Relative Consistency Examples

Set Theory · Axiom Academy

EXAMPLE A Relative Consistency Proof Prove that if ZF is consistent, then ZFC is also consistent using the constructible universe L. Excellent work! You've understood a relative consistency proof. Here's what we learned: Relative consistency: We can't prove ZFC is consistent in absolute terms, but we can show Con(ZF) → Con(ZFC) The constructible universe L: A model of ZF that can be built within any model of ZF using definability AC holds in L: L has a well-ordering of its entire universe, which implies the Axiom of Choice The conclusion: If ZF is consistent, it has a model; that model contains L; L satisfies ZFC; therefore ZFC is consistent This technique of relative consistency proofs is fundamental to understanding independence results in set theory!

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