We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain A, a codomain B and a "filter" T on finite approximations of functions from A to B, a generalised form GDC(A,B,T) of the axiom of dependent choice and dually a generalised bar induction principle GBI(A,B,T) such that: - GDC(A,B,T) intuitionistically captures the strength of • the general axiom of choice expressed as ∀a ∃b R(a, b) ⇒ ∃α ∀a R(a, α(a))) when T is a filter that derives point-wise from a relation R on A × B without introducing further constraints, • the Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set Bool (for a constructive definition of prime filter), • the axiom of dependent choice if A = ℕ, • Weak König’s Lemma if A = ℕ and B = Bool (up to weak classical reasoning) - GBI(A,B,T) intuitionistically captures the strength of • Gödel’s completeness theorem in the form validity implies provability for entailment relations if B = Bool, • bar induction when A = ℕ, • the Weak Fan Theorem when A = ℕ and B = Bool. Contrastingly, even though GDC(A,B,T) and GBI(A,B,T) smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is Bool^ℕ and B is ℕ.