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$$\text{PA} (aa)$$. (English) Zbl 0848.03018
$$\text{PA} (aa)$$ is Peano Arithmetic formulated in stationary logic. It is not at all obvious that $$\text{PA} (aa)$$ is consistent. The author shows that it is. He proves that the first-order consequences of $$\text{PA}(aa)+$$ the scheme of determinateness for stationary logic are the same as those of the full second-order arithmetic CA. Two main results are: (1) Every model of $$\text{PA} (aa)$$ is $$\omega_1$$-like; (2) Every countable model $$(M,X)$$ of CA can be elementarily extended to an $$\omega_1$$-like finitely determinate model of $$\text{PA} (aa)$$.
##### MSC:
 03C62 Models of arithmetic and set theory 03F35 Second- and higher-order arithmetic and fragments
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##### References:
 [1] Barwise, J., M. Kaufmann, and M. Makkai, “Stationary logic,” Annals of Mathematical Logic , vol. 13 (1978), pp. 171–224; (correction in Annals of Mathematical Logic , vol. 20 (1981), pp. 231–232). · Zbl 0372.02031 [2] Eklof, P., and A. Mekler, “Stationary logic of finitely determinate structures,” Annals of Mathematical Logic , vol. 17 (1979), pp. 227–270. · Zbl 0448.03025 [3] Gaifman, H., “Models and types of Peano’s arithmetic,” Annals of Mathematical Logic , vol. 9 (1976), pp. 223–306. · Zbl 0332.02058 [4] Kakuda, Y., “The role of a filter quantifier in set theory,” pp. 16–95 in Mathematical Logic and Applications , Lecture Notes in Mathematics vol. 1388, edited by J. Shinoda et al., Springer-Verlag, Berlin, 1989. · Zbl 0673.03044 [5] Kaufmann, M., Some results in stationary logic , PhD. dissertation, University of Wisconsin, 1978. · Zbl 0402.03001 [6] Kaufmann, M., “Set theory with a filter quantifier,” The Journal of Symbolic Logic , vol. 48 (1983), pp. 263–287. JSTOR: · Zbl 0518.03007 [7] Macintyre, A., “Ramsey quantifiers in arithmetic,” pp. 186–210 in Model Theory of Algebra and Arithmetic , Lecture Notes in Mathematics vol. 834, edited by L. Pacholski et al., Springer-Verlag, Berlin, 1980. · Zbl 0464.03031 [8] Mills, G., “Substructure lattices of models of arithmetic,” Annals of Mathematical Logic , vol. 16 (1979), pp. 145–180. · Zbl 0427.03057 [9] Morgenstern, C., “On generalized quantifiers in arithmetic,” The Journal of Symbolic Logic , vol. 47 (1982), pp. 187–190. JSTOR: · Zbl 0487.03019 [10] Schmerl, J. H., “Peano arithmetic and hyper-Ramsey logic,” Transactions of the American Mathematical Society , vol. 296 (1986), pp. 481–505. JSTOR: · Zbl 0626.03030 [11] Schmerl, J. H., and S. G. Simpson, “On the role of the Ramsey quantifiers in first-order arithmetic,” The Journal of Symbolic Logic , vol. 47 (1982), pp. 423–,435. JSTOR: · Zbl 0492.03015 [12] Shelah, S., “Generalized quantifiers and compact logic,” Transactions of the American Mathematical Society , vol. 204 (1975), pp. 342–364. JSTOR: · Zbl 0322.02010
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