α β-Relations and the Actual Meaning of α-Renaming

In this work we provide an alternative, and equivalent, formulation of the concept of λ-theory without introducing the notion of substitution and the sets of all, free and bound variables occurring in a term. We call α β-relations our alternative versions of λ-theories. We also clarify the actual role of α-renaming in the lambda calculus: it expresses a property of extensionality for a certain class of terms. To motivate the necessity of α-renaming, we construct an unusual denotational model of the lambda calculus that validates all structural and beta conditions but not α-renaming. The article also has a survey character.