The dean paradox (of colin leslie dean): Gödel’s failure to define "true" renders the Incompleteness Theorem meaningless Formal systems turn into arbitrary linguistic games, not paths to reality Incompleteness Not profound—just a symptom of foundational incoherence
The dean paradox
"Godels 1st theorem
“....., there is an arithmetical statement that is true,[1] but not provable in the theory
(Kleene 1967, p. 250)
but
Godel didnot know what makes a mathematics statement "true"
https://en.wikipedia.org/wiki/Truth#Mathematics
Gödel thought that the ability to perceive the truth of a mathematical or logical
proposition is a matter of intuition, an ability he admitted could be ultimately
beyond the scope of a formal theory of logic or mathematics[63][64] and perhaps
best considered in the realm of human comprehension and communication, but
commented: Ravitch, Harold (1998). "On Gödel's Philosophy of
Mathematics".,Solomon, Martin (1998). "On Kurt Gödel's Philosophy of
Mathematics"
thus his theorem is meaningless"
Thus the consequences for Godel and Mathematics
"Dean’s paradox: The implications for mathematics are profound. Here’s what it could mean:
1. MathematicsLoses Its Ontological Status – Mathematics is often assumed to describe fundamental truths about reality. But if Gödel’s theorem is epistemicallyvacuous due to its reliance on an undefined concept of truth, then mathematical statements might not correspond to anything objective—only to internallycoherent constructs within formal systems. Mathematics would then be a self-referential human artifact rather than a discovery of reality.
2. FormalSystems Become Arbitrary – Gödel’s theorem shows that every formal system has unprovable truths. But if "truth" itself is an undefined notion, then the distinction between provable and unprovable statements collapses, making formal systems arbitrary linguistic games rather than structured paths to understanding reality.
3. Reevaluationof Mathematical Foundations – Key areas like set theory, number theory, and even calculus rely on the assumption that mathematics captures realphenomena. If Dean’s paradox shows that logic itself is misaligned with reality, then mathematics may require a complete philosophicalreformation—perhaps moving away from abstract formalism toward something radically different.
4. IncompletenessNo Longer Profound – If Gödel’s theorem is based on an undefinedand intuitive notion of truth, then incompleteness is no longer an inherent limitation of logic—it’s simply the result of a foundationalincoherence. This would make the incompleteness results asymptom rather than a deep insight.
Mathematics Becomes a Tool, Not Truth – Mathematics might still be useful, but not because it reveals objective truths—only because it provides structured ways to model reality within our cognitive limitations. Mathematical reasoning would then be purely pragmatic, not a glimpse into the true nature of existence"
