Abstract
Using the techniques of reverse mathematics, we analyze the logical strength of statements similar to trichotomy and dichotomy for sequences of reals. Capitalizing on the connection between sequential statements and constructivity, we find computable restrictions of the statements for sequences and constructive restrictions of the original principles. 2010 Mathematics Subject Classification 03B30, 03F35 (primary); 03F60, 30F50 (secondary)
Highlights
We will examine several statements about real numbers and sequences of real numbers in the framework of reverse mathematics and in some formalizations of weak constructive analysis
We will concentrate on the axiom systems RCA0, WKL0, and ACA0, which are described in detail by Simpson [8]
ACA0 is strictly stronger than WKL0, and WKL0 is strictly stronger than RCA0
Summary
We will examine several statements about real numbers and sequences of real numbers in the framework of reverse mathematics and in some formalizations of weak constructive analysis. We will use quantifier free choice, QF–AC0,0 , which is defined by the same scheme but where x and y are restricted to natural number variables and A is restricted to quantifier free formulas. These systems and many others are treated in detail by Kohlenbach [7]; a short summary appears in the paper of Hirst and Mummert [5]. Since α(k) and β(k) are rationals, this formalization of equality contains only the leading universal quantifier Because we consider both classical and intuitionistic systems, we need to be especially careful in defining inequality for reals. Α is non-negative means ¬(α < 0), which is equivalent to α ≥ 0, which is equivalent to ∀k(α(k) ≥ −2−k+1)
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