tag

#incompleteness

5 pages tagged

Maths › 03. Logic and Foundations ›

24. Limits of Formal Systems

Limits of formal systems study the boundaries of what can be proved, computed, and expressed within precise mathematical frameworks, and they show that even very strong systems have inherent restrictions that cannot be removed by technical refinement alone. The chapter begins with incompleteness, where certain true statements cannot be derived from a given formal system, provided that the system is sufficiently expressive and consistent, and this reveals a gap between...
logic foundations incompleteness undecidability independence May 2, 2026
Maths › 03. Logic and Foundations ›

14.4 Implications

14.4 Implications for Formal Systems Gödel’s theorems change how we understand formal systems. They do not weaken logic. They clarify its limits and force a more precise view of what formal reasoning can achieve. Separation of Truth and Provability Before Gödel, it was natural to expect that every true mathematical statement could be proved. The incompleteness theorem shows this expectation fails for sufficiently strong systems. There exist sentences $A$ such...
logic foundations incompleteness philosophy mathematics May 2, 2026
Maths › 03. Logic and Foundations ›

14.2 First Incompleteness

14.2 First Incompleteness Theorem The goal is to construct a sentence that talks about its own provability. This requires combining arithmetization with a method for controlled self-reference. Representing Provability From the previous section, we have an arithmetic relation: $$ \mathrm{Proof}(x,y) $$ which means “$x$ codes a valid proof of the formula with code $y$”. Using this, define the provability predicate: $$ \mathrm{Prov}(y) ;\equiv; \exists x, \mathrm{Proof}(x,y) $$ So: $$ \mathrm{Prov}(\ulcorner...
logic godel incompleteness diagonalization proof theory May 2, 2026
Maths › 03. Logic and Foundations ›

14.3 Second Incompleteness

14.3 Second Incompleteness Theorem The first incompleteness theorem shows that some true statements cannot be proved. The second goes further. It shows that a system strong enough to express arithmetic cannot prove its own consistency. Formalizing Consistency A system is consistent if it does not prove a contradiction. For example, it does not prove both $A$ and $¬A$. Using arithmetization, we express this inside arithmetic. Let: $$ \mathrm{Prov}(y) $$ mean...
logic godel consistency incompleteness foundations May 2, 2026
Maths › 03. Logic and Foundations ›

14. Godel Theorems

Godel's incompleteness theorems show that sufficiently expressive formal systems have inherent limits, especially when they are able to represent basic arithmetic. The chapter begins with arithmetization of syntax, where formulas, proofs, and derivations are encoded as natural numbers, allowing a formal system to speak about its own syntactic objects. The first incompleteness theorem is then introduced, showing that any consistent and sufficiently strong formal system contains true arithmetic statements that...
logic proof theory godel theorems incompleteness formal systems May 2, 2026