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Higher order reflection principles

Published online by Cambridge University Press:  12 March 2014

M. Victoria Marshall R.
M. Victoria Marshall R.*
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile
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Extract

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(x ∈ u)”.

(A3) Subsets.

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.

(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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 474 - 489
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

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