Necessary use of induction in a reversal

Abstract

AbstractJullien'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.