A model of second-order arithmetic satisfying AC but not DC
2019 ◽
Vol 19
(01)
◽
pp. 1850013
◽
Keyword(s):
We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 (1979)].
Keyword(s):
Keyword(s):
1993 ◽
Vol 62
(1)
◽
pp. 51-64
◽
2010 ◽
Vol 16
(3)
◽
pp. 378-402
◽