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June 2011 Necessary use of Σ¹₁ induction in a reversal
Itay Neeman
J. Symbolic Logic 76(2): 561-574 (June 2011). DOI: 10.2178/jsl/1305810764

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.

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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

Information

Published: June 2011
First available in Project Euclid: 19 May 2011

zbMATH: 1218.03005
MathSciNet: MR2830416
Digital Object Identifier: 10.2178/jsl/1305810764

Subjects:
Primary: 03E75, 03F35, 03H15, 3B30

Rights: Copyright © 2011 Association for Symbolic Logic

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Vol.76 • No. 2 • June 2011
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