Here's how I visualize Goedel's incompleteness theorem (I'm not sure how "visual" this is, but bear with me): I imagine the Goedel construction over the axioms of first-order Peano arithmetic. Clearly, in the standard model, the Goedel sentence is true, so we add G to the axioms. Now we construct G' a Goedel sentence in this new set, and add G'' as an axiom. We go on and on, G''', etc. Luckily that construction is computable, so we add G^w as a Goedel sentence in this new set. We continue on and on, until we reach the first uncomputable countable ordinal, at which point we stop, because we have an uncomputable axiom set. Note that Goedel is fine with that -- you can have a complete first-order Peano arithmetic (it would have non-standard models, but it would be complete!) -- as long as you are willing to live with the fact that you cannot know if something is a proof or not with a mere machine (and yes, Virginia, humans are also mere machines).