59 research outputs found
Extended use of IST
AbstractVan den Berg, I.P., Extended use of IST, Annals of Pure and Applied Logic 58 (1992) 73–92. Internal Set Theory is an axiomatic approach to nonstandard analysis, consisting of three axiom schemes, Transfer (T), Idealization (I), and Standardization (S). We show that the range of application of these axiom schemes may be enlarged with respect to the original formulation. Not only more kinds of formulas are allowed, but also different settings. Many examples illustrate these extensions. Most concern formal aspects of nonstandard asymptotics
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom
Taxonomies of Model-theoretically Defined Topological Properties
A topological classification scheme consists of two ingredients: (1) an abstract class K of topological spaces; and (2) a taxonomy , i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class.K, is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed by the taxonomy. A space X in K is characterized within the classification scheme if whenever Y E K, and Y is equivalent to X, then Y is homeomorphic to X. As prime example, the closed set taxonomy assigns to each sentence in the first order language of bounded lattices the class of topological spaces whose lattices of closed sets satisfy that sentence. It turns out that every compact two-complex is characterized via this taxonomy in the class of metrizable spaces, but that no infinite discrete space is so characterized. We investigate various natural classification schemes, compare them, and look into the question of which spaces can and cannot be characterized within them
Topologizing Rings of Map Germs: An Order Theoretic Analysis of Germs via Nonstandard Methods
Using nonstandard analysis we define a topology on the ring of germs of
functions: . We prove that this topology
is absolutely convex, Hausdorff, that convergent nets of continuous germs have
continuous germs as limits and that, for continuous germs, ring operations and
compositions are continuous. This topology is not first countable, and, in
fact, we prove that no good first countable topology exists. We give a spectrum
of standard working descriptions for this topology. Finally, we identify this
topological ring as a generalized metric space and examine some consequences
Regularity and nearness theorems for families of local Lie groups
In this work, we prove three types of results with the strategy that,
together, the author believes these should imply the local version of Hilbert's
Fifth problem. In a separate development, we construct a nontrivial topology
for rings of map germs on Euclidean spaces. First, we develop a framework for
the theory of (local) nonstandard Lie groups and within that framework prove a
nonstandard result that implies that a family of local Lie groups that converge
in a pointwise sense must then differentiability converge, up to coordinate
change, to an analytic local Lie group, see corollary 6.1. The second result
essentially says that a pair of mappings that almost satisfy the properties
defining a local Lie group must have a local Lie group nearby, see proposition
7.1. Pairing the above two results, we get the principal standard consequence
of the above work, corollary 7.2, which can be roughly described as follows. If
we have pointwise equicontinuous family of mapping pairs (potential local
Euclidean topological group structures), pointwise approximating a (possibly
differentiably unbounded) family of differentiable (suffi- ciently approximate)
almost groups, then the original family has, after appropriate coordinate
change, a local Lie group as a limit point. The third set of results give
nonstandard renditions of equicontinuity criteria for families of
differentiable functions, see theorem 9.1. These results are critical in the
proofs of the principal results of this thesis as well as the standard
interpretations of the main results here. Following this material, we have a
long chapter constructing a Hausdorff topology on the ring of real valued map
germs on Euclidean space. This topology has good properties with respect to
convergence and composition. See the detailed introduction to this chapter for
the motivation and description of this topology
Discrete-time machines in closed monoidal categories. I
This paper develops a minimal realization theory for discrete-time machines with structure in a suitable closed monoidal category. By specifying the category a number of applications arise, most of them new. Minimal realization is stated as an adjunction between an input-output behavior functor and a realization functor. The very existence of an adjunction yields several new structural results on minimal realization. As preliminaries, certain aspects of categorical algebra are reviewed, and a theory of discrete-time transition systems is developed. The concept of an X-module and an initial object theorem are especially important. A number of examples of suitable categories is given, but discussion of the resulting machine theories is deferred to a subsequent paper
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