Contradiction rules the world
-- Hegel, The Science of Logik.
Where all contradictions meet,
there is a complete man.
-- Osho.
In 1931, the mathematician Kurt Gödel demonstrated that every mathematical system (or set of axioms) contains undecidable propositions. And if the mathematical system is completely decidable, the the mathematical system is simply incomplete. In essence, Completeness & Consistency doesnot go hand in hand. A simplified outline of Godel's brilliant Incompleteness Theorem follows: -
Suppose we have an Universal Truth Machine (UTM) supposedly capable of correctly answering any question. Now consider the following proposition: "The UTM will never say that this sentence is true." Call this sentence G, that is:
G = "The UTM will never say that this sentence, G is true."
Now, what happens when we ask the UTM whether G is true?
If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact true. So G is true (since G = "UTM will never say G is true"). We have thus produced a true statement, which UTM cannot make.
UTM is not truly universal.
Thus,
every self-referential negation is a paradox, or rather, such things neither can be asserted nor can be denied. No wonder, Ludwig Wittgenstein had once said in his Tractacus Logico Philosophicus, - "Whatever can be said at all, can be said clearly, but what cannot be said, must be passed over in silence, for that lies in the range of non-sense". & Godel Incompletes infact pertains to this "range of non-sense", that is,
a recursive function of negation is undecidable, resulting in the non-halting of a UTM..
So, for a system is to be complete (i.e., all-in-all), it has to contain all possibilities, even that of the operation of negation, be it on itself, thus, resulting in such states which are in effect undecidable; now, those very undecidable states will make the system inconsistent. So,
a complete system is always inconsistent. And to maintain consistency, the system has to be made, by leaving out precisely those self-negation states, thus, making it incomplete. This,
a consistent system is always incomplete.
So, there is a tradeoff betwen Consistency & Completeness, & this ratio of one with the other has a connection with Chaitin's Constant, Omega. This puts a limit to how much we can know & infact when applied to such things as Consciouness (with the very concept of "I" or Self, as recursive) can result in remarkable & astonishing results.
Infact this negative-recursion (or a self-referential negation) is found in many other aspects, namely -- Nagarjuna's fourfold theory of logic in Buddhist Metaphysics, the Buddhist theory of Shunyata, Zeno's Paradoxes in Pre-Socraitic Greek Philosophy, the mathematics of Fractals, the Physics of Chaos Theory, the Nietzschean concept of Eternal Recurrence, the Existential concept of Nothingness (esp that of Sartre's), Escher's paintings, Zen Koans in perspective to Linguistics, the Neti-Neti approach of Vendanta (esp pertaining to Advaita), Feedback Systems in Learning Networks ( A.I.), the Big Bang Theory, the Triad Dialectic Theory of Hegel (even in the version used by Marx in his Historic Materialism), the Quantum nature of human consciousness, Computational Theory (including the study of Algorithims), the poetry of Blake, Whitman & many others, Lao Tzu's verses in "Tao Te Ching", the loopiness in Merkaba Geometry (theories relating to the Golden Mean Spiral, with connection of Fractals & Chaos Theory), the Zero-Book concept (like that of "Waiting For Godot"), etc etc.
If something is
alive, it has to have a
Self-contradiction in it.
Does the Self then really exist, or is it just the fabric of this negative-recursion talking?
Plz comment on this.
Kind Regards,
wM.
PS. Think of the "0-0" state of the NAND flip-flop in Electronic Digital Systems.