Question

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally regular, is unprovable in ZFC. Nevertheless large cardinals were studied extensively in the last century and (apart from attempts that went too far as the Reinhardt-Cardinals) nobody ever found a contradiction to ZFC. As a consequence it seems to me that set theorists today don't consider the possibility of the non-exsistence of a large cardinal. Therefore my questions:

Why is it so unreasonable to think that the existence of large cardinals contradicts ZFC? Are there any mathematicians that do belive that large cardinals don't exist? And what are their arguments?

EDIT: I want to say thank you for all your very interesting answers and comments. It will take some time for me to fully understand them though, however I feel like I have already learned a lot due to this discussion. Thank you!

Answer 1

First of all, why is it so unreasonable to think that ZFC itself is contradictory? Because we have a good intuition about sets and we have a lot of experience with ZFC. The same basically applies to large cardinals. What I am trying to point out here is that assuming large cardinals is not much more unreasonable than passing from Peano Arithmetic to ZFC. (Edit: David Roberts points this out is his answer: If you doubt the consistency of large cardinals, why not start earlier and question the existence of the set of natural numbers?)

A lot of work has been done on the subject of large cardinals and besides Reinhardt cardinals, nothing has ever turned out to be inconsistent.
There is the so-called inner model program where one assumes the existence of a certain large cardinal and tries to build an (easily controllable) smallest model of set theory in which there is such a large cardinal and which contains all the ordinals.
The idea is that because we have a good understanding of the final inner model, we would notice during the construction of the model if there were any problems with the consistency of the large cardinal in question.

This program has worked out so far to rather high levels of the hierarchy of large cardinals.

Another important point to believe in the consistency of large cardinals is the fact that the consistency strengths of large cardinals are apparently linearly ordered.
In other words, it has not yet happened that there is a natural notion of a large cardinal which cannot be compared to the other known large cardinals in terms of consistency strength (a large cardinal is stronger that the other if the consistency of the first implies the consistency of the second). That indicates that there is a natural direction in which set theory can be strengthened, which is a remarkable (heuristic) observation.
(For certain types of large cardinal axioms it can actually be proved that they form a linearly ordered hierarchy.) I find it unlikely that we should see this linearly ordered structure above a large cardinal whose existence is actually inconsistent.

So, if there is a natural direction to strengthen our basic theory, why not go all the way and work in the strongest theory in that direction, by assuming the existence of all (consistent) large cardinals. Among other things, it has turned out that the existence of large cardinals implies a rather nice structure theory for the subsets of the real line that are definable is a certain sense (projective sets).
And as pointed out above, there are good reasons to believe in the consistency of large cardinals.

Concerning arguments against large cardinals, I would think that the main objection is that large cardinals have no effect on ordinary mathematics. But as I have pointed out in the previous paragraph, this is not entirely true. Moreover, even though this is cumbersome, most of "ordinary mathematics" can actually be carried out in weak systems of number theory. In particular, the full strength of ZFC is unnecessary most of the time.

Answer 2

I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied. [Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$." Kunen quickly afterward showed that this is inconsistent, using choice. It is not known whether choice is needed, but current research suggest that, even without choice, natural strengthenings of this axiom may be inconsistent in ZF alone.]

There are several active set theorists who do not commit themselves one way or the other to the consistency of large cardinals, but use them if necessary, and do not object on mathematical grounds to arguments that involve them. For some of them, set theory is about all possible (natural) extensions of ZFC, and there are certainly many interesting such extensions (such as V=L) that rule out large cardinals. Thus, it is not that they consider the cardinals inconsistent.

(I confess, this may be ignorance on my part.)

And I know of only two mathematicians who years ago were serious set theorists and who have expressed doubts about the consistency of (certain) large cardinals. Neither is currently active within the field, and so their position should be taken with a grain of salt, since it missed the significant results from the late 80s that could very well have forced them to reconsider.

Why do we expect ZFC to be consistent, to begin with? We expect more than mere consistency, of course, but doubting large cardinals usually means distrust in set theory as a whole. I am not a philosopher, so I will not discuss philosophical positions or justifications. A good reference for the heuristics behind the basic ZFC axioms is Penelope Maddy's wonderful paper "Believing the axioms. I", J. Symbolic Logic 53 (1988), no. 2, 481--511. Large cardinals are discussed in its follow-up, "Believing the axioms. II", J. Symbolic Logic 53 (1988), no. 3, 736--764. Maddy's two books on Platonism and Naturalism discuss extensively why the view of a set theoretic universe with large cardinals rather than not is the reasonable choice, given our current understanding. The books present several subtle technical points that can only be completely understood once one is aware of the deep connections between large cardinals and (generic) absoluteness.

