FOM: Tanaka's papers on nonstandard analysis
Stephen G Simpson
simpson at math.psu.edu
Wed Nov 12 01:13:36 EST 1997
Apropos foundations of infinitesimals, I like Yuki Tanaka's recent
Kazuyuki Tanaka, Nonstandard analysis in WKL_0, Mathematical Logic
Quarterly, 1997, vol 43 no 3.
K. Tanaka, The self-embedding theorem of WKL_0 and a non-standard
method, Ann. Pure Appl. Logic vol 84, 1997, pp 41-49.
I'm basically an epsilon-delta man, but these Tanaka papers make
infinitesimals more attractive to me, in the following ways. 1. They
show that infinitesimals do not require ultrapowers or other
high-powered set-theoretic methods. Indeed, typical nonstandard
analysis arguments can be done elegantly in a weak system which is
conservative over primitive recursive arithmetic for Pi^0_2 sentences.
2. A canonical set of infinitesimals is obtained.
The papers by Sommer and Suppes also accomplish somewhat similar
things, with primitive recursive arithmetic replaced by elementary
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