We introduce the notion of C-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties. NotationAs in common set-theoretic use, trcl(x), rank(x) denote respectively the transitive closure and the rank of a given set x. We denote by V α the sets x such that rank(x) < α and by H κ the sets x such that |trcl(x)| < κ. We use P(x), [x] κ , [x] <κ to denote the powerset, the set of subsets of size κ and the ones of size less than κ. The notation f : A → B is improperly used to denote partial functions in A × B, A B to denote the collection of all such (partial) functions, and f [A] to denote the pointwise image of A through f . We denote by id : V → V the identity map on V .We say that I ⊆ P(X) is an ideal on X whenever it is closed under unions and subsets, and feel free to confuse an ideal with its dual filter when clear from the context. We denote the collection of I-positive sets by I + = P(X) \ I.We follow Jech's approach [11] to forcing via boolean valued models. The letters B, C, D, . . . are used for set sized complete boolean algebras, and 0, 1 denote their minimal and maximal element. We use V B for the boolean valued model obtained from V and B,ẋ for the elements (names) of V B ,x for the canonical name for a set x ∈ V in the boolean valued model V B , φ B for the truth value of the formula φ.When convenient we also use the generic filters approach to forcing. The letters G, H will be used for generic filters over V ,Ġ B denotes the canonical name for the generic filter for B, val G (ẋ) the valuation map on names by the generic filter G, V [G] the generic extension of V by G. Given a set x and a model M , we denote by M [x] the smallest model of ZFC including M and containing x. Let φ be a formula. We write V B |= φ to denote that φ holds in all generic extensions V [G] with G generic for B. B/I denotes the quotient of a boolean algebra B by the ideal I, B * Ċ denotes the two-step iteration intended as the collection of B-names for elements ofĊ modulo equivalence with boolean value 1. References text for the results mentioned above are [11,19,20].Coll(κ, <λ) is the Lévy collapse that generically adds a surjective function from κ to any γ < λ. In general we shall feel free to confuse a partial order with its boolean completion. When we believe that this convention may generate misunderstandings we shall be explicitly more careful.Given an elementary embedding j : V → M , we use crit(j) to denote the critical point of j. We denote by SkH M (X) the Skolem Hull of the set X in the structure M . Our reference text for large cardinals is [13] and for generic elementary embeddings is [8].1. (Filter property) For all a ∈ V λ , F a is a non trivial filter on P(a); (Compatibility) For all
The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of k-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number k ≥ 2, Ramsey's Theorem for pairs and recursive assignments of k colors is equivalent to the Limited Lesser Principle of Omniscience for Σ 0 3 formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite k-ary tree there is some i < k and some branch with infinitely many children of index i.
In [12], Schwichtenberg showed that the System T definable functionals are closed under a rulelike version Spector's bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector's bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg's original proof, however, relies on a detour through Tait's infinitary terms and the correspondence between ordinal recursion for α < ε0 and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into T-definitions under the conditions of Schwichtenberg's theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment Ti then terms built from BR N,σ for this particular Y are definable in the fragment T i+max{1,level(σ)}+2 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.