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# PREDICATIVITY THROUGH TRANSFINITE REFLECTION

Abstract

Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.

For a set of formulas Γ, define predicative oracle reflection for T over Γ (Pred–O–RFNΓ(T)) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then

$$\forall \,\lambda < {\rm{\Lambda }}\,([\lambda |X]_T^{\rm{\Lambda }}\varphi \to \varphi ).$$

In particular, define predicative oracle consistency (Pred–O–Cons(T)) as Pred–O–RFN{0=1}(T).

Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,

$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left( {{\rm{ACA}}} \right).$$

We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic.

Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.

References
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[1] Beeson, M. and Ščedrov, A., Church’s thesis, continuity, and set theory, this Journal, vol. 49 (1984), no. 2, pp. 630–643.
[2] Beklemishev, L. D., Induction rules, reflection principles, and provably recursive functions. Annals of Pure and Applied Logic, vol. 85 (1997), pp. 193242.
[3] Beklemishev, L. D., Provability algebras and proof-theoretic ordinals, I. Annals of Pure and Applied Logic, vol. 128 (2004), pp. 103124.
[4] Beklemishev, L. D., Veblen hierarchy in the context of provability algebras, Logic, Methodology and Philosophy of Science, Proceedings of the Twelfth International Congress (Hájek, P., Valdés-Villanueva, L., and Westerståhl, D., editors), Kings College Publications, London, 2005, pp. 6578.
[5] Beklemishev, L. D., Reflection principles and provability algebras in formal arithmetic. Russian Mathematical Surveys, vol. 60 (2005), pp. 197268.
[6] Beklemishev, L. D., The Worm principle, Logic Colloquium 2002 (Chatzidakis, Z., Koepke, P., and Pohlers, W., editors), Lecture Notes in Logic 27, ASL Publications, 2006, pp. 7595.
[7] Beklemishev, L. D., On the reduction property for GLP-algebras. Doklady: Mathematics, vol. 472 (2017), no. 4.
[8] Beklemishev, L. D., Fernández-Duque, D., and Joosten, J. J., On provability logics with linearly ordered modalities. Studia Logica, vol. 102 (2014), pp. 541566.
[9] Beklemishev, L. D. and Onoprienko, A. A., On some slowly terminating term rewriting systems. Sbornik: Mathematics, vol. 206 (2015), no. 9, pp. 11731190.
[10] Boolos, G. S., The Logic of Provability, Cambridge University Press, Cambridge, 1993.
[11] Cordón-Franco, A., Fernández-Margarit, A., and Lara-Martín, F. F., Fragments of Arithmetic and true sentences. Mathematical Logic Quarterly, vol. 51 (2005), pp. 313328.
[12] Feferman, S., Systems of predicative analysis, this Journal, vol. 29 (1964), pp. 1–30.
[13] Feferman, S., Systems of predicative analysis II, this JOURNAL, vol. 33 (1968), pp. 193–220.
[14] Fernández-Duque, D., The polytopologies of transfinite provability logic. Archive for Mathematical Logic, vol. 53 (2014), no. 3–4, pp. 385431.
[15] Fernández-Duque, D. and Joosten, J. J., Models of transfinite provability logics, this Journal, vol. 78 (2013), no. 2, pp. 543–561.
[16] Fernández-Duque, D. and Joosten, J. J., The omega-rule interpretation of transfinite provability logic, (2013), arXiv, vol. 1205.2036 [math.LO].
[17] Fernández-Duque, D. and Joosten, J. J., Well-orders in the transfinite Japaridze algebra. Logic Journal of the Interest Group in Pure and Applied Logic, vol. 22 (2014), no. 6, pp. 933963.
[18] Hájek, P. and Pudlák, P., Metamathematics of First Order Arithmetic, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
[19] Hirst, J. L., A survey of the reverse mathematics of ordinal arithmetic, Reverse mMathematics 2001, Lecture Notes in Logic, vol. 21, Peters, A. K., Natick, MA, 2005, pp. 222234.
[20] Ignatiev, K. N., On strong provability predicates and the associated modal logics, this Journal, vol. 58 (1993), pp. 249–290.
[21] Japaridze, G., The polymodal provability logic, Intensional Logics and Logical Structure of Theories: Material from the Fourth Soviet-Finnish Symposium on Logic, Metsniereba, Telaviv, 1988, pp. 1648, In Russian.
[22] Joosten, J. J., . ${\rm{\Pi }}_1^0$ -ordinal analysis beyond first-order arithmetic. Mathematical Communications, vol. 18 (2013), pp. 109121.
[23] Kreisel, G. and Lévy, A., Reflection principles and their use for establishing the complexity of axiomatic systems. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 97142.
[24] Leivant, D., The optimality of induction as an axiomatization of arithmetic, this Journal, vol. 48 (1983), pp. 182–184.
[25] Schmerl, U. R., A fine structure generated by reflection formulas over primitive recursive arithmetic, Logic Colloquium ’78 (Mons, 1978) (Boffa, M., Dalen, D., and Mcaloon, K., editors), Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam, 1979, pp. 335350.
[26] Simpson, S. G., Friedman’s research on subsystems of second-order arithmetic, Harvey Friedman’s Research in the Foundations of Mathematics (Harrington, L., Morley, M., Ščedrov, A, and Simpson, S. G., editors), North-Holland, Amsterdam, 1985, pp. 137159.
[27] Simpson, S. G., Subsystems of Second Order Arithmetic, Cambridge University Press, New York, 2009.
[28] Simpson, S. G. and Smith, R. L., Factorization of polynomials and ${\rm{\Sigma }}_1^0$ induction. Annals of Pure and Applied Logic, vol. 31 (1986), pp. 289306.
[29] Tait, W., Finitism. Journal of Philosophy, vol. 78 (1981), pp. 524546.
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