To: HangnJudge
a basic tenet of logic is that a thing can not be used to explain itself...reason cannot explain itself The first is not true (sounds like a misunderstanding of Godel's theorems). The second is not logically true, although reason has certainly not yet explained itself.
33 posted on
08/10/2005 3:42:24 PM PDT by
beavus
(Hussein's war. Bush's response.)
To: beavus
From Wikipedia
Gödel's first incompleteness theorem shows that any such system that allows you to define the natural numbers is necessarily incomplete: it contains statements that are neither provably true nor provably false.
The existence of an incomplete system is in itself not particularly surprising. For example, if you take Euclidean geometry and you drop the parallel postulate, you get an incomplete system (in the sense that system does not contain all the true statements). An incomplete system can mean simply that you haven't discovered all the necessary axioms.
What Gödel showed is that in most cases, such as in number theory or real analysis, you can never discover the complete list of axioms. Each time you add a statement as an axiom, there will always be another statement out of reach.
You can add an infinite number of axioms; for example, you can add all true statements about the natural numbers to your list of axioms, but such a list will not be a recursive set. Given a random statement, there will be no way to know if it is an axiom of your system. If I give you a proof, in general there will be no way for you to check if that proof is valid.
Gödel's theorem has another interpretation in the language of computer science. In first-order logic, theorems are recursively enumerable: you can write a computer program that will eventually generate any valid proof. You can ask if they satisfy the stronger property of being recursive: can you write a computer program to definitively determine if a statement is true or false? Gödel's theorem says that in general you cannot.
Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's program towards a universal mathematical formalism. The generally agreed upon stance is that the second theorem is what specifically dealt this blow. However some believe it was the first, and others believe that neither did.
This is not actually what I was referencing, but interesting reading however.
What I was referencing was the fundamental logic error of a thing being used in defining itself.
The color blue can not be used to describe "BLUE", it must be defined fully in terms not including the thing to be defined (Circular Logic)
Hence, though we see the universe through the lens of reason,
the lens (Reason) cannot look at itself and define itself.
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