How do set theorists measure the internal plausibility of large cardinal assumptions, beyond their usefulness in proving results? The point of the inner model program is to develop fine structural ("L like") models for large cardinals. These models are canonical in several precise ways, and have a rich internal structure that many set theorist take as evidence of the consistency of the large cardinals under consideration. Thanks to its advances, we have a much clearer view of the set theoretic universe nowadays (for example, we now have the different covering lemmas, and several generic invariance results) than when the program began, motivated by what we now call Gödel's program.

The program has currently reached well past Woodin cardinals, but is not yet at the level of supercompact cardinals. This can be interpreted as saying that, using the strongest tools currently at our disposal, we are fairly certain of the consistency of, say "there is a Woodin limit of Woodin cardinals". Time will tell whether the program will reach supercompactness. If it doesn't, this will provide us with strong evidence of their inconsistency, though I am not sure anybody actually expects this to be the outcome.

John Steel and Tony Martin have over the years refined something they call "the speech", were they explain their position towards large cardinals. It is well worth reading, and trying to summarize it in a few lines would be an injustice. It can be found in these two postings to the FOM list: 1, 2 (the notation here is $P_T =$ set of $\Pi^0_1$ consequences of $T$), and in the papers from the "Does mathematics need new axioms?" panel, see here. (Steel's own views are also presented in some detail in Maddy's books.)

At the risk of not being balanced, let me point out some highlights: We have a coherent picture of the universe of sets, with large cardinals. We can, within this picture, interpret theories where there are no such cardinals. However, we do not have such a coherent picture in the opposite direction. The consequences of large cardinals, at the arithmetic level (and more, as we climb up through the hierarchy) are compatible. The arithmetic consequences of any natural extension of ZFC fall somewhere within this hierarchy (as far as the theories we can currently analyze), even if the theory does not mention large cardinals. In fact, determinacy statements, incompatible with choice, also fall within this hierarchy and are bi-interpretable with large cardinals (again, as far as those theories we can currently analyze).

Large cardinals provide us with generic absoluteness, and generic absoluteness, a natural requirement if we are interested in understanding the projective theory of the reals, requires the consistency of large cardinals. See this answer for a bit more on this issue; let me emphasize that this is not some technical or artificial requirement, but rather a natural extension of basic results in classical descriptive set theory.

Large cardinals seem inherently necessary to mathematical practice, not just set theory. Harvey Friedman has written extensively on this issue.

In short: We have a very clear measure of progress understanding large cardinals and their consequences. By this measure, we can now understand many set theoretical issues that do not involve large cardinals but for which they are necessary in deeper ways (not just consistency-wise). This measure actually requires the large cardinals, we do not have anything like that without them. This measure is meaningful even in settings that are not set theoretical, and seems unavoidable even within mathematical practice (though it is perhaps too soon to tell how significant it will be at the end for "practicing mathematicians"). We do not have any serious mathematical model where large cardinals would be inconsistent, however, we have a serious program of research that would ultimately teach us that, were this the case. The program has provided us, instead, with many positive results (we have nice inner models for measurability, for strong cardinals, for Woodin cardinals, we have nice inner models of models of determinacy, that capture the large cardinals that provide us with the consistency of the determinacy statements).

To conclude, we understand (motivate/explain) large cardinals within the larger context of reflection principles, the simplest of which follow already from ZFC. (So, we have a natural generating principle for them.) On the other hand, I know of no objections to large cardinals beyond "they are too large" or "they do not feel right", neither of which seems mathematical to me. The first also seems particularly artificial.

The only 'program' towards their inconsistency that I am aware of has instead produced many interesting consequences for the partition calculus at the level of $0^\sharp$ (and is perhaps responsible for the early theory of $0^\sharp$ itself).

