Classification by formalization degree. The idea is: the more formal it becomes the easier it is to transition from “cheap talk” to paradoxes, from paradoxes to theorems, from theorems to theories.
Degrees:
FA) Unformalized Semantic / Natural Language Level: linguistic or conceptual paradoxes.
“Talk is cheap, it takes money to buy whiskey”.
Example: “This sentence is false”
- Problem: no stable semantics, unrestricted reference.
- Outcome: Paradoxes reveal flaws in ordinary language reasoning.
FB) Semi-formal Semantic Level (via labeling or name indirection)
Example: Label Q: “Sentence with label Q is false” Example: Cards paradox, Yablo’s paradox.
These paradoxes often involve a “shift” in referential layers — like using one level of labeling (e.g. “names,” descriptions) to talk about another. Delayed self-reference avoids immediate collapse.
FC) Naive Formal Level (unrestricted comprehension) [Naive formalization / naive predicates / set-theoretic self-ref]
Example: Russell’s paradox, Cantor’s Theorem, Formalized Grelling-Nelson (H(A) = ¬A(A), A<-H) etc.
Truth values exist, predicates behave “as expected”, unrestricted comprehension. The playground of classical diagonalization in naive systems.
FD) Arithmetized Self-Reference / Gödelian Level
Example: Gödel’s incompleteness, Tarski’s theorem
Self-reference is constructed, not assumed. The system is well-defined, and paradoxes become theorems (incompleteness, undefinability/Tarski).
FE?) Temporal / Dynamical Level (procedural truth evaluation)
Example: Halting problem, Kripke’s fixed point, Gupta/Belnap
What makes this interesting is that paradoxes dissolve when treated non-instantaneously. For example: - The Liar doesn’t collapse if truth evolves in stages. - Paradoxes of information (like self-prediction, Crocodile, Unexpected Hanging) can be tamed if we take into account that the future is possible and not necessary.
Arguably, this is not a formalization degree, but a switch to the temporal modality.
FF) Algorithmic Complexity Level
This adds quantitative constraints, and reveals that some truths can’t be compressed or proven if they exceed a system’s descriptive power.
Example: Chaitin’s incompleteness: a finite analog of Gödel; Berry paradox reinterpreted via descriptional length.
FG) Topological / Categorical Level (Lawvere, S4)
Example: Lawvere’s theorem abstracts diagonalization to categories, topology, etc.
Arguably, while other degrees [ except for the Temporal one ] add more and more precision, this one leads to better generalization.