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A variant of Mathias forcing that preserves \(\mathsf{ACA}_0\). (English) Zbl 1269.03020
This paper formulates and applies \(F_\sigma\)-Mathias forcing, an adaptation of set-theoretic Mathias forcing specially devised to preserve properties of models of second-order arithmetic. For example, the author proves that if \(\mathcal N\) is a model of ACA\(_0\) and \(G\) is generic for \(F_\sigma\)-Mathias forcing, then \(\mathcal N [G]\) is also a model of ACA\(_0\). This statement also holds with ACA\(_0\) replaced by WKL\(_0\) plus induction for \(\Sigma^0_2\) formulas. The final section demonstrates the use of this forcing machinery in obtaining conservation and cone-avoidance theorems.
The article supplies a conceptual bridge from the set-theoretic treatment of Mathias forcing of J. E. Baumgartner [Lond. Math. Soc. Lect. Note Ser. 87, 1–59 (1983; Zbl 0524.03040)] and A. Blass [Ann. Pure Appl. Logic 109, No. 1–2, 77–88 (2001; Zbl 0980.03055)] to the computability-theoretic results of P. A. Cholak et al. [J. Symb. Log. 66, No. 1, 1–55 (2001; Zbl 0977.03033); corrigendum ibid. 74, No. 4, 1438–1439 (2009; Zbl 1182.03107)] and D. D. Dzhafarov and C. G. Jockusch jun. [J. Symb. Log. 74, No. 2, 557–578 (2009; Zbl 1166.03021)].

MSC:
03B30 Foundations of classical theories (including reverse mathematics)
03E40 Other aspects of forcing and Boolean-valued models
03F35 Second- and higher-order arithmetic and fragments
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