Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile
In  and  there is a development of a class theory, whose axioms were formulated by Bernays and based on a reflection principle. See . These axioms are formulated in first order logic with ∈:
(A2) Class specification. If ϕ is a formula and A is not free in ϕ, then
Note that “x is a set“ can be written as “∃u(x ∈ u)”.
Note also that “B ⊆ A” can be written as “∀x(x ∈ B → x ∈ A)”.
(A4) Reflection principle. If ϕ(x) is a formula, then
where “u is a transitive set” is the formula “∃v(u ∈ v) ∧ ∀x∀y(x ∈ y ∧ y ∈ u → x ∈ u)” and ϕ Pu is the formula ϕ relativized to subsets of u.
(A6) Choice for sets.
We denote by B 1 the theory with axioms (A1) to (A6).
The existence of weakly compact and -indescribable cardinals for every n is established in B 1 by the method of defining all metamathematical concepts for B 1 in a weaker theory of classes where the natural numbers can be defined and using the reflection principle to reflect the satisfaction relation; see . There is a proof of the consistency of B 1 assuming the existence of a measurable cardinal; see  and . In  several set and class theories with reflection principles are developed. In them, the existence of inaccessible cardinals and some kinds of indescribable cardinals can be proved; and also there is a generalization of indescribability for higher-order languages using only class parameters.
The purpose of this work is to develop higher order reflection principles, including higher order parameters, in order to obtain other large cardinals.
(Received December 05 1987)