Answer 3

Most of the answers have addressed the "consistency" part of the original question, "Why is it so unreasonable to think that the existence of large cardinals contradicts ZFC?" There's another part, about existence, "Are there any mathematicians that do belive that large cardinals don't exist?" I'd like to point out that belief in the existence of large cardinals and belief in their consistency are not necessarily the same thing. In the first place, someone who accepts the real existence of the natural numbers but not of fancier sets might well believe that large cardinal axioms are consistent with ZFC (that's an arithmetical belief), but not believe in the existence of large cardinals or even of small cardinals like $\aleph_1$. In the second place, even a Platonist might believe the consistency of some axioms for reasons other than a belief in their truth. For example, I consider it very likely that large cardinal axioms are consistent (for reasons others have explained in their answers) up through the cardinals for which there is a clear (to the experts) inner model theory, and even that these axioms hold in certain proper-class size inner models. On the other hand, I've never been able to persuade myself of the actual existence of even subtle cardinals (which are so small that they're consistent with $V=L$). My difficulty here is that the definition of "subtle" (like that of "measurable" and many others) seems to include a combinatorial aspect quite different from mere largeness (though of course the definitions imply a certain largeness). I'm generally OK with mere largeness, but I don't see why there should really exist cardinals with these additional combinatorial properties. (A plausible-looking picture of the real world might be that there really aren't any measurable cardinals, but that all singular cardinals are measurable (and more) in some inner models.)

Answer 4

You asked:

Why is it so unreasonable to think that the existence of large cardinals contradicts ZFC?

It's not "unreasonable," any more than it's "unreasonable" to disbelieve the Riemann Hypothesis. If you systematically poll people and ask them at what point in the hierarchy do they start developing acrophobia (i.e., doubting that infinities of that "size" exist), you'll get answers all along the spectrum. The doubts, however, are usually of the form, "I don't feel that there's enough evidence to make me believe in them," rather than any concrete arguments against their existence. Once in a while you may run into someone who believes $V=L$ strongly enough to use that as a reason to disbelieve in measurable cardinals, but that's pretty rare.

What you're observing in the literature is that large cardinals lead to a very fruitful and insightful theory. Assuming that they don't exist, on the other hand, doesn't seem to lead to very interesting results. So that's why the literature is biased in this direction. Similarly, you'll find lots of papers that assume the Riemann Hypothesis or that assume $P \ne NP$, and build up a complicated conjectural picture of the mathematical universe on that basis, even though it's theoretically possible that the whole picture could collapse like a house of cards if the underlying assumption were shown to be false. The reason is that the picture is a compelling one that "feels true" and that generates many ideas and corollaries that can sometimes be verified unconditionally. Assuming the opposite hypotheses, however, rarely leads anywhere—though in the cases where it does lead somewhere, people don't hesitate to mention it.

Answer 5

Nelson (and at the linked question, we see also Doyle and Conway) is famous for not believing in the existence of $\aleph_0$, which is the cardinal of a limit ordinal (i.e. $\omega$), so can be considered large, but not large from a usual set theory point of view. He would be called a finitist in this respect. More precisely, his axioms of arithmetic do not presuppose the existence of a natural numbers object, and do not show it either.

Edit: In fact Nelson is an ultrafinitist, in that he doubts the existence of large natural numbers, and gives a combinatorial example (see my comment below) of a number the finiteness of which he questions (this corrects a mis-statement on my behalf on the original version of the question, where I called Nelson an ultrafinitist for not 'believing in' $\aleph_0$).

Answer 6

According to this comment by François Dorais, Petr Vopěnka doubted the largest of large cardinal axioms, and gave an example of a statement he thought was false, yet was implied by some large cardinal axioms. No one has ever found a proof that his statement -- now known as Vopěnka's principle -- is false, and now it is regarded as just another large cardinal axiom.

There's more detail in the thread, but I'll give a quick summary: A set or class of graphs is "rigid" if there exists no nontrivial graph homomorphisms between graphs in the set. Vopěnka proved that there exist sets of rigid graphs of arbitrarily large cardinality, and conjectured that there must be a proper class of graphs with the same property. If anyone constructs such a class, then that proves that the largest of the large cardinal axioms are inconsistent with ZFC.

Answer 7

Erinna, in your question you use the word 'exist'. In the philosophy of mathematics, that is a word that is part of discussion. If you follow a Plato philosophy, then there is a perfect mathematical universe, were all the objects 'exists'. Then you can talk about the existence of large cardinals. If you remove those objects from that universe, then that is probably considered a loss by many.

However, such philosophy has problems. The main problem is how this mathematical universe interacts with our daily world. Without such interaction, we can not access such universe. Of course, I have some personal opinions here, and one can have a long discussion.

In some philosophies one tries to have a more limited mathematics, with only mathematical objects with a clear meaning. In such philosophy, (large) cardinals are not part of that limited mathematics. However, that does not mean that other constructs are entirely rejected. They can still be mental constructs, or meta-mathematics. Constructs that can be used to do mathematics that has more clear meaning.

As said, with philosophy you can have lot of discussion and there are many views.

I think a better question is whether '(large) cardinals must be part of the fundamentals of mathematics'. I strongly say no to that question, although I do not object against (large) cardinals as mental construct.

Lucas