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Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).

Their non-existence is consistent with axioms of usual mathematics.

It is provable that some of them don't exist at all.

They show many unusual strange properties.

$\vdots$

These are a part of arguments which could be used against large cardinal axioms, but many set theorists not only believe in the existence of large cardinals, but also refute every statement like $V=L$ which is contradictory to their existence.

What makes large cardinal axioms reasonable enough to add them to set of axioms of usual mathematics? Is there any particular mathematical or philosophical reason which forces/convinces us to accept large cardinal axioms? Is there any fundamental axiom which is philosophically reasonable and implies the necessity of adding large cardinals to mathematics? Is it Reflection Principle which informally says "all properties of the universe $V$ should be reflected in a level of von Neumann's hierarchy" and so because within $\text{ZF-Inf}$ the universe $V$ is infinite we should add large cardinal $\omega$ (which is inaccessible from finite numbers) by accepting large cardinal axiom $\text{Inf}$ and because $V$ is a model of (second order) $\text{ZFC}$ we should accept existence of inaccessible cardinals to reflect this property to $V_{\kappa}$ for $\kappa$ inaccessible and so on?

Question. I am searching for useful mathematical, philosophical,... references which investigate around possible answers of above questions.

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Isn't it a little arrogant to propose some anti-large cardinal axiom like V=L? It's as if you know all there is to know about the structure of the world. Large cardinals are a way of acknowledging our limitations in comparison to the huge complicated world out there. –  Monroe Eskew Apr 25 at 3:43
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@Monroe Some have argued that point on the other side. For example, Stephen Simpson compares large cardinal skepticism with religious skepticism. –  Joel David Hamkins Apr 25 at 4:05
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Nobody forces you to accept anything. If you don't want to accept large cardinal axioms, you don't have to. –  Asaf Karagila Apr 25 at 4:15
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While I think large cardinals are fun, I don't feel particularly forced to have an opinion about their existence. It would be way cool if they could be explained in terms of computation. For instance, inaccessible cardinals correspond to type-theoretic universes, and Mahlo cardinals can be seen as a very strong induction principle in type theory. But what about even larger cardinals? Do they have a computational meaning? That would "force" me personally to regard them seriously. –  Andrej Bauer Apr 25 at 7:39
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He has said it during some talks I've seen, and so you might find it in his slides. For example, see pages 10-11 of the slides from his talk at the 2009 NYU conference: personal.psu.edu/t20/talks/nyu0904/nyu.pdf, also personal.psu.edu/t20/talks/nyu0904/nyu-slides.pdf. –  Joel David Hamkins Apr 25 at 11:03

6 Answers 6

The line of reasoning you mention at the end of your post, firmly in support of large cardinals, was first argued forcefully in

  • W. N. Reinhardt, “Remarks on reflection principles, large cardinals, and elementary embeddings,” Proceedings of Symposia in Pure Mathematics, Vol 13, Part II, 1974, pp. 189-205

and the ideas are further discussed, explained and basically supported in

These articles have now a rather large literature of discussion and criticism in the philosophy of set theory. To get started, you might find further resources on the reading list of my recent course NYU Philosophy of Set Theory. One can now find numerous articles arguing on any given side of each issue.

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Your answer is quite helpful. Thank you very much. –  user47697 Apr 25 at 11:19

This is only a partial answer, but Harvey Friedman has a research program to find concrete $\Pi^0_1$ sentences that are purely combinatorial (i.e., make no reference to concepts from logic such as axioms or formal systems) that can be deduced from a large cardinal axiom and that imply the consistency of a (slightly weaker) large cardinal axiom. The $\Pi^0_1$ statement can of course be partially verified by direct computation, so if you convince yourself that it is true, then the large cardinal axiom helps "explain" why it is true. I believe that Friedman has carried out his program up to and including subtle cardinals; see this post on the Foundations of Mathematics mailing list, for example.

I believe Friedman is optimistic that his program can in principle be carried out for any large cardinal axiom, but at present I believe he has no natural, explicit $\Pi^0_1$ statements that require (say) measurable cardinals to prove.

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In Friedman's examples, do you know if the consistency of a given large cardinal axiom enough to prove the given $\Pi^0_1$ statements, or does it actually require the existence of a given large cardinal? –  Jesse Elliott Dec 11 at 12:28
    
@JesseElliott : Usually what is needed is 1-consistency. If you think about it, there's no way that a large cardinal could be required by an arithmetical statement S in the strongest possible sense that its existence is actually implied by S, because there are models of true arithmetic in which there is no inaccessible cardinal. –  Timothy Chow Dec 11 at 18:37

There are (possibly) two questions here:

  1. Why should we believe that large cardinal axioms are consistent?

  2. 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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Regarding your comment: "Most people find it natural to believe the assumptions that they use in their day-to-day work…" I would venture to guess, from the point of view of human psychology, that one common trait of successful mathematicians is the ability to manage their level of belief in various unproved statements---strengthening it as they attempt to achieve proof; gutting it as they attempt to achieve disproof. –  Lee Mosher Apr 26 at 14:21

