Abstract

In [1] and [2] there is a development of a class theory, whose axioms were formulated by Bernays and based on a reflection principle. See [3]. These axioms are formulated in first order logic with ∈:Extensionality.Class specification. Ifϕis a formula andAis not free inϕ, thenNote that “xis a set“ can be written as “∃u”.Subsets.Note also that “B⊆A” can be written as “∀x”.Reflection principle. Ifϕis a formula, thenwhere “uis a transitive set” is the formula “∃v ∧ ∀x∀y” andϕPuis the formulaϕrelativized to subsets ofu.Foundation.Choice for sets.We denote byB1the theory with axioms to.The existence of weakly compact and-indescribable cardinals for everynis established inB1by the method of defining all metamathematical concepts forB1in a weaker theory of classes where the natural numbers can be defined and using the reflection principle to reflect the satisfaction relation; see [1]. There is a proof of the consistency ofB1assuming the existence of a measurable cardinal; see [4] and [5]. In [6] 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.