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CHAT: Goedel (was: Re: Láadan and woman's speak)

From:Dennis Paul Himes <dennis@...>
Date:Saturday, May 20, 2000, 19:00
Peter Clark <pclark@...> wrote:
> One parting comment: would someone mind explaining me what > Goedel's theorem is?
Goedel proved several theorems. The most famous were his Completeness Theorem and his two Incompleteness Theorems. These all apply to first order logic, i.e. logical systems in which quantification over objects, but not sets of objects, is allowed. I spent several semesters studying these theorems in graduate school. That was a while ago, but hopefully I can describe them without mangling them too badly. The Completeness Theorem states that a statement is true in a given theory if and only if it is provable in that theory. "True" meaning true in all models of the theory. One Incompleteness Theorem (I forget which is the First and which is the Second) states that a consistent theory cannot prove its own consistency. The other Incompleteness Theorem states that any consistent theory which is able to express arithmetic has undecidable statements. "Undecidable" means neither provable nor refutable, and therefore, by the Completeness Theorem, neither true nor false. (Although the statement he exhibits in the proof is intuitively a true statement, and provably so in a stronger theory, so you may hear the theorem described as "Any strong enough theory has true statements which cannot be proven.". (The statement he exhibits is a formalization of a variation of the Liar's Paradox, "This statement is not provable.".)) =========================================================================== Dennis Paul Himes <> Disclaimer: "True, I talk of dreams; which are the children of an idle brain, begot of nothing but vain fantasy; which is as thin of substance as the air." - Romeo & Juliet, Act I Scene iv Verse 96-99