Joel Hamkins recently wrote an excellent post on Kelley-Morse set theory (${\rm KM}$) right here on Boolesrings. I commented on the post about the variations one finds of what precisely is included in the ${\rm KM}$ axioms. I claimed that the two most commonly encountered versions are easily seen to be equivalent and I am going to make that quick argument below.

$

\newcommand{\GBC}{\rm{GBC}}

\newcommand{\ORD}{\rm{ORD}}

\newcommand{\KM}{\rm{KM}}

\newcommand{\ZFC}{\rm{ZFC}}

$

It is actually quiet surprising how little there is to be found about $\KM$ on the web. I had to dig through pages of Google search on $\KM$ (tediousness pays off) before I stumbled on a very interesting paper about forcing over models of $\KM$ by Ronaldo Chuaqui [1]. In a recent post, I discussed some folklore results about class forcing over models of $\GBC$ that adds classes but not sets. Apparently, many of the standard forcing arguments can be done over models of $\KM$ where new sets and classes are added simultaneously while preserving $\KM$ to the forcing extension. This will probably make a future post, once I have figured out the details. But for now back to what precisely is $\KM$.

There are two types of semantics for set theories with classes. A set theory with classes can be axiomatized in first-order logic by viewing classes as elements of the model and sets as those particular classes that happen to be $\in$-members of some other classes. Alternatively we can consider models in a two-sorted language with separate variable for sets and classes. Even though the particular approach one takes does not make much difference, let’s fix here on the two-sorted language approach, so that a model of set theory with classes has the form $\langle M,\in,S\rangle$ where $M$ is the sets and $S$, some collection of subsets of $M$, is the classes.

$\KM$’s more widely studied cousin the Gödel-Bernays axioms ${\rm GBC}$ use a class existence principle that limits comprehension to formulas quantifying only over sets. Shoenfield showed that $\GBC$ is conservative over $\ZFC$, meaning that any statement about sets that follows from $\GBC$ already follows from $\ZFC$. His argument was proof theoretic, but with forcing this is easy to see because an arbitrary model of $\ZFC$ can be extended to a model of $\GBC$ without adding sets by forcing the existence of a global well-order class. The axioms of $\KM$ include full class comprehension: any collection that is definable by a formula in the two-sorted language is a class. The use of full class comprehension was suggested by several logicians among them Tarski, Quine, and Mostowski before a coherent version of the axioms appeared in the appendix of Kelley’s topology textbook [2]. $\KM$ is not conservative over $\ZFC$ because it proves for instance $\text{Con}(\ZFC)$, $\text{Con}(\text{Con}(\ZFC))$ and much more, as Joel argued in his post.

Here is a version of $\KM$ that is essentially the one given by Kelley and adapted in work on $\KM$ such as the forcing paper I mentioned earlier. So let’s call it the *common version*. The axioms for sets are Extensionality, Pairing, Infinity, Union, Powerset, Regularity. The axioms for classes are:

**Extensionality****Full Class Comprehension**: If $\varphi(x)$ is any formula in the two-sorted language with class parameters, then the collection of all those sets such that $\varphi(x)$ holds is a class.**Replacement**: If $F$ is a class function and $a$ is a set, then the range of $F\upharpoonright a$ is a set.**Global well-order**: there is a 1-1 and onto class function $\varphi:\ORD\to V$.

In place of (4), Kelley assumed that there is a global choice function $F$ on $V-\emptyset$ such $F(x)\in x$. But it is easy to argue that this is equivalent to the existence of a well-order of $V$. Clearly $F$ can be used to well-order every $V_{\xi+1}-V_\xi$ and then choose one such well-order for every $\xi$. The well-orders are then glued together to form a well-order of $V$.

A slightly different axiomatization is found in the Wikipedia entry on $\KM$. So let’s call it the *Wikipedia version*. The two versions differ on the class part in that Replacement and Global well-order are replaced by the *Limitation of Size principle* stating that a class $C$ is proper if and only if there is a 1-1 function from $V$ to $C$. While I think Joel favors the common version, I am tempted by the, in my opinion, elegantly formulated Limitation of Size principle. The principle seems to nicely capture the idea that the classness of a class should be confirmed by a witness. Let’s finish with a quick argument that the Limitation of Size principle implies and is implied by Replacement together with Global well-order.

**Theorem**: Every model of the common version of $\KM$ is a model of the Wikipedia version of $\KM$ and conversely.

**Proof**: Suppose that the common version holds and let’s verify the Limitation of Size principle. Suppose that $C$ is a proper class. Fix a global well-order $\varphi:\ORD\to V$. Use $\varphi$ to define an enumeration $\langle c_\xi\mid\xi\in\ORD\rangle$ of $C$. Define $F:V\to C$ by $F(x)=c_{\varphi(x)}$. Clearly $F$ is 1-1. Next, suppose that $F:V\to C$ is 1-1. By using a global well-order to further shrink $C$ if necessary, we can assume that $F$ is onto. Let $F^{-1}:C\to V$. If $C$ is a set, then, by Replacement, the range of $F^{-1}$ is a set as well, but this is impossible.

