Variants of Kelley-Morse set theory

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.

It is actually quiet surprising how little there is to be found about ${\rm KM}$ on the web. I had to dig through pages of Google search on ${\rm KM}$ (tediousness pays off) before I stumbled on a very interesting paper about forcing over models of ${\rm KM}$ by Ronaldo Chuaqui [1]. In a recent post, I discussed some folklore results about class forcing over models of ${\rm GBC}$ that adds classes but not sets. Apparently, many of the standard forcing arguments can be done over models of ${\rm KM}$ where new sets and classes are added simultaneously while preserving ${\rm KM}$ 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 ${\rm KM}$.

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.

${\rm KM}$'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 ${\rm GBC}$ is conservative over ${\rm ZFC}$, meaning that any statement about sets that follows from ${\rm GBC}$ already follows from ${\rm ZFC}$. His argument was proof theoretic, but with forcing this is easy to see because an arbitrary model of ${\rm ZFC}$ can be extended to a model of ${\rm GBC}$ without adding sets by forcing the existence of a global well-order class. The axioms of ${\rm KM}$ 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]. ${\rm KM}$ is not conservative over ${\rm ZFC}$ because it proves for instance $\text{Con}({\rm ZFC})$, $\text{Con}(\text{Con}({\rm ZFC}))$ and much more, as Joel argued in his post.

Here is a version of ${\rm KM}$ that is essentially the one given by Kelley and adapted in work on ${\rm KM}$ 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:

  1. Extensionality
  2. 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.
  3. Replacement: If $F$ is a class function and $a$ is a set, then the range of $F\upharpoonright a$ is a set.
  4. Global well-order: there is a 1-1 and onto class function $\varphi:{\rm ORD}\to V$.

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 ${\rm KM}$. 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 ${\rm KM}$ is a model of the Wikipedia version of ${\rm KM}$ 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 $\varphi:{\rm ORD}\to V$. Use $\varphi$ to define an enumeration $\langle c_\xi\mid\xi\in{\rm ORD}\rangle$ 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 ${\rm ORD}$. 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. $\square$

References

  1. R. Chuaqui, “Forcing for the impredicative theory of classes,” J. Symbolic Logic, vol. 37, pp. 1–18, 1972.
  2. J. L. Kelley, General topology. New York: Springer-Verlag, 1975, p. xiv+298.