Gödelian phenomenology is characterized by its incompleteness: The subject is "always" already there but never in its totality. The life we lead must be incomplete. It may be that only in death is totality achieved; unfortunately, the subject will never witness this. For death is the ultimate Godelian paradox.
Kurt Gödel was a relatively obscure mathematician who existed on the margins of the Vienna Circle until formulating one of the most important mathematical statements of all time (and consequently moving out of obscurity). This formulation (there were, in fact, two related theorems) challenged--well, refuted--accepted thinking about the consistency and completeness of mathematical systems. Also, and of much more interest to me, were the philosophical implications of Gödelian incompleteness.
The paradox. In general, a paradox exists when logic requires that the mind draw contradictory or inconsistent conclusions. Assumptions or axioms stipulated as true, when considered together, are logically inconsistent. Individually they make sense. Together they they make nonsense. Nonsensical, as I mean it, does not imply untruth. but merely a conclusion or conclusions that do not fit within the parameters of Aristotelian logic. Aristotelian logic is encapsulated by the syllogistic form. One syllogism with which many are familiar is:
All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.
Taken together, the major and minor premises require the conclusion. Syllogisms don't tell us what is true about the world, but only what conclusion must follow, given particular premises. In any case, Gödel revolutionized logic and the philosophy of mathematics (specifically work by Russell and Hilbert). I am neither a logician nor a mathematician and won't try to delve too deeply into those realms.
Gödel's 1st incompleteness theorem is akin to the liar's paradox, which is of the form "all Cretans are liars" and is attributed to Epimenides, himself a Cretan. If a Cretan uttered that sentence, then a paradox is suggested. If the sentence is true it is false, and if it is false it is true. Or take: "this very sentence is false." The paradox is clear (which makes it confusing!). As Rebecca Goldstein as written, Gödel essentially stated that "This very statement is not provable within this system."
My interest is in the implications of this paradox for subjective, conscious experience. That is, in a phenomenological application of Gödelian incompleteness. I will expound on this in subsequent entries.
[Note: My understanding of Gödel's work is indebted to Rebecca Goldstein's excellent Incompleteness: The Proof and Paradox of Gödel] http://www.rebeccagoldstein.com/books/incompleteness/index.html