Reverse mathematics and the ACC

Stephen G. Simpson Department of Mathematics Pennsylvania State University http://www.math.psu.edu/simpson/ simpson@math.psu.edu

January 24, 2014

This is my abstract for Computability Theory and Foundations of Mathematics (CTFM 2014), Tokyo Institute of Technology, February 17–20, 2014.

In abstract algebra, a ring is said to satisfy the ACC (ascending chain condition) if it has no infinite ascending sequence of ideals. A famous theorem of Hilbert, 1890, says that polynomial rings with finitely many indeterminates satisfy the ACC. There is also a similar theorem for noncommuting indeterminates, due to J. C. Robson, 1978. In 1988 I performed a reverse-mathematical analysis of the theorems of Hilbert and Robson, proving that they are equivalent over RCA_{0} to the well-orderedness of ω^{ω} and ω^{ω^{ω}} respectively. Now I perform a similar analysis of a theorem of E. Formanek and J. Lawrence, 1976. Let S be the group of finitely supported permutations of the natural numbers. Let K[S] be the group ring of S over a countable field K of characteristic 0. Formanek and Lawrence proved that K[S] satisfies the ACC. I now prove that the Formanek/Lawrence theorem is equivalent over RCA_{0} to the well-orderedness of ω^{ω}. I also show that, in all of these reverse-mathematical results, RCA_{0} can be weakened to RCA_{0}^{∗}. This recent work was done jointly with Kostas Hatzikiriakou.

In addition, I make some remarks concerning reverse mathematics as it applies to Hilbert’s foundational program of finitistic reductionism. It is significant that RCA_{0} and WKL_{0} and even WKL_{0} + Σ_{2}^{0}-bounding are conservative for Π_{2}^{0} sentences over PRA, while Σ_{2}-induction and the well-orderedness of ω^{ω} are not.