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Comparing hierarchies of total functionals
 Logical Methods in Computer Science
, 2005
"... In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem i ..."
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In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem is if these two hierarchies coincide. We will show that this coincidence problem is equivalent to the statement that the topology on the KleeneKreisel continuous functionals of a fixed type induced by all continuous functions into the reals is zerodimensional for each type. As a tool of independent interest, we will construct topological embeddings of the KleeneKreisel functionals into both the extensional and the intensional hierarchy at each type. The embeddings will be hierarchy embeddings as well in the sense that they are the inclusion maps at type 0 and respect application at higher types. 1
mlq header will be provided by the publisher On Local NonCompactness in Recursive Mathematics
"... theory ..."
Realizability Models Refuting Ishihara’s Boundedness Principle
, 2011
"... In [Ish92] H. Ishihara introduced the socalled boundedness principle BDN which claims that every countable pseudobounded subset of N is bounded. Here S ⊆ N is called pseudobounded iff for every sequence a ∈ SN there exists an n ∈ N such that ak < k for all k ≥ n. 1 Obviously, the principle BDN is ..."
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In [Ish92] H. Ishihara introduced the socalled boundedness principle BDN which claims that every countable pseudobounded subset of N is bounded. Here S ⊆ N is called pseudobounded iff for every sequence a ∈ SN there exists an n ∈ N such that ak < k for all k ≥ n. 1 Obviously, the principle BDN is
Realizability Models Refuting Ishihara’s Boundedness Principle
"... Ishihara’s Boundedness Principle BDN was introduced in [Ish92] and has turned out to be most useful for constructive analysis, see e.g. [Ish01]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We constr ..."
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Ishihara’s Boundedness Principle BDN was introduced in [Ish92] and has turned out to be most useful for constructive analysis, see e.g. [Ish01]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from N N to N is sequentially continuous and in which the axiom of choice from N N to N holds. Since the latter is known to be inconsistent with the statement that all functions from N N to N are continuous these models refute BDN.