This is one of a series of draft papers on set theory that I did in the late 1960’s. None were ever published. This presents a weak zigzag theory that establishes alternative lines of development for strong theories. The main body is faithful to the original except for minor editorial changes. I have added a commentary at the end.

# A limitation in the construction of zigzag theories

When Lord Russell surveyed the ruins that naive set theory had fallen into under the impact of the paradoxes he propounded four lines of cure: the no class theory, the theory of types, the theory of limitation of size, and the zigzag theory. To date the principal example of a zigzag theory is Quine’s system NF. NF has often been rejected as a proper system for foundations because of various features that are considered to be anomalous and idiosyncratic.

One of the features that is commonly accounted to be peculiar is the existence of so called non-cantorian sets, a feature due to the lack of a function that carries every set into its unit set. It might be supposed that this is an accidental feature that could be eliminated by a minor revision. The results given here suggest that this is not true, that the presence or absence of the unit set function is one of the major dividing lines in the construction of zigzag theories.

To show this we will first define a rather elementary axiomatic set theory, system A. This system is one sorted and has a set of all sets. System A has six axioms; the axiom of extensionality, the axiom of the null set, the axiom of complements, the axiom of unit sets, and the axiom of finite intersections. Relationsand functions will be taken as sets of ordered pairs with function values on the right. Any reasonable definition of ordered pair may be used; U(x,y) denotes the union of x and y, and I(x,y) denotes the intersection of x and y. The axioms for system A are:

```     1.  (∀x,y) (x=y <-> ∀z (x∈z <-> y∈z)
2.  (∃x)(∀y) (y ~∈ x)
3.  (∀x)(∃y)(∀z)  (z∈x <-> ~ z∈y)
4.  (∀x)(∃y)(∀z)  (z∈y <-> z=x)
5.  (∀x,y)(∃z)(∀u) (u∈z <->  u∈x & u∈y)
```
To this system we might consider adding any or all of the following axioms:
```     6.  (∀x)(∃y)(∀u,v) ([uv]∈x <-> v∈y)
7.  (∃f)(∀u,v) ([uv]∈f <-> u=v)
8.  (∃f)(∀u,v) ([uv]∈f <-> v={u})
9.  (∀f,g)(∃h)(∀x,y,z) ([xy]∈f & [xz]g <-> (∃u)(u=U(y,z) & [xu]∈h))
```
Axiom 6 is the axiom of the domain; its effect is to enable sets of ordered pairs be treated as functions. Axiom 7 is the axiom of the identity function, and axiom 8 is the axiom of the unit set function. Axiom 9 is the axiom of the relationship value union; it can be formulated in a variety of ways, each of which expresses a general principle that boolean operations are permitted on relation values. The latitude available to us in constructing axiomatic set theories compatible with system A is considerably restricted by the following two results.

### Theorem 1:

In the system obtained by adjoining axioms 6 and 7 to the system A the Von Neumann successor function is not a set, i.e.

```~(∃f)(∀u,v)([uv]∈f <-> v = U(u,{u})
```
Proof: Suppose there were such an f. Let g be the intersection of f and the identity function. Then
```(∀u,v) ([uv]#8712;g <-> u=v & v=U(u,{u})
```
Applying the axiom of the domain to g yields the set r such that
```(∀x)(x∈r <-> x = U(x,{x}).  But x=U(x,{x})<-> x∈x.
```
The complement of r is the Russell set. QED.

### Theorem 2:

The system obtained by adjoining axioms 6,7,8 and 9 to system A is inconsistent.

Proof: Applying axiom 9 to the unit set function and the identity function yields the Von Neumann Successor function. QED.

If the Wiener-Kuratovski definition of ordered pairs is used than it can be shown that system A is consistent and that adding 7 and any two of axioms 6,8, and 9 produces consistent systems. Hence there are three major alternatives in constructing zigzag theories, depending on which one of the three axioms we choose to omit. Omitting the unit set function leads to NF or to systems that are very similar in character to NF. The remaining two alternatives have not been studied to any extent, and systems based on them are worthy of investigation.

### Commentary

This little paper is rather interesting (to me anyway) and I have a few comments.

As an initial minor note, when I read the axioms for system A, I asked, “what happened to the pair set axioms””, but then I realized that it follows from the unit set axiom, the axiom of complements, and the intersection axiom. If x and y are two sets, then the intersection of the complements of {x} and {y} is the complement of {x,y}.

I believe (though I haven’t proved it) that the Von Neumann successor function can be rescued by adding the condition ~ u∈u, i.e.,

```     ~(∃f)(∀u,v)([uv]∈f & ~u∈u<-> v = U(u,{u})
```
Also it should be noted that the unit set function is the Zermelo successor function. In a general way, it appears that the successor functions are incompatible with the identity function.

It is unfortunate that in the final paragraph the paper simply said “it can be shown”; however much the reasoning can be reconstructed. As a note the original paper assumed that axiom 7 (the identity function) was part of system A. This doesn’t make a great deal of sense. Axiom 6 is required for treating sets of ordered pairs and relationships; the other axioms are ineffective without axiom 6. The question should be, which of 7,8, and 9 can be deleted, or alternately, how could 9 be weakened so as to retain consistency with having both 7 and 8.

This page was last updated December 1, 2006.