9.º a 12.º gradoFoundationsPlanificado
Russell's paradox and why sets need axioms
Naive comprehension, the idea that every property defines a set, is inconsistent: the set of all sets that are not members of themselves cannot exist. Axiomatic set theory replaces comprehension with safer principles such as separation.
math.found.russell-paradox
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Se basa en
Set-builder notation, quantifiers and infinite setsProof: direct, contrapositive, contradiction and induction
Requisitos previos obligatorios, hasta el fondo: 10
Aprenderás a
- State the naive comprehension principle and derive Russell's contradiction from it.
- Explain how the separation axiom, which only forms {x ∈ A : P(x)} inside an existing set A, blocks the paradox, and deduce that there is no set of all sets.
- Distinguish naive set theory, a useful informal language, from axiomatic set theory, a precise theory.
Definición
- Naive comprehension: for every property P there is a set {x : P(x)}.
- Russell's set is R = {x : x ∉ x}.
- Separation: for every set A and property P, {x ∈ A : P(x)} is a set.
Teorema
- Naive comprehension is inconsistent: it gives R ∈ R if and only if R ∉ R.
- With separation, for any set A the set {x ∈ A : x ∉ x} is not an element of A; hence no set contains every set.
- Ordinary mathematics can be carried out in axiomatic set theories such as ZFC that avoid the paradox.
Errores frecuentes
- Thinking the paradox shows mathematics is inconsistent; it shows naive comprehension is.
- Treating Russell's paradox and the liar paradox as the same argument; they are analogous, not identical.
- Believing ZF avoids the paradox by banning x ∈ x. ZF does prove x ∉ x, but from the separate axiom of foundation; the paradox is blocked by restricting comprehension to separation.
Descubrimientos opcionales
- ZFC: axiomatic set theory Zermelo proposed axioms avoiding the paradox in 1908; with the axiom of replacement (Fraenkel, Skolem) and the axiom of choice these became ZFC.Sendero del Asombro →
- Type theory and propositions as types Russell's own response was a theory of types, which stratifies objects by type; modern type theories descend in part from this idea.
- ← Collections
- ← Sets
Lecturas complementarias
- Paul R. Halmos, Naive Set Theory (1960)
- Herbert B. Enderton, Elements of Set Theory (1977)