Buchholzの導出可能性条件
References
Def:
$ \mathbf{B_m} \colon T \vdash \bigwedge_{0<i<m} \varphi_i \to \varphi_m \implies S \vdash \bigwedge_{0<i<m} \mathrm{Pr}_T(\ulcorner \varphi_i \urcorner) \to \mathrm{Pr}_T(\ulcorner \varphi_m \urcorner)
ただし
$ m \ge 1
$ \varphi_1,\dots,\varphi_mは任意の論理式とする.
$ Tは$ Sの拡大理論
remark:
the formula $ \Phi(x) is intended to denote some provability predicate of $ T. However, we deal with more general situations, that is,$ \Phi(x) may not be any provability predicate of $ T
3. $ \bf B3 \implies \bf D2
4. 次は同値.
a. $ \bf D1, D2
b. 任意の$ m \ge 1について$ \mathbf{B_m}
c. $ \bf D1と$ m \ge 3について$ \bf B_m
d. $ \bf \Delta_0 Cと$ m \ge 3について$ \bf B_m