Godel's first Theorem states that,
In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers, one can construct a statement that can be neither proved nor disproved within that system.
Here a formal system conforms to first-order logic, which permits the formulation of quantified statements such as " there is at least one X such that..." as opposed to higher-order logic which supports statements of the type "For All X, ......".
And a formal system is said to be consistent if none of its proven theorems can also be disproven within that system. Therefore a system is consistent if it lacks contradiction.
So a crude rewording of Godel, goes something like "In a system which does not contradict itself, it is possible to construct a statement(Not a theorem) which is neither provable nor disprovable." Ofcourse, for this statement to be invoked The system has to be a formal system conforming to first order logic capable of defining natural numbers. Rephrasing again, we can take it that in such a system, there will always be some statement which we cannot determine the truthfulness of.
Godel's second theorem states that,
Any consistent system cannot be used to prove its own consistency.
Rephrase: Take a system such as the one described above; Now take that systems rules and axioms. Godel tells us that, by working with those alone we cannot prove that the system does not contradict itself.
So Godel's first tells us that given a specific type of system we cannot assign a truth value to everything in that system. Godel's second tells us that, given such a system we cannot vouch for the non-contradictory nature of the said system by appealing only to elements contained within the system.
See, not only is the application of Godel limited, Godel does NOT apply to the completeness of systems.Completeness is another story altogether. And Godel does not preclude the existence of complete systems. What Godel questions is, our ability to claim the consistency of such systems, i.e The Hilbert Program and hence quite a huge chunk, of science.
So the argument put forth was, in a nutshell, pure bollocks.Kudos to forge for sorting me out.
It is so easy to stray <sigh > The temptation is just too much. See some vague familiarity in a situation and label it with some theory or the other, slap! bang! whammo! without taking any care to notice the real question at hand nor the applicability of the theory.
I have done penance.
urped by gumz @ 7:03 PM