It is useful in the category theory, in particular, in its applications to algebraic geometry. "Small" categories (whose objects form a set) are much nicer than "large categories" (whose objects mere form a "class", whatever it means). In algebraic geometry one wants to consider categories like Sets, Schemes and so on as small --- and technically it is done using Grothendieck's "Axiom of universe", which is equivalent to existence of strongly inacessible cardinals large than a given cardinal, see http://en.wikipedia.org/wiki/Grothendieck_universe

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I think this is a good reason. Non set-theoretic reasons for believing in set-theoretic concepts are the most compelling, by far. –  goblin May 23 at 13:39

Let me give an argument, not that we should believe large cardinal axioms or their consistency, but rather that regardless of our belief in consistency we should still be interested in results around them.

First, large cardinals are uniquely useful in analyzing principles of strong consistency strength. That is, we know from experience that there are many principles, with consistency strength greater than $ZFC$, for which large cardinals function as a useful organizational principle. This is especially important if I'm agnostic about the consistency of large cardinals (which I am), since then I'm also agnostic about a bunch of other fairly natural principles and want a nice "yardstick" to organize my knowledge of them.

Even if I actively believe, say, that "there is an inaccessible" is inconsistent with $ZFC$, playing with large cardinals is still useful to me: if I believe inaccessibles are inconsistent with $ZFC$, then I also must believe that "DC + every set of reals is Lebesgue measurable" is inconsistent with $ZF$; the point is, there are reasonably natural philosophical viewpoints which reject inaccessibles - say, believing that the Inner Model Hypothesis is true - which yield, via arguments around large cardinals, philosophical positions against other principles for which no such natural viewpoint is known to exist.

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Do we have examples of axioms A and B, not obviously about large cardinals, for which Con(A) $\rightarrow$ Con(B) can only / best / most easily be proved via theorems about large cardinals? Such an example would strengthen this argument substantially. –  Matt F. Apr 26 at 21:08
    
@MattF. An example is that $\mathbf{\Pi}^1_1$-determinacy implies $<\omega^2$-$\mathbf{\Pi}^1_1$-determinacy. The only known proof goes by showing that the first assumption implies the existence of sharps for reals, and that the sharps imply the stronger determinacy assumption. Other examples of such transfer theorems in descriptive set theory are also known at higher consistency strength. See here for more on this. –  Andres Caicedo Apr 26 at 21:19
    
@AndresCaicedo, that is an interesting result, which was new to me. But Ralf Schindler in that very reference says: "Point is: We don’t know another, 'direct', proof. The only proof known goes through the study of L." He argues there that anyone who cares about determinacy should care about inner models, and does not argue there that they should care about large cardinals specifically. For me "$0^\sharp$ exists" is easier to understand and think about without large cardinals. –  Matt F. Apr 26 at 21:42
    
@MattF. Sure. The point is that unless you are in a situation of very limited interest or too specialized to be a general phenomenon, you are not going to interpolate from $A$ to $B$ using large cardinals in $V$ but rather in inner models. –  Andres Caicedo Apr 26 at 21:51
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@MattF., this is the result mentioned in the final paragraph of my answer - the relevant source is "Can you take Solovay's inaccessible away?" (link.springer.com/article/10.1007%2FBF02760522) by Saharon Shelah (specifically, this is the paper that showed that Con(ZF+DC+everything measurable)$\implies$Con(ZFC+inaccessible); the converse direction had been proved by Solovay math.wisc.edu/~miller/old/m873-03/solovay.pdf). Zero sharp in my comment is kind of a red herring; the point is that inaccessibles exactly capture "everything's measurable," and zero sharp is stronger. –  Noah S Apr 26 at 23:48

This is a personal opinion rather than an answer (in fact another personal opinion of mine is that this kind of question cannot have a meaningful objective answer).

Compare this situation with Euclidean geometry. It is not quite correct to ask whether one should believe in the fifth postulate or not. This is because with the current state of knowledge there is no problem at all to deal with all possible versions of it. And in fact, already situations when the status of the fifth postulate varies from point to point are very well understood.

In set theory also, already state of knowledge is ripe to study if not all, then at least significant amount of possibilities which can arise from various combinations of large cardinal (and several other important) axioms. And in fact it is perfectly meaningful to consider and study mathematical structures which allow for variability of the status of these axioms similarly to the variation of curvature on a geometric surface.

I believe that in such circumstances the question of belief becomes obsolete. It is true that in physics one may believe that the universe is positively or negatively curved, or flat. But this is because we are placed inside this universe. In case of mathematics, we are not placed inside any particular model of set theory, hence we are not forced to choose. Certainly some models are distinguished among the rest by some special properties, like flat geometry is distinguished among the rest of geometries, but that's all one can say I think.

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