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In 1975 H. Friedman introduced two statements of hyperarithmetic analysis, SL0 (sequential limit system) and ABW0 (arithmetic BolzanoWeierstrass), which are motivated by standard and well-known theorems from analysis such as the Bolzano-Weierstrass theorem for Fσ and Gδ sets of reals. In this article we characterize the reverse mathematical strength of ABW0 by comparing it to most known theories of hyperarithmetic analysis. In particular we show that, over RCA0 + IΣ1, SL0 is equivalent to Σ 1 1 − AC0, and that ABW0 is implied by Σ1 − AC0, and implies weak− Σ1 − AC0. We then use Steel’s method of forcing with tagged trees to show that ABW0 is incomparable with INDEC (i.e. Jullien’s Theorem) and Δ1 − CA0. This makes ABW0 the first theory of hyperarithmetic analysis that is known to be incomparable with other (known) theories of hyperarithmetic analysis. We also examine the reverse mathematical strength of the Bolzano-Weierstrass theorem in the context of open, closed, Fσ , Gδ, and other types of sets.

Comparing theorems of hyperarithmetic analysis with the arithmetic Bolzano-Weierstrass theorem