proof of Fodor’s lemma

If we let f-1:κP(S) be the inversePlanetmathPlanetmathPlanetmath of f restricted to S then Fodor’s lemma is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the claim that for any function such that αf(κ)α>κ there is some αS such that f-1(α) is stationary.

Then if Fodor’s lemma is false, for every αS there is some club set Cα such that Cαf-1(α)=. Let C=Δα<κCα. The club sets are closed under diagonal intersection, so C is also club and therefore there is some αSC. Then αCβ for each β<α, and so there can be no β<α such that αf-1(β), so f(α)α, a contradictionMathworldPlanetmathPlanetmath.

Title proof of Fodor’s lemma
Canonical name ProofOfFodorsLemma
Date of creation 2013-03-22 12:53:19
Last modified on 2013-03-22 12:53:19
Owner Henry (455)
Last modified by Henry (455)
Numerical id 4
Author Henry (455)
Entry type Proof
Classification msc 03E10