From BolzanoWeierstraß to ArzelàAscoli
Abstract
We show how one can obtain solutions to the ArzelàAscoli theorem using suitable applications of the BolzanoWeierstraß principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of instances of the ArzelàAscoli theorem and a variant of it. Let AA be the statement that each equicontinuous sequence of functions f_n: [0,1] > [0,1] contains a subsequence that converges uniformly with the rate 2^k and let AA_weak be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that AA is instancewise equivalent over RCA_0 to the BolzanoWeierstraß principle BW and that AA_weak is instancewise equivalent over WKL_0 to BW_weak, and thus to the strong cohesive principle StCOH. Moreover, we show that over RCA_0 the principles AA_weak, BW_weak + WKL and StCOH + WKL are equivalent.
 Publication:

arXiv eprints
 Pub Date:
 May 2012
 arXiv:
 arXiv:1205.5429
 Bibcode:
 2012arXiv1205.5429K
 Keywords:

 Mathematics  Logic;
 03F60 (Primary) 03D80;
 03B30 (Secondary)
 EPrint:
 Mathematical Logic Quarterly, vol. 60 (2014), no. 3, pp. 177183