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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: antonio@math.berkeley.edu
JAMES WALSH
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720, USAE-mail: walsh@math.berkeley.edu
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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

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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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References

REFERENCES

Andrews, U., Cai, M., Diamondstone, D., Lempp, S., and Miller, J. S., On the structure of the degrees of relative provability. Israel Journal of Mathematics, vol. 207 (2015), no. 1, pp. 449–478.10.1007/s11856-015-1182-8CrossRefGoogle Scholar
Beklemishev, L. D., Provability logics for natural turing progressions of arithmetical theories. Studia Logica, vol. 50 (1991), no. 1, pp. 107–128.10.1007/BF00370390CrossRefGoogle Scholar
Beklemishev, L. D., Iterated local reflection versus iterated consistency. Annals of Pure and Applied Logic, vol. 75 (1995), no. 1–2, pp. 25–48.10.1016/0168-0072(95)00007-4CrossRefGoogle Scholar
Beklemishev, L. D., Proof-theoretic analysis by iterated reflection. Archive for Mathematical Logic, vol. 42 (2003), no. 6, pp. 515–552.10.1007/s00153-002-0158-7CrossRefGoogle Scholar
Beklemishev, L. D., Provability algebras and proof-theoretic ordinals, I. Annals of Pure and Applied Logic, vol. 128 (2004), no. 1–3, pp. 103–123.10.1016/j.apal.2003.11.030CrossRefGoogle Scholar
Beklemishev, L. D., Reflection principles and provability algebras in formal arithmetic. Russian Mathematical Surveys, vol. 60 (2005), no. 2, p. 197.10.1070/RM2005v060n02ABEH000823CrossRefGoogle Scholar
Feferman, S. and Spector, C., Incompleteness along paths in progressions of theories, this Journal, vol. 27 (1962), no. 4, pp. 383–390.Google Scholar
Friedman, S.-D., Rathjen, M., and Weiermann, A., Slow consistency. Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 382–393.10.1016/j.apal.2012.11.009CrossRefGoogle Scholar
Harrison, J., Recursive pseudo-well-orderings. Transactions of the American Mathematical Society, vol. 131 (1968), no. 2, pp. 526–543.10.1090/S0002-9947-1968-0244049-7CrossRefGoogle Scholar
Joosten, J. J., Turing–Taylor expansions for arithmetic theories. Studia Logica, vol. 104 (2016), no. 6, pp. 1225–1243.10.1007/s11225-016-9674-zCrossRefGoogle Scholar
Pour-El, M. B. and Kripke, S., Deduction-preserving “recursive isomorphisms” between theories. Fundamenta Mathematicae, vol. 61 (1967), pp. 141–163.10.4064/fm-61-2-141-163CrossRefGoogle Scholar
Schmerl, U. R., A fine structure generated by reflection formulas over primitive recursive arithmetic. Studies in Logic and the Foundations of Mathematics, vol. 97 (1979), pp. 335–350.10.1016/S0049-237X(08)71633-1CrossRefGoogle Scholar
Shavrukov, V. Y. and Visser, A., Uniform density in Lindenbaum algebras. Notre Dame Journal of Formal Logic, vol. 55 (2014), no. 4, pp. 569–582.10.1215/00294527-2798754CrossRefGoogle Scholar
Slaman, T. A. and Steel, J. R., Definable functions on degrees, Cabal Seminar 81–85 (Kechris, A. S., Martin, D. A., and Steel, J. R., editors), Springer, Berlin, 1988, pp. 37–55.10.1007/BFb0084969CrossRefGoogle Scholar