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In the meantime, interesting developments were taking place in classical mathematics.

A theory of different types of infinities, known as the theory of sets, was developed by

Georg Cantor in the 1880's. The theory was powerful and beautiful, but intuition-defying.

Before long, a variety of set-theoretical paradoxes had been unearthed. The situation was

very disturbing, because just as mathematics seemed to be recovering from one set of

paradoxes-those related to the theory of limits, in the calculusalong came a whole new

set, which looked worse!

The most famous is Russell's paradox. Most sets, it would seem, are not members of

themselves-for example, the set of walruses is not a walrus, the set containing only Joan

of Arc is not Joan of Arc (a set is not a person)-and so on. In this respect, most sets are

rather "run-of-the-mill". However, some "self-swallowing" sets do contain themselves as

members, such as the set of all sets, or the set of all things except Joan of Arc, and so on.

Clearly, every set is either run-of-the-mill or self-swallowing, and no set can be both.

Now nothing prevents us from inventing R: the set of all run-o,-the-mill sets. At first, R

might seem a rather run-of-the-mill invention-but that opinion must be revised when you

ask yourself, "Is R itself "a run-of-the-mill set or a self-swallowing set?" You will find

that the answer is: "R is neither run-of-the-mill nor self-swallowing, for either choice

leads to paradox." Try it!

But if R is neither run-of-the-mill nor self-swallowing, then what is it? At the very

least, pathological. But no one was satisfied with evasive answers of that sort. And so

people began to dig more deeply into the foundations of set theory. The crucial questions

seemed to be: "What is wrong with our intuitive concept of 'set'? Can we make a rigorous

theory of sets which corresponds closely with our intuitions, but which skirts the

paradoxes?" Here, as in number theory and geometry, the problem is in trying to line up

intuition with formalized, or axiomatized, reasoning systems.

A startling variant of Russell's paradox, called "Grelling's paradox", can be made using

adjectives instead of sets. Divide the adjectives in English into two categories: those

which are self-descriptive, such as "pentasyllabic", "awkwardnessful", and "recherche",

and those which are not, such

Introduction: A Musico-Logical Offering

20

as "edible", "incomplete", and "bisyllabic". Now if we admit "non-selfdescriptive" as an

adjective, to which class does it belong? If it seems questionable to include hyphenated

words, we can use two terms invented specially for this paradox: autological (= "self-

descriptive"), and heterological (= "non-self-descriptive"). The question then becomes:

"Is 'heterological' heterological?" Try it!

There seems to he one common culprit in these paradoxes, namely self-reference, or

"Strange Loopiness". So if the goal is to ban all paradoxes, why not try banning self¬

reference and anything that allows it to arise? This is not so easy as it might seem,

because it can be hard to figure out just where self-reference is occurring. It may be

spread out over a whole Strange Loop with several steps, as in this "expanded" version of

Epimenides, reminiscent of Drawing Hands:

The following sentence is false.

The preceding sentence is true.

Taken together, these sentences have the same effect as the original Epimenides paradox:

yet separately, they are harmless and even potentially useful sentences. The "blame" for

this Strange Loop can't he pinned on either sentence-only on the way they "point" at each

other. In the same way, each local region of Ascending and Descending is quite

legitimate; it is only the way they are globally put together that creates an impossibility.

Since there are indirect as well as direct ways of achieving self-reference, one must figure

out how to ban both types at once-if one sees self reference as the root of all evil.

Banishing Strange Loops

Russell and Whitehead did subscribe to this view, and accordingly, Principia

Mathematica was a mammoth exercise in exorcising Strange Loops from logic, set

theory, and number theory. The idea of their system was basically this. A set of the

lowest "type" could contain only "objects" as membersnot sets. A set of the next type up

could only contain objects, or sets of the lowest type. In general, a set of a given type

could only contain sets of lower type, or objects. Every set would belong to a specific

type. Clearly, no set could contain itself because it would have to belong to a type higher

than its own type. Only "run-of-the-mill" sets exist in such a system; furthermore, old R-

the set of all run-of-the-mill sets-no longer is considered a set at all, because it does not

belong to any finite type. To all appearances, then, this theory of types, which we might

also call the "theory of the abolition of Strange Loops", successfully rids set theory of its

paradoxes, but only at the cost of introducing an artificial-seeming hierarchy, and of

disallowing the formation of certain kinds of sets-such as the set of all run-of-the-mill

sets. Intuitively, this is not the way we imagine sets.

The theory of types handled Russell's paradox, but it did nothing about the Epimenides

paradox or Grelling's paradox. For people whose

Introduction: A Musico-Logical Offering

21

interest went no further than set theory, this was quite adequate-but for people interested

in the elimination of paradoxes generally, some similar "hierarchization" seemed

necessary, to forbid looping back inside language. At the bottom of such a hierarchy

would be an object language. Here, reference could be made only to a specific domain-

not to aspects of the object language itself (such as its grammatical rules, or specific

sentences in it). For that purpose there would be a metalanguage. This experience of two