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tomatoes).  Examples of non normal sets would be the set of things except tomatoes, the set of bibli 

ographies, and the set of things that can be contemplated.  To be more specific, the set of things except

tomatoes is not, itself, a tomato; likewise the set of all things that can be contemplated can, itself, be

contemplated.  Hence, both sets are members of themselves.  Now, Russell asks us to consider the set

of all normal sets.  Is it normal or non normal?  Well, if it is normal, it is a member of itself, since it is

the set of all sets that are not members of themselves.  But in that case, it is non normal, since, then,

it is a member of itself.  Going the other way, if the set is non normal, it is a member of itself (by the

definition of a "non normal set"), but that means it is not a member of itself, because it, of course, is

the set of all sets that are not members of themselves.  Basically, the only time that this set in question

can be normal is when it is non normal.   This is a troublesome monkey wrench that mathematicians

had to deal with early in the twentieth century in their attempts to systematize arithmetic.  And it was

the self reference at the heart of this and other paradoxes   "Strange Loops", as Douglas Hofstadter calls

them   that was seen as the bane of these attempts early on.  Certainly Bertrand Russell, in proposing

his paradox, attested to their existence.  In fact, Hofstadter notes, Russell and Whitehead's "Principia

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

theory."4 The way they did that was by distinguishing between types of sets.  Sets of the lowest type

were not allowed to contain sets, but only objects.  If a set were one type above this kind, it could con 

tain either objects or sets of the lowest type, but nothing else.  One can then surmise that any set can

contain only objects or sets of a less inclusive type than itself.  Thus, our previous set of all normal sets

no longer even exists under this stipulation of types of sets.  So the theoretical revision resolves the

paradox, though "only at the cost of introducing an artificial seeming hierarchy, and of disallowing the

formation of certain kinds of sets,"5 rather like trading completeness for consistency.  Hofstadter seems

to espouse the same feeling as Godel, as likely do many of us, that "the solution suggested by Whitehead

and Russell, that a proposition cannot say something about itself, is too drastic."6 Russell himself was

even suspect of the position, saying "the whole effect of the doctrine of types is negative: it forbids cer 

tain inferences which would otherwise be valid."7

Let us now move on to the ideas of completeness and consistency, an understanding of which

is necessary to grasp the significance of Godel's theorems.  These are certainly not difficult concepts.

Consider any axiomatic system.  It will have a number of axioms (foundational propositions) formed

from a finite set of symbols, and it will have a set of rules    "rules of inference"   in accordance with

which theorems are formed.  Now, the purpose of this axiomatic system (or formal system) is to gener 

ate some truths in a particular field of inquiry.  A system such as this is called "deductively complete"

if every truth within that field of inquiry can, in theory, be generated by using the system's axioms and

rules.  To put it differently, deductive completeness is a system's ability to show whether or not each

formula of the system is demonstrable. At the turn of the twentieth century, this notion of complete 

ness seemed naturally on its way to the mathematical logicians.  Indeed, before Godel, ". . . it was taken

as a matter of course that a complete set of axioms for any given branch of mathematics can be assem 

bled .  At worst, [the set] could be made complete simply by adding a finite number of axioms to the

original list."8

Now let us look at consistency.  Bluntly, a system is consistent if everything it produces is true.

Elements of Godel's Incompleteness in Axiomatic Systems      Jared Balkman

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