Reverse Mathematics is a program in foundations of mathematics initiated by Friedman ([1,2]) and developed extensively by Simpson ([4]). Its aim is to determine which minimal axioms prove theorems of ordinary mathematics. Nonstandard Analysis plays an important role in this program ([3, 5]). We consider Reverse Mathematics where equality is replaced by the predicate ≈, i.e. equality up to infin...

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...

The sheaves over the category of lters, with a joint cover topology, serve as a universe of sets where nonstandard analysis can be developed along constructive principles. In this paper we show that all functions between internal sets in this model are internal. We also illustrate how this model can be used by giving a constructive nonstandard proof of the Implicit Function Theorem.

A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched. §

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...

Kohlenbach’s proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel’s pioneering work on the so-called unwinding of proofs. The proof mining of classical mathematics is rather restricted in scope due to the existence of sentences without computational content which are provable from the law of excluded mid...

Using techniques of nonstandard analysis Abraham Robinson showed that it is possible to represent each Schwartz distribution T as an integral T () = R f, where f is some nonstandard smooth function. We show that the theory based on this representation can be developed within a constructive setting.

Smarandache (2003) introduced a new set-valued fuzzy logic called (nonstandard) neutrosophic by using Robinson’s nonstandard analysis. However, its definition involved many errors including the illegal use of In this paper, we provide rigorous logic. All in original are addressed. We then point out some paradoxes Finally formulate with no

Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the constr...

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out -mathematically speakingfor its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intu...