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CA0 and order types of countable ordered groups

  • Reed Solomon (a1)

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Reverse mathematics uses subsystems of second order arithmetic to determine which set existence axioms are required to prove particular theorems. Surprisingly, almost every theorem studied is either provable in RCA0 or equivalent over RCA0 to one of four other subsystems: WKL0, ACA0, ATR0 or CA0. Of these subsystems, CA0 has the fewest known equivalences. This article presents a new equivalence of C0 which comes from ordered group theory.

One of the fundamental problems about ordered groups is to classify all possible orders for various classes of orderable groups. In general, this problem is extremely difficult to solve. Mal'tsev [1949] solved a related problem by showing that the order type of a countable ordered group is ℤαε where ℤ is the order type of the integers, ℚ is the order type of the rationals, α is a countable ordinal, and ε is either 0 or 1. The goal of this article is to prove that this theorem is equivalent over RCA0 to CA0.

In Section 2, we give the basic definitions and notation for RCA0, ACA0 and CA0 as well as for ordered groups. For more information on reverse mathematics, see Friedman, Simpson, and Smith [1983] or Simpson [1999] and for ordered groups, see Kokorin and Kopytov [1974] or Fuchs [1963]. Our notation will follow these sources. In Section 3, we show that CA0 suffices to prove Mal'tsev's Theorem and the reversal is done over RCA0 in Section 4.

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Kokorin, A.I. and Kopytov, V.M. [1974], Fully ordered groups, Halsted Press.
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