatively, you can think of increasingly strong large cardinal axioms as imposing increasing amounts of structure on $V$, giving it extra desirable properties.

One can still ask, if we're just looking for "natural conjectures," why do large cardinal axioms stand out? Why aren't there lots of other, equally prominent axioms? Well, as a matter of fact, there are other axioms that set theorists study, but large cardinals stand out because they somewhat miraculously seem to line up nicely in a single hierarchy. This fact is not at all obvious a priori (for example, it is not at all obvious from the definition of a measurable cardinal that it must be inaccessible). To some people, it suggests that the large cardinal axioms are "true," but even if you don't believe in truth in this sense, the large cardinal axioms bring a lot of order to what might otherwise be a chaotic universe. Instead of a huge mess of "natural conjectures," we find that many other questions can be more-or-less answered by showing that they lie sandwiched in between two well-known large cardinal axioms.

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