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Higher order reflection principles

  • M. Victoria Marshall R. (a1)

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 ∈:

(A1) Extensionality.

(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(xu)”.

(A3) Subsets.

Note also that “BA” can be written as “∀x(xBxA)”.

(A4) Reflection principle. If ϕ(x) is a formula, then

where “u is a transitive set” is the formula “∃v(uv) ∧ ∀xy(xyyuxu)” and ϕPu is the formula ϕ relativized to subsets of u.

(A5) Foundation.

(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 [1]. There is a proof of the consistency of B1 assuming 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.

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[3] P. Bernays , On the problem of schemata of infinity in axiomatic set theory, Sets and classes. On the work of Paul Bernays ( G. H. Müller , editor), North-Holland, Amsterdam, 1976, pp. 121172.

[4] R. Solovay , W. Reinhardt and A. Kanamori , Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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