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Topological aspects of suitable theories

Published online by Cambridge University Press:  20 January 2009

H. Simmons
H. Simmons
Affiliation:
University of Aberdeen
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Abstract

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Roughly speaking a suitable theory is a theory T together with its formal provability predicate Prv (.). A pseudo-topological space is a boolean algebra B which carries a derivative operation d and its associated closure operation c. Thus we can pretend that B is a topological space. We show that the Lindenbaum algebra B(T) of a suitable theory becomes, in a natural way, a pseudotopological space, and hence we can translate properties of T into topological language, as properties of B(T). We do this translation for several properties of T, including (1) satisfying Gödel's first theorem, (2) satisfying Löb's theorem and (3) asserting one's own inconsistency. These correspond to the topological properties (1) having an isolated point, (2) being scattered, (3) being discrete.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 4 , September 1975 , pp. 383 - 391
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Macintyre, A. and Simmons, H., Gödel's diagonalization technique and related properties of theories, Colloq. Math. 28 (1973), 165–180.Google Scholar
(2) Aull, C. E. and Thron, W. J., Separation axioms between T0 and T1, Indag. Math. 24 (1963), 26–37.Google Scholar