paradoxes and intuitionist logic
Sun, 15 Aug 1999 14:31:48 -0400
Epimenides Paradox, also known as the liar paradox or the paradox of
self-reference is attributed to Epimenides, a Cretan who made one immortal
statement: "All Cretans are liars." A sharper version is simply "This
statement is false."
This paradox is a result of self-reference. And the issue of self-reference
within an axiomatic formal calculus and its consequences have been laid out
neatly by Kurt Godel and are formalized by his infamous result: Godel's
Theorem. And what Godel showed was that any formal system is a member of one
of two groups: either less complex than number theory, or at least as
complex as number theory. And here's the rub: if a formal system is less
complex than number theory, then it won't be very useful, and if it is as
complex as number theory, then it is incomplete because of the system's
ability to refer to itself.
So it gets you coming and going; either way your formal system is incomplete
because there are always statements, both true and false, that cannot be
proven either way within any particular formal system.
I definitely suggest you read _Godel, Escher, Bach_ for a lengthy discussion
of essentially this issue.