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A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC
- Part of
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- Journal:
- Bulletin of Symbolic Logic / Volume 28 / Issue 3 / September 2022
- Published online by Cambridge University Press:
- 09 June 2022, pp. 451-465
- Print publication:
- September 2022
-
- Article
- Export citation
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We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory
$T_0$
extending 
$\mathsf {RCA}_0$
and axiomatizable by a 
$\Pi ^1_{k+2}$
sentence, and for any 
$n\geq k+1$
, 
$$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}$$
where T is
$$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \end{align*}$$
$T_0$
augmented with full induction, and 
$\mathrm {TI}_{\varPi ^1_n}(\varepsilon _0)^{-}$
denotes the schema of transfinite induction up to 
$\varepsilon _0$
for 
$\varPi ^1_n$
formulas without set parameters.
