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PREDICATIVITY THROUGH TRANSFINITE REFLECTION

Published online by Cambridge University Press:  08 September 2017

ANDRÉS CORDÓN-FRANCO ,
DAVID FERNÁNDEZ-DUQUE ,
JOOST J. JOOSTEN  and
FRANCISCO FÉLIX LARA-MARTÍN
ANDRÉS CORDÓN-FRANCO
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE AND ARTIFICIAL INTELLIGENCE UNIVERSIDAD DE SEVILLA SEVILLE, SPAINE-mail:acordon@us.es
DAVID FERNÁNDEZ-DUQUE
Affiliation:
CENTRE INTERNATIONAL DE MATHÉMATIQUES ET D’INFORMATIQUE UNIVERSITY OF TOULOUSE TOULOUSE, FRANCE and DEPARTMENT OF MATHEMATICS INSTITUTO TECNOLÓGICO AUTÓNOMO DE MÉXICO, MEXICO MEXICO CITY, MEXICOE-mail:david.fernandez@irit.fr
JOOST J. JOOSTEN
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BARCELONA BARCELONA, SPAINE-mail:jjoosten@ub.edu
FRANCISCO FÉLIX LARA-MARTÍN
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE AND ARTIFICIAL INTELLIGENCE, UNIVERSIDAD DE SEVILLA SEVILLE, SPAINE-mail:fflara@us.es
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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.

Keywords

03F4597E3003F03second-order arithmeticreflection principlesprovability logic
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 787 - 808
Copyright
Copyright © The Association for Symbolic Logic 2017 

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