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Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below:

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So if we start with a concept of number and play the game of naming the largest one, does Kunen inconsistency draw the boundary -so to speak- of what we can conceive of as a number? In the blurb of Upper Attic it is said:

Welcome to the upper attic, the transfinite realm of large cardinals, the higher infinite, carrying us upward from the merely inaccessible and indescribable to the subtle and endlessly extendible concepts beyond, towards the calamity of inconsistency.

My question is how can we understand this notion of KI and more importantly why is it at the top or what is the limit of conception of number? Does tradinotional notions of common sense logic break down at certain point? What is the rationale behind the choice of words "calamity of inconsistency"?

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Mathematics in general, and set theory and large cardinals in particular have little to do with common sense. –  Asaf Karagila Feb 13 '12 at 23:37
The calamity of inconsistency refers to defining a cardinal so large that its existence can be shown to be inconsistent with the usual axioms of set theory. –  Brian M. Scott Feb 13 '12 at 23:53
Thanks for referring to Cantors attic! cantorsattic.info –  JDH Feb 14 '12 at 1:03
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3 Answers

up vote 7 down vote accepted

I wrote the blurb that you quoted from Cantor's attic about the calamity of inconsistency, and what I had in mind was the following (although I may have overdone the purple prose there). One of the principal features of the large cardinal hierarchy is the fact that it is strictly increasing in consistency strength as one moves higher in the hiearchy. Although a naive critic may find fault with this situation---after all, we are not able to prove even the consistency of the large cardinals we study, let alone their actual existence---nevertheless a more informed perspective leads to the view that this is precisely the situation we seek. Gödel's incompleteness theorem shows us that any theory that we can describe is superceded in consistency strength by a strictly stronger theory, and how fortunate that we find such a hierarchy of consistency strength, not merely in the meta-logic assertions of consistency, but in the highly natural statements of infinite combinatorics expressed by the large cardinal concepts of infinity. Thus, the large cardinal hierarchy is exactly what we had known must exist on the basis of the incompletness theorem---an extremely tall hierarchy of consistency strength---but which finds its substantial essence, not in twisted self-referential mathematical knots, but in genuine, robust mathematical concepts of infinity.

So, we have the smallish concepts of infinity at the level of inaccessible cardinals and the like, and then moving up through the Mahlo cardinals, the weakly compact cardinals, the Ramsey cardinals, the measurable cardinals, the strong cardinals, the strongly compact and supercompact cardinals, the almost huge, the huge and the super-huge cardinals, and so on forming the higher portion of the large cardinal hierarchy. As the large cardinals become stronger and stronger, the assertions that those concepts are even consistent with the axioms of set theory become increasingly strong statements. The strongest possible statement, from which any other statement follows as a matter of logic, is an inconsistent statement. Thus, the large cardinals reach toward what I called the calamity of inconsistency, the hypothesis from which all statements become trivial. The Kunen inconsistency is the first and most famous refutation of any large cardinal axiom, and so it sits atop the large cardinal hierarchy.

It is conceivable, and consistent with everything that we know so far, that one might find inconsistency lower in the large cardinal hierarchy. Indeed, we already know that it is relatively consistent with the axioms of ZFC to hold that one might find inconsistency at any given desired level of the large cardinal hierarchy. This is an immediate consequence of the incompletness theorem, since if a given large cardinal assertion is consistent with ZFC, then the assertion that it is inconsistent over ZFC is consistent with the assertion that it is true.

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Dear JDH, I don't think the very last clause of your answer parses. Best wishes, –  Matt E Feb 14 '12 at 4:47
Thanks! I've fixed it. –  JDH Feb 14 '12 at 4:58
Thanks! Interested reader can further do research here a paper I found after reading this answer. –  Sniper Clown Feb 15 '12 at 3:22
You can also see the slides for a talk I gave on the subject at the Kurt Goedel Research Center in Vienna at jdh.hamkins.org/kunen-inconsistency-vienna-2011. –  JDH Feb 15 '12 at 3:45
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Set theory, and in general logic, has a big portion devoted to "how far can we stretch this theory?" and since inconsistencies in FOL appear instantly, there is no "a little bit of contradiction".

