FOM: computable non-orderable abelian group?

Jeff Hirst jlh at
Wed Nov 7 15:49:02 EST 2001


V. Kanovei asked:

>is there a computable torsion-free abelian groop,
>which is not computably-orderable ?

Downey and Kurtz proved that there is a computable torsion
free abelian group with no computable full order in their
article "Recursion theory and ordered groups," in the
Annals of Pure and Applied Logic, vol 32 (1986), pp. 137-151.

A related result by Hatzikiriakou and Simpson shows that
over the weak base system RCA0, the system WKL0 is equivalent
to the statement "every countable torsion free abelian group
is an O-group (i.e. has a linear ordering)."

Both of these results are cited in Reed Solomon's nice
article "Reverse mathematics and fully ordered groups,"
with appeared in the Notre Dame Journal of Formal Logic,
vol 39, number 2, Spring 1998, pages 157-189.  (Reed also
included a number of new interesting results of his own.)


-Jeff Hirst
Jeff Hirst   jlh at
Professor of Mathematics
Appalachian State University, Boone, NC  28608
vox:828-262-2861    fax:828-265-8617

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