Now suppose that the Wikipedia version holds and let’s verify that Replacement and Global well-order axioms hold. Global well-order follows nearly immediately because there is a 1-1 function from $V$ to $\ORD$. For Replacement, suppose that $F$ is a class function. Let $a$ be a set and let $b$ be the range of $F\upharpoonright a$. By using a global well-order to shrink $a$ if necessary, we can assume that $F$ is 1-1 on $a$. Let’s assume towards a contradiction that $b$ is proper class. Then there is a 1-1 function $G:V\to b$. But then $G\circ F^{-1}:V\to a$ is 1-1, contradicting that $a$ is a set. $\square$

[Bibtex]

```
@article {chuaqui:km,
AUTHOR = {Chuaqui, Rolando},
TITLE = {Forcing for the impredicative theory of classes},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {37},
YEAR = {1972},
PAGES = {1--18},
ISSN = {0022-4812},
MRCLASS = {02K05},
MRNUMBER = {0327516 (48 \#5858)},
MRREVIEWER = {D. Pincus},
}
```

```
@book {kelley:topology,
AUTHOR = {Kelley, John L.},
TITLE = {General topology},
NOTE = {Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.],
Graduate Texts in Mathematics, No. 27},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1975},
PAGES = {xiv+298},
MRCLASS = {54-XX},
MRNUMBER = {0370454 (51 \#6681)},
}
```

Thanks for making this very clear, Vika! You mention that you think I prefer what you call the common axiomatization, and that is true. Part of my reason for that is that I find there to be relatively little direct philosophical support for the limitation of size principle—why should all proper classes be the same size? Of course, this is provable in KM once you have the global AC principle and replacement, but I find the appeal to “limitation of size” to be essentially obscuring the set-theoretic issues, rather than clarifying them. It hides the difference between AC and global AC, whereas the common approach to KM makes this difference quite explicit and clear. But since the two axiomatizations are equivalent, I suppose I should not protest too much!

Hmm, I guess I have to agree that there is no a priori reason to suppose that all classes are the same size, but that is just such an ingrained intuition for me. On a separate note, I wonder whether the restrictions for class forcing to preserve KM are the same as for preserving ZFC or whether something stronger is needed. I haven’t been able to figure out the argument in the paper yet.

Let’s discuss, but I believe that the ideas in our current project on the natural strengthening of KM will show that some natural class forcing notions will not always preserve KM. We’ve already got it failing in an ultrapower, and forcing is a kind of ultrapower, if one thinks of the Boolean ultrapower.

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Regarding the proof that one gets a global well-ordering from a global choice function, one can also argue directly as in proving WOP from AC, namely, one simply chooses the next element in the order from among the minimal-rank sets not yet previously chosen.

Of course, you are right!

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Thank you for this very interesting post!

Regarding class forcing over KM: I’m currently writing my PhD thesis with Sy Friedman about “higher-oder” forcings over KM (class and hyperclass forcing). In the first part of the thesis we show how class forcing can be defined over KM using different names for sets and classes. In contrast to the definition of class forcing in ZFC (as gives by S. Friedman) here we don’t need the pretameness condition for proving the Definability Lemma, but we still need it to show that the axioms are preserved. The main difference between our approach and Chuaqui’s is that we restrict the forcing notion one can use to get the preservation of the axioms (by using tameness) whereas he restricts the kind of generic one can have for a forcing notion. (We have a paper for this in preparation and it should be ready for submission in the summer). We are also working on showing how to define “hyperclass” forcing over KM (that is a forcing whose conditions are classes). This seems to be much more complicated and we need an extension of the KM axioms, including a class-bounding principle and a form of dependent choice for classes.

Thanks for the great comment, Carolin! So you prove that tame forcing preserves ${\rm KM}$, right? What is the class bounding principle and what is the dependent choice axiom you are using for hyperclass forcing? Is there any relationship between Chuaqui’s condition and tameness?

Yes, exactly, we have proven that KM is preserved under tame class forcings. The class-bounding principle is the same as the class collection you, Joel and Thomas are using in your work on “a natural strengthening of KM” (I just read Joel’s entry here). The dependent choice says that if for all sequences of classes X with ordinal-length there is a Y such that phi(X, Y) then we can find for all X an ordinal-length sequence Z such that Z_0 = X and for all i in the ordinals phi(Z restricted to i, Z_i), where Z restricted to i is the sequence of previously “chosen” Z_j, j<i. We need the last axiom for technical reasons when we are showing that a forcing we would like to use is pretame. Let me get back to you for the part about Chuaqui, I don’t have the book at hand at the moment.

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