A geometric note on Russell's paradox

A lengthier discussion of Russell's paradox was found on my web page on vacuous truth before it vanished into the vacuum left by the demise of Geocities. It is there that I discuss banishing 'actual infinity' and replacing it with construction algorithms. However, such an approach, though countering other paradoxes, does not rid us of Russell's, which is summed up with the question: If sets are sorted into two types: those that are elements of themselves ('S is the name of a set that contains sets named for letters of the alphabet' is an example) and those that are not, then what type of set is the set that contains sets that are elements of themselves?

---

Here we regard the null set as the initial set and build sets from there, as in:

Step 0: { }

Step 1: { { { } },{ } }

Using an axiom of infinity, we can continue this process indefinitely, leading directly to a procedure for building an abstraction of all countable sets; indirectly, noncountable sets can also be justified.

In Russell's conundrum, some sets are elements of themselves.

So let us regard a plane as pre-existent (null space or some such) and regard Set sub 0, the empty set, as a circle surrounding nothing. Set sub 1 we picture as a concentric circle around set 0. Now those two circles express the element represented as {{ }}.

We might also say the two circles express the set that contains the element { }.

In that case, we may wish to say that the set {{ }} = the element {{ }}.

Suppose we establish two construction algorithms for sets of concentric circles. In algorithm A, the outer circle is a solid line, so that the null space container is not construed to be part of the set. In algorithm B, the outer circle is a dashed line, so that the null space container is construed to be part of the set.

At each step we add a new enclosing circle and require that it be either solid or dashed.

So as step n is taken to infinity, we find that the null space container vanishes. That is, the circles fill the plane.

In that case, the container is non-existent or undefined, so that the rule that requires that it be either part of the set or not part of the set is not applicable.

We have assumed that the initial circle surrounds something that is not an element -- as with { }. But we really must give a rule that says the inmost area is not an element. We could have a rule that permits not only the outmost but the inmost circle to be dashed. In that case the inner void would be considered a part (element) of the set. Though the void in a null set is, for sound reasons, not permitted to be construed as an element, it is still useful to see the relationship of admitting the inner void to Russell's paradox.

---

Russell's paradox is usually disposed of by resort to an axiom prohibiting a set from being an element of itself. But we can see that sets and elements can be identical except for the rule prohibiting a set belonging to itself.

For example, element {{ }} in a set-building algorithm X is identical to the set containing the element { } in set-building algorithm Y.

It seems that the axiom while necessary remains questionable from the standpoint of logical consistency.

---

First published ca. 2000