The proof, published by Kurt Gödel in 1931, of the existence of formally undecidable propositions in any formal system of arithmetic. More precisely, his first incompleteness theorem states that in any formal system S of arithmetic,there will be a sentence P of the language of S such that if S is consistent, neither P nor its negation can be proved in S. ... [The second incompleteness] theorem states that the consistency of a formal system of arithmetic cannot be proved by means formalizable within that system.
(Antony Flew: A Dictionary of Philosophy, St. Martin's Press 1979)

It [Goedel's Theorem] proves that there exist meaningful mathematical statements that are neither provable nor disprovable, now or ever - neither provable nor disprovable, that is, not simply because human thought or knowledge is insufficiently advanced but because the very nature of logic renders them incapable of resolution, no matter how long the human race survives or how wise it becomes.
(Alfred Adler in: The World Treasury of Physics, Astronomy, and Mathematics (edited by Timothy Ferris), 1991, p. 439)

The ordinary view of consciousness is, that it is local to every individual. If we take this as a fact, we will never be able to explain consciousness completely, because now we ran into Goedel's Theorem of the incompleteness of any self-referential system.
In a nutshell, Goedel's theorem states that for any formal system there are certain self-referencing assertions about the system that cannot be evaluated as either wholly true or false. They remain insoluble for our human reasoning. This paradox is originally attributed to the Cretan Epimenides who presented the statement "I am lying" as being undecidable concerning truth or falsity. If it is true, that I am lying, then the statement is false, and if it is false, that I am lying, then the statement is true. (see Essay Self-Referentiality of Reflective Thought)

see also Self-Referentiality.

See, thro' this air, this ocean, and this earth,
All matter quick, and bursting into birth.
Above, how high, progressive life may go!
Around how wide! how deep extend below!
Vast Chain of Being! which from God began,
Natures ethereal, human, Angel, Man,
Beast, bird, fish, insect; what no eye can see,
No glass can reach: from Infinite to thee,
From thee to Nothing! - On superior pow'rs
Were we to press, inferior might on ours;
Or in the full creation leave a void,
Where, one step broken, the great scale's destroy'd:
From Nature's Chain whatever link you strike,
Tenth or ten thousandth, breaks the chain alike.
(Alexander Pope: Essay on Man, Epistle 1)

The result was the conception of the plan and structure of the world, which, through the Middle Ages and down to the late eighteenth century, many philosophers, most men of science, and, indeed, most educated men, were to accept without question - the conception of the universe as a "Great Chain of Being," composed of an immense, or ... of an infinite, number of links ranging in hierarchical order from the meagerest kind of existents, which barely escape non-existence, through "every possible" grade up to the ens perfectissiumum - or, ... to the highest possible kind of creature, between which and the Absolute Being the disparity was assumed to be infinite - every one of them differing from that immediately above and that immediately below it by the "least possible" degree of difference.
(Arthur O. Lovejoy: The Great Chain of Being, Harvard University Press)