Open AccessOn the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness
Author(s) -
Ulrich Kohlenbach
Publication year - 1997
Publication title -
brics report series
Language(s) - English
Resource type - Journals
eISSN - 1601-5355
pISSN - 0909-0878
DOI - 10.7146/brics.v4i23.18949
Subject(s) - arithmetic function , mathematics , mathematical proof , monotone polygon , omega , function (biology) , arithmetic , class (philosophy) , base (topology) , discrete mathematics , pure mathematics , computer science , mathematical analysis , artificial intelligence , geometry , evolutionary biology , biology , physics , quantum mechanics
In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems Tn^omega in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in Tn^omega -proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As corollaries we obtain new conservation results for fragments of analysis over fragments of arithmetic which strengthen known purely first-order conservation results.