Necessary use of Σ¹₁ induction in a reversal

Jullien’s indecomposability theorem 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 Σ 1 1 induction, it was shown by Neeman to have strength strictly between weak Σ 1 choice and ∆ 1 comprehension. We prove in this paper that Σ 1 induction is needed for the reversal of INDEC, that is for the proof that INDEC implies weak Σ 1 choice. This is in contrast with the typical situation in reverse mathematics, where reversals can usually be refined to use only Σ 1 induction. §