Contingency

Contingency and the in-completibility of reality#

• Dummett's indefinitely extensible concepts is just Peirce's concept localized to mathematics, but with some hard arguments for its necessity given.
• Descriptive set theory illustrates how the continuum is inexhaustible: a full hierarchy of infinites is needed to decide subsets of reals on it. 
• This is best visualized by determinacy concretized in terms of games.
• Note that descriptive set theory, via Wadge hierarchy, it has connection to algorithmic complexity. Also it is the oldest branch of set theory; it subsumes under its area of study the origin of set theory. (Cantor's work on trigonometric series and derived sets.)
• Determinacy and game: links to constructivism via Samuel Abramsky's game semantics.
• Diagonal argument: this can certainly be refined or restructured to showcase the inexhaustibility of concepts. In set theory, it works best with Scott-Potter set theory, which makes explicit how the cumulative hierarchy is generated and is in the process of being generated, and that this process cannot halt unless the law of non-contradiction is abandoned.
• Contingency is not randomness neither unpredictability, but diagonal argument is ontic, in that logically it is inadmissible to have a fully actualized reality.

But these cannot "prove" contingency. The point is, still, as I've said, concretise self-realization in the semantic level, and by means of giving a working theory of truth and meaning building a robust ontology.