Slaman recently proved that Σₙ collection is provable from Δₙ induction plus exponentiation, partially answering a question of Paris. We give a new version of this proof for the case n = 1, which only requires the following very weak form of exponentiation: " exists for some y sufficiently large that x is smaller than some primitive recursive function of y".
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-6,
author = {Neil Thapen},
title = {A note on D1 induction and S1 collection},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {79-84},
zbl = {1082.03050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-6}
}
Neil Thapen. A note on Δ₁ induction and Σ₁ collection. Fundamenta Mathematicae, Tome 185 (2005) pp. 79-84. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-6/