EDIT: the following is an edit to the below-mentioned scheme which was proven to be inconsistent by Andreas Blass [see comments], here is a salvage:
Hereditary comprehension schema: if ϕ(x,k)ϕ(x,k) is a formula in which both and only symbols ‘‘x",‘‘k"‘‘x",‘‘k" occur free, and those only occur free, and if ϕ(x,v)ϕ(x,v) is the formula obtained from ϕ(x,k)ϕ(x,k)by merely replacing all occurrences of the symbol ‘‘k"‘‘k" by the symbol ‘‘v"‘‘v", then
∀v[v≠∅∧∃k∀x(ξϕ(x,k)↔x∈v)→∃w∀x(ξϕ(x,v)→x∈w)]∀v[v≠∅∧∃k∀x(ξϕ(x,k)↔x∈v)→∃w∀x(ξϕ(x,v)→x∈w)];
is an axiom.
where ξϕ(x,a)⟺∃t[x∈t∧∀q∈t(q⊂t∧ϕ(q,a))]ξϕ(x,a)⟺∃t[x∈t∧∀q∈t(q⊂t∧ϕ(q,a))]
Prior post:
Hereditary comprehension schema: if ϕ(x,k)ϕ(x,k) is a formula in which both and only symbols ‘‘x",‘‘k"‘‘x",‘‘k" occur free, and those only occur free, and if ϕ(x,v)ϕ(x,v) is the formula obtained from ϕ(x,k)ϕ(x,k)by merely replacing all occurrences of the symbol ‘‘k"‘‘k" by the symbol ‘‘v"‘‘v", then
∀v[∃k∀x(ξϕ(x,k)→x∈v)→∃w∀x(ξϕ(x,v)→x∈w)]∀v[∃k∀x(ξϕ(x,k)→x∈v)→∃w∀x(ξϕ(x,v)→x∈w)];
is an axiom.
where ξϕ(x,a)⟺∃t[x∈t∧∀q∈t(q⊂t∧ϕ(q,a))]ξϕ(x,a)⟺∃t[x∈t∧∀q∈t(q⊂t∧ϕ(q,a))]
In English: ξξ here is taken to represent "hereditarily", so ξϕξϕ means 'hereditarily ϕϕ' so the schema reads:
"for any set vv if there exists a set kk such that all hereditarily ϕϕ objects with respect to parameter kk, are elements of vv, then there exists a set ww such that all objects that are hereditarily ϕϕ objects with respect to parameter vv, are elements of ww".
The crude idea is that a parameterized hereditarily property that can be contained in a set for some parameter, would also be set containable for all other parameters.
If it is consistent to add this to axioms of ZF-InfinityZF-Infinity, then this would prove the existence of inaccessibles. And I think this is the case even if we add it to Z-InfinityZ-Infinity. However I highly doubt its consistency.
Is there a counter-example involved with this axiom schema within the context of adding it to axioms of Z-InfinityZ-Infinity?