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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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##### References:
 [1] Baumgartner, J.E.: Iterated forcing. In: Surveys in Set Theory, London Mathematical Society Lecture Note Series, vol. 87. Cambridge University Press, Cambridge, pp. 1–59 (1983) [2] Blass A.R.: Needed reals and recursion in generic reals. Ann. Pure Appl. Logic (Dedicated to Petr Vopěnka) 109(1–2), 77–88 (2001) · Zbl 0980.03055 · doi:10.1016/S0168-0072(01)00043-4 [3] Cholak P.A., Jockusch C.G. Jr., Slaman T.A.: On the strength of Ramsey’s theorem for pairs. J. Symb. Logic. 66(1), 1–55 (2001) · Zbl 0977.03033 · doi:10.2307/2694910 [4] Daguenet, M.: Propriété de Baire de $$\beta$$ N muni d’une nouvelle topologie et application àla construction d’ultrafiltres. Séminaire Choquet, 14e année (1974/75), Initiation à l’analyse, exp. no. 14. Secrétariat Mathématique, Paris, p. 3 (1975) [5] Dzhafarov D.D., Jockusch C.G. Jr.: Ramsey’s theorem and cone avoidance. J. Symb. Logic. 74(2), 557–578 (2009) · Zbl 1166.03021 · doi:10.2178/jsl/1243948327 [6] Ellentuck E.: A new proof that analytic sets are Ramsey. J. Symb. Logic 39, 163–165 (1974) · Zbl 0292.02054 · doi:10.2307/2272356 [7] Hirschfeldt D.R., Shore R.A.: Combinatorial principles weaker than Ramsey’s theorem for pairs. J. Symb. Logic 72(1), 171–206 (2007) · Zbl 1118.03055 · doi:10.2178/jsl/1174668391 [8] Kohlenbach, U.: Higher order reverse mathematics. In: Reverse Mathematics 2001, Lecture Notes Logic, vol. 21. Association for Symbolic Logic, La Jolla, CA, pp. 281–295 (2005) · Zbl 1097.03053 [9] Mazur K.: $${F_$$\backslash$$sigma}$$ -ideals and $${$$\backslash$$omega_1$$\backslash$$omega_1\^*}$$ -gaps in the Boolean algebras P($$\omega$$)/I. Fund. Math. 138(2), 103–111 (1991) [10] Seetapun D., Slaman T.A.: On the strength of Ramsey’s theorem. Notre Dame J. Formal Logic (Special issue: Models of arithmetic) 36(4), 570–582 (1995) · Zbl 0843.03034 · doi:10.1305/ndjfl/1040136917 [11] Simpson S.G.: Subsystems of Second Order Arithmetic, 2nd edn. Perspectives in Logic. Cambridge University Press, Cambridge (2009)
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