Source: J. Symbolic Logic
Volume 76, Issue 2
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
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