Determinacy in strong cardinal models
نویسنده
چکیده
We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A Det(Π1-IND)⇒there exists an inner model with a strong cardinal. Theorem B Det(AQI)⇒there exist type-1 mice and hence inner models with proper classes of strong cardinals where Π1-IND (AQI) is the pointclass of boldface Π 1 1-inductive (respectively arithmetically quasi-inductive) sets of reals. §
منابع مشابه
Determinacy from Strong Compactness of Ω1
In the absence of the Axiom of Choice, the “small” cardinal ω1 can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that ω1 is Xstrongly compact (where X is any set) if there is a fine, countably complete measure on ℘ω1(X). Working in ZF + DC, we prove that the ℘(ω1)-strong compact...
متن کاملCalculus on strong partition cardinals
The assumption of [HM] and our assumption here, the existence of a strong partition cardinal, is moderately special. On the one hand, it violates the Axiom of Choice and is not relatively consistent with ZF (unlike AC and its negation). On the other hand, under the Axiom of Determinacy (AD), such cardinals are abundant and consistent with countable choice and DC, the principle of Dependent Choi...
متن کاملDeterminacy for games ending at the first admissible relative to the play
Let o(κ) denote the Mitchell order of κ. We show how to reduce long games which run to the first ordinal admissible in the play, to iteration games on models with a cardinal κ so that (1) κ is a limit of Woodin cardinals; and (2) o(κ) = κ. We use the reduction to derive several optimal determinacy results on games which run to the first admissible in the play. Given a set C ⊂ R1 consider the fo...
متن کاملUltrafilters and Large Cardinals
This paper is a survey of basic large cardinal notions, and applications of large cardinal ultrafilters in forcing. The main application presented is the consistent failure of the singular cardinals hypothesis. Other applications are mentioned that involve variants of Prikry forcing, over models of choice and models of determinacy. My talk at the Ultramath conference was about ultrafilters and ...
متن کاملThe Determinacy of Context-Free Games
We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by realtime 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic GaleStewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of ω-languages accepted by 1-counter Büchi automata is equivalent to the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Symb. Log.
دوره 76 شماره
صفحات -
تاریخ انتشار 2011