Refuting Ehrenfeucht conjecture on rigid models
A. Ehrenfeucht conjectured that for countable first order theories \(T\) the set \(RS(T)=\{\lambda:T\) has a rigid model of cardinality \(\lambda\}\) is always convex, its lower bound is \(\omega\) or \(\omega_1\) and its upper bound \(\infty\) or smaller than \(\beth_{\omega_1}\). In the paper it is proved that for any \(\Sigma_2\)-sentence \(\chi\) of pure second order logic there is a first order sentence \(\psi\) such that \(RS(\psi)=\{\lambda:\lambda=\omega\) or \(\sum\limits_{\varkappa<\lambda}\varkappa^\omega=\lambda\) and \(\lambda\models\chi\}\). Thus Ehrenfeucht's conjecture is disproved.
scientific article; zbMATH DE number 3559564
scientific article