Morse–Kelley set theory

Morse–Kelley set theory or Kelley–Morse set theory (MK or KM) is a set theory with proper classes properly extending the usual set theory ZF. It is a first order theory (though it can be confused with second-order ZF). Its primitive predicates are equality and membership. General objects of this theory are called classes. Objects which are elements are called sets: i.e., "x is a set" is defined as $(\exists y.x \in y)$ . A class which is not a set is a proper class. Although this is a first-order theory, it is a common practice (not followed here) to use language resembling that of second-order logic, that is, to distinguish between class variables (often capitalized) and set variables. It is not properly a two-sorted theory, though, since a set is identified with the class with the same extension.

A convenient set of axioms for this theory (which entails choice in an amusing way, following von Neumann) is the following:

axiom of extensionality: classes with the same elements are the same.
axiom of class comprehension: for any formula $\phi$ , there is a class $\{x \mid \phi\}$ whose elements are exactly those sets x such that $\phi$ .
axiom of pairs: for any sets x and y, there is a set $\{x,y\}$ whose elements are exactly x and y. In terms of these unordered pairs, we can define the usual Kuratowski ordered pair and use class comprehension to show that relations and functions on sets can be defined as usual, thus providing support for the following axiom.
axiom of limitation of size: a class C is a proper class if and only if there is a bijection between C and the class V of all sets.
axiom of power set: the class P(A) of all subsets of a set A is a set.
axiom of union: the class $\bigcup A$ of all elements of elements of a set A is a set.
axiom of infinity: there is a set I which contains the empty set as an element and contains $y \cup \{y\}$ as an element for each element y of I.
axiom of foundation: Each nonempty class is disjoint from at least one of its elements.

Important comments on the axioms follow.

The form of the axiom of class comprehension is the reason that this theory is stronger than the usual theory of proper classes NBG which extends ZFC only conservatively: in the latter theory, class comprehension is restricted to formulas containing no quantifiers over classes (all quantifiers are there restricted to particular classes, or equivalently to the universe of sets; the other axioms can be taken to be exactly the ones given here). This theory is impredicative while the weaker theory is predicative. In the weaker theory, it is possible to replace the scheme of class comprehension with finitely many instances; this is not possible here (Kelley-Morse set theory cannot be finitely axiomatized).

No axiom asserting the sethood of the empty class is required, though of course one could be added (and minor perturbations of our formulation would cause this to be necessary). Notice that limitation of size plus the fact that I is a set (so the universe is not empty) does the trick here.

The axiom of global choice follows from the axiom of limitation of size, since the class of von Neumann ordinals cannot be a set, so must be a proper class, so must be the same size as the universal class, so the universal class can be well-ordered. If the humor of this is not appreciated, this can be replaced by an axiom asserting that the range of any class function whose domain is included in a set must itself be a set (i.e., any class smaller than or the same size as a set is a set); this will serve all other functions of Limitation of Size without causing Choice to hold. One could also add Choice in any of its usual local forms after restricting Limitation of Size (for example, one could stipulate that any particular set can be well-ordered).

The set I is not called $\omega$ because it might actually be any larger set; Limitation of Size (in either form) is then used to show the existence of $\omega$ .

This theory is known to be strictly stronger than ZFC (it proves the consistency of ZFC!)

A theory which can be confused with Morse-Kelley set theory is second-order ZF or ZFC, in which the logic is extended to second-order logic (representing second-order objects in set language rather than predicate language). The language of this theory is similar (though a technical distinction must be drawn between a set and the class with the same extension), and its resources for practical proof on the level of syntax are almost identical (identical if the strong form of Limitation of Size is added), but its semantics are quite different: whatever is true in all models in which the classes are all the collections of the sets is valid (because we are in second-order logic) so questions such as CH (and many others) are "decided" in an arguably not very practical sense. The theory of any rank of the cumulative hierarchy, up to the first inaccessible, is uniquely determined in second-order ZF.

This theory became well known through its appearance (not in this exact form: Kelley's axioms are given below) as an appendix in a book on General Topology by J. L. Kelley (1955). An apparently equivalent system, but formulated in an idiosyncratic formal language rather than in standard First-Order logic, was presented by A. P. Morse in A Theory of Sets (1965).

The basic idea was also proposed by A. Mostowski and D. Lewis. W. V. Quine and Hao Wang introduced the impredicative theory of classes as well, but over New Foundations rather than the usual set theory ZFC.

The axioms from the appendix of Kelley's General topology

Kelley does not use two different sorts of variables; he uses the same shape of variables for classes generally and for classes known to be sets.

The axioms and definitions that follow are exactly as in Kelley, with some minor typographical modifications. The intervening remarks are ours. We do not include all definitions (most notations used are familiar). The domain and range of a function f are written simply "domain f" and "range f" in the original text (that is Kelley's notation); this is not a grammatical error on our part. The full text of Kelley's appendix is very much worth examining.

It is useful to be aware that set abstracts $\{x : A\}$ ("the class of all sets x such that A") are for Kelley part of his primitive logical language.

I. Axiom of extent For each x and each y it is true that x=y if and only if for each z, $z \in x$ when and only when $z \in y$ .
Definition x is a set if and only if for some y, $x \in y$ .
II. Classification axiom-scheme An axiom results if in the following $\alpha$ and $\beta$ are replaced by variables, A by a formula A

and B by the formula obtained from A by replacing each occurrence of the variable which replaced

\alpha

by the variable which replaced

\beta

: For each

\beta

\beta \in \{\alpha:A\}

if and only if

\beta

is a set and B.

III. Axiom of subsets If x is a set, there is a set y such that for each z, if $z \subseteq x$ , then $z \in y$ .

Kelley deduces from this not only Power Set but Separation: for any class z which is a subclass of the set x, the class z is an element of the set y provided by the axiom and so is a set.

IV. Axiom of union If x is a set and y is a set then so is $x \cup y$ .

The role of this axiom is to allow the proof of Pairing: the singleton $\{x\}$ of a set x is a set because it is a subclass of the power set of x (two applications of axiom III); in combination with axiom IV, we see that $\{x,y\}$ is a set if x and y are sets. We can then define pairs, relations and functions as usual.

V. Axiom of substitution If f is a function and domain f is a set, then range f is a set.

The function f in axiom V is a class function: this is the axiom of Replacement, in effect.

VI. Axiom of amalgamation If x is a set, then $\bigcup x$ is a set.
VII. Axiom of regularity If $x \neq \emptyset$ there is a member y of x such that $x \cap y = \emptyset$ .
VIII. Axiom of infinity For some y, y is a set, $\emptyset \in y$ and $x \cup \{x\} \in y$ whenever $x \in y$ .
Definition c is a choice function if and only if c is a function and $c(x) \in x$ for each member x of domain c.

It is worth noting that Kelley uses axiom VIII to deduce the sethood of $\emptyset$ (and so the existence of any sets at all): the extensive discussion of sets in the appendix before this point is entirely hypothetical.

IX. Axiom of choice There is a choice function c whose domain is $U - \{\emptyset\}$ (U being the universal class of all sets).

Note that this theory, like the one we present above, has a kind of global choice.

References

Morse, Anthony P. (1965) A Theory of Sets, Academic Press
Lemmon, E. J. (1986) Introduction to Axiomatic Set Theory, Routledge & Kegan Paul
Kelley, John L. (1955) General Topology, D. van Nostrand.

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The axioms from the appendix of Kelley's General topology

References

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