In the present paper we introduce a constructive theory of nonstandard arithmetic in higher types. The theory is intended as a framework for developing elementary nonstandard analysis constructively. More specifically, the theory introduced is a conservative extension of HA” + AC. A predicate for distinguishing standard objects is added as in Nelson’s internal set theory. Weak transfer and idea...

This paper gives an overview of linear programming methods for solving standard and nonstandard Markovian control problems. Standard problems are problems with the usual criteria such as expected total (discounted) rewards and average expected rewards; we also discuss a l~articular class of stochastic games. In nonstandard problems there are additional considerations as side constraints, multip...

In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman’s lemma and show that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a ...

We show that each of the five basic theories of second order arithmetic that play a central role in reverse mathematics has a natural counterpart in the language of nonstandard arithmetic. In the earlier paper [HKK1984] we introduced saturation principles in nonstandard arithmetic which are equivalent in strength to strong choice axioms in second order arithmetic. This paper studies principles ...

Limits can be placed on nonstandard neutrino physics when big bang nucleosynthesis (BBN) calculations employing standard neutrino physics agree with the observationally inferred primordial abundances of deuterium (D), He, He, and Li. These constraints depend most sensitively on the abundances of D and He. New observational determinations of the primordial D and/or He abundances could force revi...

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak Kö...

There are various ways to evaluate the Generative Lexicon (GL). One is to see to what extent it accounts for what we find in text corpora. This has not previously been done, and this chapter presents a first foray. The experiment looks at the “nonstandard” uses of words found in a sample of corpus data: “nonstandard” is defined as not matching a literal reading of any of the word’s dictionary d...

This article describes a nonstandard quantum group that may be used to derive a positive formula for the plethysm problem, just as the standard (Drinfeld-Jimbo) quantum group can be used to derive the positive Littlewood-Richardson rule for arbitrary complex semisimple Lie groups. The sequel [GCT8] gives conjecturally correct algorithms to construct canonical bases of the coordinate rings of th...

We present a search for the pair production of a narrow nonstandard-model strongly interacting particle that decays to a pair of quarks or gluons, leading to a final state with four hadronic jets. We consider both nonresonant production via an intermediate gluon as well as resonant production via a distinct nonstandard-model intermediate strongly interacting particle. We use data collected by t...

The bounded functional interpretation of arithmetic in all finite types is able to interpret principles like weak König’s lemma without the need of any form of bar recursion. This interpretation requires the use of intensional (rule-governed) majorizability relations. This is a somewhat unusual feature. The main purpose of this paper is to show that if the base domain of the natural numbers is ...