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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 › 03. Logic and Foundations ›

7.5 ZF and ZFC

Set theory becomes a foundation for mathematics only after its basic principles are stated as axioms. Informal set language is useful, but it can lead to contradictions if every definable collection is allowed to be a set. The Zermelo Fraenkel axioms provide a controlled formal framework in which sets may be built, compared, and used without allowing unrestricted comprehension. Motivation A naive view says that every collection described by a...
logic set theory ZF ZFC axioms 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