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#consistency

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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 ›

13.4 Consistency Proofs

13.4 Consistency Proofs A formal system should avoid proving contradictions. Consistency is the property that a system does not collapse into proving everything. At the simplest level, a system is consistent if there is some formula $A$ such that the system does not prove both $A$ and $¬A$. More commonly, we say that a system is consistent if there is no formula $A$ such that: $$ \vdash A $$ and...
logic proof theory consistency foundations May 2, 2026
Maths › 03. Logic and Foundations ›

8.4 Consistency Results

A consistency result explains that one formal theory does not lead to contradiction, usually relative to the consistency of another theory. In set theory, such results are especially important because many central principles cannot be proved or refuted from the standard axioms alone. The typical form is a relative consistency statement. Instead of proving absolutely that a theory is consistent, one proves that if a base theory is consistent, then...
logic set theory consistency inner models constructibility May 2, 2026
Maths › 00. General Mathematics ›

02.3 Consistency and Completeness

2.3 Consistency and Completeness A formal system is judged partly by what it can prove and partly by what it avoids proving. Two central properties are consistency and completeness. Consistency means the system does not prove contradictions. A system is consistent if there is no statement P such that both P and ¬P are provable. not both: ⊢ P and ⊢ ¬P An inconsistent system is unusable for ordinary mathematics...
mathematics logic foundations consistency completeness May 2, 2026