[FOM] links between nonstandard analysis and formal logic and set theory

Stephen G Simpson simpson at math.psu.edu
Mon Jan 16 11:17:27 EST 2006

Dana Scott asks:

 > What work in Reverse Mathematics has been done in connection with
 > Nonstandard Analysis?

Kazuyuki Tanaka has shown that certain methods of nonstandard analysis
can be accommodated within WKL_0, a relatively weak subsystem of
second-order arithmetic which plays a large role in reverse
mathematics.  See:

  K. Tanaka, The self-embedding theorem of WKL_0 and a non-standard
  method, Annals of Pure and Applied Logic 84, 1997, pp. 41-49.

There are also two recent papers by H. Jerome Keisler, a leading
expert and contributor to nonstandard analysis:

  H. J. Keisler, Nonstandard arithmetic and reverse mathematics, 2005,
  25 pages, Bulletin of Symbolic Logic, to appear.

  H. J. Keisler. The strength of nonstandard analysis, 2005, to

In the first of these papers, Keisler develops a close relationship of
mutual interpretability between (a) certain formal systems for
nonstandard analysis, (b) the so-called "big five" subsystems of
second-order arithmetic: RCA_0, WKL_0, ACA_0, ATR_0, Pi^1_1-CA_0.  The
"big five" subsystems are of great importance in reverse mathematics,
as explained in my book Subsystems of Second Order Arithmetic and in
some of my earlier papers.

Some relevant older papers concerning the relationship between
nonstandard analysis and reverse mathematics are:

  C. W. Henson, M. Kaufmann, and H. J. Keisler. The strength of
  nonstandard methods in arithmetic. Journal of Symbolic Logic, 49,
  1984, pp. 1039-1058.

  C. W. Henson and H. J. Keisler. On the strength of nonstandard
  analysis, Journal of Symbolic Logic 51, 1986, pp. 377-386.

as well as Tanaka's paper mentioned above.

By the way, my book Subsystems of Second Order Arithmetic remains the
standard reference on reverse mathematics.  The first edition,
published in 1999 by Springer-Verlag, sold out quickly, but the second
edition is in galley proofs and will be published in 2006 by the
Association for Symbolic Logic.

A recent advance in reverse mathematics is:

  Carl Mummert and Stephen G. Simpson, Reverse mathematics and Pi^1_2
  comprehension, Bulletin of Symbolic Logic, 11, 2005, pp. 526-533.

This paper (a) initiates the reverse mathematics of general topology,
(b) extends reverse mathematics far beyond the "big five", to Pi^1_2

Name: Stephen G. Simpson

Research interest: foundations of mathematics

Professional affiliation: Professor of Mathematics, Pennsylvania State

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