Implementing sets in $\lambda$-calculus Both Set theory and $\lambda$-calculus are considered to be valid foundations for mathematics. Since these are both equivalent (in the sense that any structure that can be implemented in set theory can be implemented in $\lambda$-calculus, and vice-versa), is there a way to implement sets in $\lambda$-calculus?
I am sorry to inform you that the lambda calculus (Church, 1932) as a foundation of mathematics was shown to be logically inconsistent (Kleene and Rosser, 1935). Therefore your question is moot.
Sets can be implemented as lists in $\lambda$-calculus. The order of the elements in the list can be ignored by allowing only one type of function on the list - a membership test. All lists that produce the same results on performing membership tests are equivalent.