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E

LEMENTS OF

G

ODEL

'

S

I

NCOMPLETENESS IN

A

XIOMATIC

S

YSTEMS

|  J

ARED

B

ALKMAN

A tale is told of the philosopher B. Spinoza (1632 77), who, in his youth and innocent of mathematics,

picked up a text in Euclidean geometry.  He marveled at the theorems displayed at book's end, and he

proceeded to page backwards, wondering how the earlier proven theorems were generated.  Much to

his amazement, the starting points   the axioms and postulates   were as easy as pie to understand.

A common goal of philosopher/mathematicians of the twentieth century was to demonstrate that the

whole of mathematics, in all of its arcane slender, is derivable from logic systems that were developed

in the early 1900's.  Unhappily, this quest was ill fated, and these dreamers' monumental effort  

glimpsed, for example in B. Russell's and A.N. Whitehead's Principia Mathematica   went for naught.

In this essay, written in Mr. Charles Roll's honors logic class, Jared Balkman explores the principal rea 

son for the end of this dream.  That this essay was written by a freshman student taking his first course

in logic indicates much about Jared, of course; but, equally, it bespeaks the kind of instruction he received,

Elements of Godel's Incompleteness in Axiomatic Systems      Jared Balkman

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