- Author
- Andrés Cordón-Franco,
**David Fernández-Duque (UGent)**, Joost J Joosten and Francisco Félix Lara-Martín - Organization
- Abstract
- Let T be a second-order arithmetical theory, Lambda a well-order, lambda < Lambda and X subset of N. We use [lambda vertical bar X](T)(Lambda)phi as a formalization of "phi is provable from T and an oracle for the set X, using omega-rules of nesting depth at most lambda". For a set of formulas Gamma, define predicative oracle reflection for T over Gamma (Pred-O-RFNG(T)) to be the schema that asserts that, if X subset of N, Lambda is a well-order and phi is an element of Gamma, then for all lambda < Lambda ([lambda vertical bar X](T)(Lambda)phi -> phi ). In particular, define predicative oracle consistency (Pred-O-Cons(T)) as Pred-O-RFN.({0= 1}) (T). Our main result is as follows. Let ATR(0) be the second-order theory of Arithmetical Transfinite Recursion, RCA(0)(*) be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then, ATR(0) RCA(0)(*) + Pred-O-Cons(RCA(0)(*)) = RCA(0)(*) + Pred-O-RFN Pi 12 (ACA). We may even replace RCA(0)(*) by the weaker ECA(0), the second-order analogue of Elementary Arithmetic. Thus we characterize ATR(0), a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.
- Keywords
- second-order arithmetic, reflection principles, provability logic, PROVABILITY ALGEBRAS, PRINCIPLES, INDUCTION, SYSTEMS, ORDINALS, LOGIC

## Citation

Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-8566405

- MLA
- Cordón-Franco, Andrés et al. “Predicativity Through Transfinite Reflection.”
*JOURNAL OF SYMBOLIC LOGIC*82.3 (2017): 787–808. Print. - APA
- Cordón-Franco, A., Fernández-Duque, D., Joosten, J. J., & Lara-Martín, F. F. (2017). Predicativity through transfinite reflection.
*JOURNAL OF SYMBOLIC LOGIC*,*82*(3), 787–808. - Chicago author-date
- Cordón-Franco, Andrés, David Fernández-Duque, Joost J Joosten, and Francisco Félix Lara-Martín. 2017. “Predicativity Through Transfinite Reflection.”
*Journal of Symbolic Logic*82 (3): 787–808. - Chicago author-date (all authors)
- Cordón-Franco, Andrés, David Fernández-Duque, Joost J Joosten, and Francisco Félix Lara-Martín. 2017. “Predicativity Through Transfinite Reflection.”
*Journal of Symbolic Logic*82 (3): 787–808. - Vancouver
- 1.Cordón-Franco A, Fernández-Duque D, Joosten JJ, Lara-Martín FF. Predicativity through transfinite reflection. JOURNAL OF SYMBOLIC LOGIC. 2017;82(3):787–808.
- IEEE
- [1]A. Cordón-Franco, D. Fernández-Duque, J. J. Joosten, and F. F. Lara-Martín, “Predicativity through transfinite reflection,”
*JOURNAL OF SYMBOLIC LOGIC*, vol. 82, no. 3, pp. 787–808, 2017.

@article{8566405, abstract = {Let T be a second-order arithmetical theory, Lambda a well-order, lambda < Lambda and X subset of N. We use [lambda vertical bar X](T)(Lambda)phi as a formalization of "phi is provable from T and an oracle for the set X, using omega-rules of nesting depth at most lambda". For a set of formulas Gamma, define predicative oracle reflection for T over Gamma (Pred-O-RFNG(T)) to be the schema that asserts that, if X subset of N, Lambda is a well-order and phi is an element of Gamma, then for all lambda < Lambda ([lambda vertical bar X](T)(Lambda)phi -> phi ). In particular, define predicative oracle consistency (Pred-O-Cons(T)) as Pred-O-RFN.({0= 1}) (T). Our main result is as follows. Let ATR(0) be the second-order theory of Arithmetical Transfinite Recursion, RCA(0)(*) be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then, ATR(0) RCA(0)(*) + Pred-O-Cons(RCA(0)(*)) = RCA(0)(*) + Pred-O-RFN Pi 12 (ACA). We may even replace RCA(0)(*) by the weaker ECA(0), the second-order analogue of Elementary Arithmetic. Thus we characterize ATR(0), a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.}, author = {Cordón-Franco, Andrés and Fernández-Duque, David and Joosten, Joost J and Lara-Martín, Francisco Félix}, issn = {0022-4812}, journal = {JOURNAL OF SYMBOLIC LOGIC}, keywords = {second-order arithmetic,reflection principles,provability logic,PROVABILITY ALGEBRAS,PRINCIPLES,INDUCTION,SYSTEMS,ORDINALS,LOGIC}, language = {eng}, number = {3}, pages = {787--808}, title = {Predicativity through transfinite reflection}, url = {http://dx.doi.org/10.1017/jsl.2017.30}, volume = {82}, year = {2017}, }

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