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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 ›

9.1 Forcing

Forcing is a method for constructing new models of set theory from old ones by adjoining new sets in a controlled way, and its main purpose is to show that certain statements cannot be decided from a given list of axioms, such as ZFC, provided those axioms are consistent. The guiding idea is that we begin with a model $M$ of set theory and a partially ordered set $\mathbb{P}$ whose...
logic set theory forcing independence May 2, 2026
Maths › 03. Logic and Foundations ›

8.5 Independence Phenomena

An independence result shows that a given statement cannot be proved or refuted from a given system of axioms, assuming that the system itself is consistent. In set theory, many natural and important statements turn out to be independent of the standard axioms. The study of independence phenomena reveals that the axioms of ZF and ZFC do not determine a single complete picture of the universe of sets. Instead, different...
logic set theory independence forcing constructibility May 2, 2026
Maths › 03. Logic and Foundations ›

8. Axiomatic Systems

Axiomatic set theory studies the principles that govern the universe of sets, and it asks which mathematical statements follow from the chosen axioms, which statements require additional assumptions, and which statements cannot be decided from the usual axioms alone. The chapter begins with the Axiom of Choice, which asserts that one may choose an element from each set in a family of nonempty sets, and this principle has many consequences...
logic set theory axioms choice independence May 2, 2026
Maths › 00. General Mathematics ›

02.4 Independence

2.4 Independence Phenomena A statement is independent of an axiom system when the system proves neither the statement nor its negation. Independence is a precise form of undecidability. T ⊬ φ T ⊬ ¬φ Here T is a theory and φ is a sentence in the language of that theory. The notation says that φ cannot be derived from T , and ¬φ cannot be derived from T either. Independence...
mathematics logic foundations independence axioms May 2, 2026