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Maths › 03. Logic and Foundations ›

14.5 Extensions

14.5 Extensions and Refinements Gödel’s original theorems admit stronger forms and alternative formulations. These refinements clarify exactly which assumptions are needed and how incompleteness behaves across different systems. Rosser’s Improvement Gödel’s first theorem used a strengthened consistency assumption. Rosser showed that this can be weakened. Instead of the original Gödel sentence, Rosser constructs a sentence $R$ that balances proofs of a statement and its negation. Informally, $R$ says: “if there...
logic godel proof theory computability foundations 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