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and it reflects most favorably on the kind of student who enrolls in our Honors Program.

Richard W. Behling

Professor of Philosophy

For the 2500 years since the Greeks, mathematicians have had the axiomatic system   a method

of accepting unproven propositions as true, and then deriving all other true propositions within that

system from those foundational propositions or axioms.  In the beginning of the twentieth century, it

was this system that was employed by several metamathematicians, at the urging of David Hilbert, with

the ultimate goal of codifying all mathematical reasoning.  Bertrand Russell and Alfred North Whitehead

are most noteworthy here, with their gigantic and well respected Principia Mathematica that claimed to

derive all mathematics from logic   the codification that was sought.  The work, published between 1910

and 1913, " carried out in minute detail a program intended to prove that all of pure mathematics can

be derived from a small number of fundamental logical principles."1 Russell's reason for writing the

Principia was to affirm his belief that there is no distinction between mathematics and logic, and he

attempted to answer the question posed by Hilbert about "whether it can be proved that the axioms of

arithmetic are consistent   that a finite number of logical steps based upon them can never lead to con 

tradictory results."2 But with such an intricate leviathan, it was difficult to say whether the Principia

really brought, contradiction free, this complete formalization of reasoning to the field.  

No definite answers in the affirmative came, nor would they, since, in 1931, Czechoslovakian 

born Kurt Godel provided a rather twisted and complex   but solid   answer in the negative.  His

Incompleteness Theorems shed light upon some key deficiencies in "Principia Mathematica and Related

Systems" that could not be corrected by any modification.  The fact was that any axiomatic system, by

being an axiomatic system, had a certain hole within itself.  What is perhaps most astonishing is that,

according to Godel, the stronger the system seems in terms of its consistency and completeness, the

stronger it shows its own inconsistency by producing a well formed formula that states the inconsis 

tency of the system.  What his insight amounted to was this: one could prove the consistency of a sys 

tem only if that system were incomplete (that is if the system could not generate the theorem stating

inconsistency).  Furthermore, the most powerful (that is, the most convincingly complete) systems that

had been developed, along with any system sufficiently powerful (or complete), were shown by Godel

invariably to be capable of producing theorems that express their own.  Enter the notion of self refer 

ence.  All of this may sound incredibly cloudy and mysterious; some exposition, therefore, is due with

regard to these concepts of incompleteness, inconsistency, and self reference, concepts so integral to

so much of Godel's work.

First, to give flavor to the idea of self reference and to the twisted complications it creates,

Bertrand Russell's famous paradox relating to set theory provides a good, accessible introduction.

Russell proposes that there are two types of sets.  Specifically, either a set will contain itself as a mem 

ber or it will not.  Here is the self reference.  Nagel and Newman call the former "non normal sets" and

the latter "normal."3 Examples of normal sets would be the set of tomatoes and the set of windows.

Clearly, the set of tomatoes is not, itself, a tomato (and, thus, it is not a member of itself, the set of

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Elements of Godel's Incompleteness in Axiomatic Systems      Jared Balkman