What do I mean? Well, we work in a certain framework, this framework has rules of inferences which do not allow "a little bit" of inconsistency, but rather explode once inconsistencies are introduced.

Godel's theorem shows that if our theory is strong enough then if we can effectively decide which axiom is in the theory, the theory will not be complete. In particular it will not be able to prove its own consistency.

ZFC is such theory, it has the properties needed for Godel's incompleteness theorem, and in particular has the capabilities of expressing its own consistency. Therefore it cannot prove it. Indeed we always have to assume that ZFC is consistent.

Large cardinals are additional axioms, those are strong axioms which often prove the consistency of ZFC. This is quite a leap of faith, so to speak, and we always have to be wary that we do not introduce contradictions. Contradictions are of course assumptions from which we can prove everything, in particular the consistency of ZFC.

When we add the axiom "There exists an inaccessible cardinal" we may have added an inconsistency, but we did not find such yet (modulo some claims which are unverified). This is a very mild form of large cardinal, but it is an additional axiom nonetheless.

We can add more, two inaccessible. Infinitely many inaccessible cardinals, or even a proper class thereof. Those still do not yield a clear contradiction and thus we are still okay with believing those axioms are consistent.

Going even further we reach the point of measurable cardinals. There we find a subtle point which is expressed in Kunen's inconsistency's theorem. Namely, measurable cardinals are critical points of elementary embeddings of the universe into a subclass.

If we assume this embedding to be onto the universe then we have derived a contradiction. This is something which we do not want, so while we cannot approximate it by "little inconsistency" or "a bit of contradiction" we have to approach from a different angle. By studying the exact reason from which we derive the contradiction we are able to write axioms which are closer to it but not strong enough to prove it.

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Classical logic has a law called the principle of explosion, also known by its latin name, ex falso quodlibet: from contradiction anything follows. So if one has an inconsistent theory then one can derive any statement, including, of course, any statement $\psi$ in the language of set theory, such as any large cardinal statement you like (and its negation).

Inconsistency is calamitous precisely because of this property: if a statement and its negation are both theorems, then the false is on a par with the true, and mathematics fails to describe anything (since inconsistent theories have no models).

Of course, this general property of classical logic doesn't address the context of set theory and large cardinals, but the situation here is merely a special case. Supposing for the sake of argument that the ZFC axioms are consistent, we can extend them by adding new axioms, including those that assert the existence of this or that large cardinal.

Kunen showed that assuming both ZFC and the existence of a Reinhardt cardinal (the critical point $\kappa$ of a non-trivial elementary embedding $j : V \rightarrow V$) one could derive an inconsistency, and so in this sense it forms an upper bound for the programme of adding large cardinal axioms to ZFC: at this point, an inconsistency with ZFC is reached.

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I agree with most of your answer. But one minor point is that Kunen actually proved his theorem in Kelly-Morse set theory, rather than ZFC, because the theorem involves a class quantifier. It is also possible to prove it in Goedel-Bernays GBC set theory. In ZFC, one can seem only to rule out definable classes $j$, and this is a considerably easier result, not requiring the axiom of choice. These issues are explained in my recent article with Norman Perlmutter and Greg Kirmayer at jdh.hamkins.org/generalizationsofkuneninconsistency. –  JDH Feb 14 '12 at 1:01
@JDH: I'm not particularly familiar with this, so this may be a silly question: what goes wrong if we add a free class variable $J$ to the signature ZFC along with a scheme that $J$ is an elementary embedding - presumably that extended theory is inconsistent? –  Carl Mummert Feb 14 '12 at 4:51
You can express elementarity as a scheme that way, but you don't get a critical point for the embedding, unless you have ZFC in the language with $j$, that is, unless $j$ is a class. So having such a $j$ carries no large cardinal strength without extra hypotheses. Nevertheless, your suggestion is how one formalizes the Wholeness Axiom, with those extra hypotheses. Meanwhile, Reinhardt cardinals can be expressed without that formalism in GBC and KM, as we discuss in my paper. –  JDH Feb 14 '12 at 4:57
@JDH thanks for the clarification! I shall definitely check out your article. –  Benedict Eastaugh Feb 14 '12 at 12:10
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