The foundations of mathematics destroyed: Tarski Godel ZFC-by colin leslie dean
1)Tarski never gets to define truth 2) Godel is logically invalid as the axiom he uses bans his G statement which is used to prove his theorem 3)ZFC bans itself and allows what it bans internally inconsistent
take
1) Tarski never gets to define truthean argues that Tarski’s semantic theory of truth collapses due to infinite regress. Tarski requires a metalanguage to define truth for an object language, but that metalanguage in turn needs its own metalanguage, and so on indefinitely. Thus, truth is never actually defined—every effort to formally anchor “truth” leads to an endless regress, leaving mathematical truth fundamentally undefined in formal terms
2)A) Godel “Any effectively generated theory capable of expressing elementary arithmetic cannot beboth consistent and complete. In particular, for any consistent, effectively generated formaltheorythat proves certain basic arithmetic truths, there is an arithmetical statement that istrue,[1] but not provable in the theory (Kleene 1967, p. 250) but Godel cant tell us what makes a mathematical statement true,thus his theorem is meaningless
B) Gödel is logically invalid—the axiom he uses bans his G statementean claims Gödel's incompleteness theorems are built on a logical error. Gödel uses the G-statement (“G cannot be proved to be true within theory T”), which is impredicative—it makes reference to the totality it belongs to. Dean asserts that Gödel's proof relies on the axiom of reducibility in the original Principia Mathematica—but this very axiom was introduced to ban impredicative statements like the G-statement. Therefore, Gödel's logical framework bans the key statement needed for his proof, rendering the theorem self-defeating and logically invalid by its own foundational rules
3) A) ZFC bans itself: the axiom of separation bans impredicative statements yet is impredicative itself-thus it bans itself
B) ZFC allows what it bans—internally inconsistentean points out a structural contradiction in the axiom schema of separation (or specification) in ZFC set theory. While the axiom was intended to prohibit impredicative set definitions (to avoid paradoxes like Russell’s), it actually permits impredicative formulas, since the defining property φ(x) can quantify over all sets, possibly including the set being defined. This means ZFC both claims to ban impredicativity and simultaneously allows it, creating a formal inconsistency within its own foundational structure.
Dean argues that this is not a superficial problem, but a catastrophic collapse—ZFC’s attempts at resolving paradox actually recreate the conditions for paradox.Dean's overall position is that these foundational systems (Tarski, Gödel, ZFC) are not merely flawed on the margins; they are fundamentally self-contradictory and collapse under close philosophical scrutiny.
This suggests that mathematics, logic, and formal semantics are incoherent as systems of “truth”, and should be regarded as pragmatic tools for survival and utility rather than as vehicles of absolute knowledge.the foundations of mathematics—logic (via the Tarski semantic theory of truth), Gödel’s incompleteness, and ZFC set theory—collapse, then mathematics loses its claim to epistemic certainty, coherence, and truth.
Key consequences: End of mathematical certainty: If logic itself is flawed, there is no reliable basis for mathematical proof, consistency, or truth. The formal systems of mathematics become "painted veils"—useful but ultimately ungrounded constructs.
Epistemological crisis: All knowledge built upon logical or formal reasoning—including mathematics, science, and philosophy—becomes questionable and unstable. We can no longer trust proofs, axioms, or definitions to represent true or coherent knowledge.
Mathematics as fiction: Mathematical structures and proofs are reduced to pragmatic tools, valuable only for practical utility rather than as genuine representations of truth. Mathematics becomes "dead" as a discipline of knowledge, surviving only as a "useful fiction".
Collapse of other fields: Since physics, engineering, computer science, logic, and even ethical frameworks are rooted in mathematics, their rational foundations fail as well, leading to a profound intellectual void.No reliable authority or model: There are no universally trusted rules for reasoning, calculation, or systemic inquiry. Attempts to repair or rebuild the foundations (via new axioms, meta-languages, or empirical tweaks) would inevitably founder on the same paradox.
Philosophical implicationean's thesis implies the "death of reason" for all intellectual systems, far more radical than any previous philosophical skepticism—it claims that human logical faculties (“monkey-brain” logic) cannot capture the structure of reality.
All efforts at knowledge—rationalism, empiricism, metaphysics—fail to escape the foundational contradiction exposed by the Dean paradox.Thus, if Dean is right, mathematics (and all knowledge systems based on logic) must be seen as provisional, ultimately meaningless structures that do not guarantee truth or coherence, but only serve pragmatic and survivalistic functions in human life.
see
The Collapse of Mathematical Foundations: How Dean’s Paradox Exposes the Incoherence of Logic, Gödel, ZFC, and Truth (Tarski)
ean argues that Tarski’s semantic theory of truth collapses due to infinite regress. Tarski requires a metalanguage to define truth for an object language, but that metalanguage in turn needs its own metalanguage, and so on indefinitely. Thus, truth is never actually defined—every effort to formally anchor “truth” leads to an endless regress, leaving mathematical truth fundamentally undefined in formal terms
