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ON THE INEVITABILITY OF THE CONSISTENCY OPERATOR

Published online by Cambridge University Press:  14 March 2019

ANTONIO MONTALBÁN  and
JAMES WALSH
ANTONIO MONTALBÁN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720, USAE-mail: [email protected]
JAMES WALSH
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720, USAE-mail: [email protected]
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Abstract

We examine recursive monotonic functions on the Lindenbaum algebra of $EA$. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and $\left( {\varphi \wedge Con\left( \varphi \right)} \right)$. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of $Con$ that bounds f everywhere, then f must be somewhere equal to an iterate of $Con$.

Keywords

03F3003F40consistencyreflection principles
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 205 - 225
Copyright
Copyright © The Association for Symbolic Logic 2019 

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