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Necessary use of \(\Sigma ^1_{1}\) induction in a reversal. (English) Zbl 1218.03005

In an earlier paper [J. Math. Log. 8, No. 1, 93–119 (2008; Zbl 1186.03071)], the author showed that, in \(\text{RCA}_0\)+\(\Sigma^1_1\)-induction, Jullien’s indecomposability theorem implies weak \(\Delta^1_1\)-choice. In the paper under review he provides a model which shows that it is not possible to obtain this result when weakening \(\Sigma^1_1\)-induction to \(\Delta^1_1\)-induction.

MSC:

03B30 Foundations of classical theories (including reverse mathematics)
03E75 Applications of set theory
03F35 Second- and higher-order arithmetic and fragments

Citations:

Zbl 1186.03071
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References:

[1] -determinacy, comprehension and induction 72 pp 452– (2007)
[2] Combinatorial principles weaker than Ramsey’s theorem for pairs 72 pp 171– (2007) · Zbl 1118.03055
[3] DOI: 10.1016/0003-4843(78)90026-8 · Zbl 0404.03020
[4] Subsystems of second order arithmetic (2009) · Zbl 1181.03001
[5] DOI: 10.1142/S0219061306000517 · Zbl 1105.03061
[6] DOI: 10.1002/malq.200710081 · Zbl 1172.03008
[7] Journal of Mathematical Logic 8 pp 93– (2009)
[8] DOI: 10.2178/bsl/1174668219 · Zbl 1129.03024
[9] Logic Colloquium ’80(Prague, 1980) 108 pp 239– (1982)
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