The Godel statement represents itself, completely, via diagonalization.
Unfortunately I'm not equipped to discuss Godel in depth.
All I can do is argue by simple analogy, that is, it takes N>>1 neurons in the brain to mentally represent the idea of a neuron. Therefore the brain cannot represent itself.
Now it's true that the representation of the system can be reduced in complexity by good theories, but given our severe limitations in working memory and abstract spatial conceptualization both of which are necessary to understand a complex system, we are orders of magnitude from having the capacity to understand even a reduced version of ourselves.
In other words, any representation of the brain that is so reduced as to be singularly understandable by a single human will be so abstract as to have surrendered much of its explanatory power.
-Brad
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