FOM: Tanaka's papers on nonstandard analysis

Stephen G Simpson simpson at
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
recursive arithmetic.

-- Steve

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