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Enriched stratified systems for the foundations of category
"... This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much ..."
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This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have
A NOTE ON STANDARD SYSTEMS AND ULTRAFILTERS
"... Let (M, X)  = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an endextension N  = T of M such that the subsets of ..."
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Let (M, X)  = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an endextension N  = T of M such that the subsets of M coded in N are precisely those in X. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T.
A NOTE ON STANDARD SYSTEMS AND ULTRAFILTERS
, 804
"... Abstract. Let (M, X)  = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an endextension N  = T of M such that the s ..."
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Abstract. Let (M, X)  = ACA0 be such that PX, the collection of all unbounded sets in X, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an endextension N  = T of M such that the subsets of M coded in N are precisely those in X. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T. The standard system of a model M of PA (the first order formulation of Peano arithmetic) is the collection of standard parts of the parameter definable subsets of M, i.e., sets of the form X ∩ ω, where X is a parameter definable set of M, and ω is the set of natural numbers. It turns out that the standard system tells you a lot about the model; for example, any two countable recursively saturated models of the same completion of PA with the same standard system are isomorphic. A natural question to ask is then which collections of subsets of the natural numbers are standard systems. This problem has become known as
NONPRINCIPAL ULTRAFILTERS, PROGRAM EXTRACTION AND HIGHER ORDER REVERSE MATHEMATICS
"... Abstract. We investigate the strength of the existence of a nonprincipal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a nonprincipal ultrafilter on N exists and let ACAω 0 be the higher order extension of ACA0. We show that ACAω 0 + (U) is Π1 2conservative ..."
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Abstract. We investigate the strength of the existence of a nonprincipal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a nonprincipal ultrafilter on N exists and let ACAω 0 be the higher order extension of ACA0. We show that ACAω 0 + (U) is Π1 2conservative over ACAω 0 and thus that ACAω 0 + (U) is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly Π1 2 statement ∀f ∃g Aqf(f, g) in ACAω 0 + (U) a realizing term in Gödel’s system T can be extracted. This means that one can extract a term t ∈ T, such that ∀f Aqf(f, t(f)). In this paper we will investigate the strength of the existence of a nonprincipal ultrafilter over fragments of higher order arithmetic. We will classify the consequences of this statement in the spirit of reverse mathematics. Furthermore, we will provide a program extraction method. Let (U) be the statement that a nonprincipal ultrafilter on N exists. Let RCA ω 0, ACA ω 0 be the extensions of RCA0 resp. ACA0 to higher order arithmetic as introduced