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Expressing cardinality quantifiers in monadic second-order logic over chains

Published online by Cambridge University Press:  12 March 2014

Vince Bárány,
Łukasz Kaiser and
Alexander Rabinovich
Vince Bárány
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Ox1 3QD, Oxford, UK, E-mail: vbarany@comlab.ox.ac.uk
Łukasz Kaiser
Affiliation:
Laboratoire d'Informatique Algorithmique: Fondements et Applications & CNRS, Université Paris Diderot, 75205 Paris, France, E-mail: kaiser@liafa.jussieu.fr
Alexander Rabinovich
Affiliation:
Tel Aviv University, The Blavatnik School of Computer Science, Ramat Aviv, Tel Aviv 69978, Israel, E-mail: rabinoa@post.tau.ac.il
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Abstract

We investigate the extension of monadic second-order logic of order with cardinality quantifiers “there exists uncountably many sets such that…” and “there exists continuum many sets such that … ”. We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders.

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011, pp. 603 - 619
Copyright
Copyright © Association for Symbolic Logic 2011

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References

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