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Necessary use of induction in a reversal

Published online by Cambridge University Press:  12 March 2014

Itay Neeman
Itay Neeman*
Affiliation:
Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095-1555, USA, E-mail: ineeman@math.ucla.edu
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Abstract

Jullien's indecomposability theorem (INDEC) states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base system RCA0 plus induction, it was shown by Neeman to have strength strictly between weak choice and comprehension. We prove in this paper that induction is needed for the reversal of INDEC. that is for the proof that INDEC implies weak choice. This is in contrast with the typical situation in reverse mathematics, where reversals can usually be refined to use only induction.

Keywords

Reverse Mathematicsinductionweak choicelinear orderingsscatteredindecomposable
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 561 - 574
Copyright
Copyright © Association for Symbolic Logic 2011

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