An honest question - what is the problem with inconsistency in using lambda calculus as a programming language and not as foundation of mathematics?
"""The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.[7][a] The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox.[8][9]"""
https://en.wikipedia.org/wiki/Lambda_calculus#History
By virtue of all programming being resource-bounded computations can be interpreted in Linear logic [1].
Linear logic is paraconsistent [2] not consistent.
Untyped Lambda calculus is essentially an assembly language and Assembly languages treat their own code as data [3]. Contradictions arrise when a system can mutate its own state. That is the power and pitfall of reflection [4]
"""The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.[7][a] The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox.[8][9]""" https://en.wikipedia.org/wiki/Lambda_calculus#History