### Necessary use of Σ¹₁ induction in a reversal

Itay Neeman
Source: J. Symbolic Logic Volume 76, Issue 2 (2011), 561-574.

#### 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 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.

First Page:
Primary Subjects: 3B30, 03F35, 03H15, 03E75
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1305810764
Digital Object Identifier: doi:10.2178/jsl/1305810764
Mathematical Reviews number (MathSciNet): MR2830416

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