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 ϕPu is the formula ϕ relativized to subsets of u.
(A6) Choice for sets.
We denote by B1 the theory with axioms (A1) to (A6).
The existence of weakly compact and -indescribable cardinals for every n is established in B1 by the method of defining all metamathematical concepts for B1 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 B1 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.
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