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Higher order reflection principles

Published online by Cambridge University Press:  12 March 2014

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

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(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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1]Chuaqui, R., Bernays' class theory, Mathematical logic: proceedings of the first Brazilian conference (Campinas, 1977; Arruda, A. I.et al., editors), Lecture Notes in Pure and Applied Mathematics, vol. 39, Marcel Dekker, New York, 1978, pp. 3155.Google Scholar
[2]Chuaqui, R., Axiomatic set theory. Impredicative theory of classes, North-Holland, Amsterdam, 1981.Google Scholar
[3]Bernays, P., On the problem of schemata of infinity in axiomatic set theory, Sets and classes. On the work of Paul Bernays (Müller, G. H., editor), North-Holland, Amsterdam, 1976, pp. 121172.CrossRefGoogle Scholar
[4]Solovay, R., Reinhardt, W. and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.CrossRefGoogle Scholar
[5]Hanf, W. and Scott, D., Classifying inaccessible cardinals, Notices of the American Mathematical Society, vol. 8 (1961), p. 445 (abstract 61T-240).Google Scholar
[6]Gloede, K., Reflection principles and indescribability, Sets and classes. On the work of Paul Bernays (Müller, G. H., editor), North-Holland, Amsterdam, 1976, pp. 277323.Google Scholar
[7]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Higher set theory (proceedings, Oberwolfach, 1977; Müller, G. H. and Scott, D. S., editors), Lecture Notes in Mathematics, vol. 669, Springer-Verlag, Berlin, 1978, pp. 99275.Google Scholar