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 Σ¹₁ induction, it was shown by Neeman to have strength strictly between weak Σ¹₁ choice and Δ¹₁ comprehension. We prove in this paper that Σ¹₁ induction is needed for the reversal of INDEC, that is for the proof that INDEC implies weak Σ¹₁ choice. This is in contrast with the typical situation in reverse mathematics, where reversals can usually be refined to use only Σ⁰₁ induction.
Itay Neeman. "Necessary use of Σ¹₁ induction in a reversal." J. Symbolic Logic 76 (2) 561 - 574, June 2011. https://doi.org/10.2178/jsl/1305810764