Robert A. Herrmann

ULTRALOGICS AND PROBABILITY MODELS

We show how nonstandard consequence operators, ultralogics, can generate the general informational content displayed by probability models. In particular, a probability model that predicts that a specific single event will occur and those models that predict that a specific distribution of events will occur.

Fractals and ultrasmooth microeffects

In this paper, a portion of the nonstandard methods first introduced by Nottale and Schneider [J. Math. Phys. 25, 1296 (1984)] for the investigation of fractal behavior are replaced by methods employing polysaturated enlargements. This process yields internal functions that are ultrasmooth and have a standard part equal to the original fractal. These ultrasmooth nonstandard functions also possess a well‐behaved * integral length concept. Additionally, a method is presented that shows that if certain behavior is modeled after a simple finitely discontinuous step function, then this function is also the standard part of an internal hypersmooth nonstandard approximation.

The best possible unification for any collection of physical theories

We show that the set of all finitary consequence operators defined on any nonempty language is a join-complete lattice. This result is applied to various collections of physical theories to obtain an unrestricted standard supremum unification. An unrestricted hyperfinite ultralogic unification for sets of physical theories is also obtained.

Nonstandard topological extensions

This paper investigates the nonstandard theory of filters on a non-empty meet-semi-lattice of sets and applies this theory to the general study of topological extensions Y for a space X. In particular, we apply this theory to Baire and quasi- H -closed extensions as well as Wallman type compactifications. Whereas these extensions have previously teen obtained and studied as types of ultrafilter extensions, we study them as subsets of an enlargement of X. Since X ⊂ Y ⊂ ◯ and the elements of X and Y - X are of the same set-theoretic type, these extensions appear more natural from the nonstandard viewpoint.

An operator equation and relativistic alterations in the time for radioactive decay

In this paper, using concepts from the nonstandard physical world, the linear effect line element is derived. Previously, this line element was employed to obtain, with the exception of radioactive decay, all of the experimentally verified special theory relativistic alterations in physical measures. This line element is now used to derive, by means of separation of variables, an expression that predicts the same increase in the decay time for radioactive material as that predicted by the Einstein time dilation assumption. This indicates that such an increase in lifetime can be attributed to an interaction of the radioactive material with a nonstandard electromagnetic field rather than to a basic time dilation.

Preconvergence compactness and P-closed spaces

In this article the major result characterizes preconvergence compactness in terms of the preconvergence closedness of second projections. Applying this result to a topological space (X,T) yields similar characterizations for H-closed, nearly compact, completely Hausdorff-closed, extremely disconnected Hausdorff-closed, Urysohn-closed, S-closed and R-closed spaces, among others. Moreover, it is established that the s-convergence of Thompson (i.e. rc-convergence) is equivalent to topological convergence where the topology has as a subbase the set of all regular-closed elements of T.

Erratum to “Hyperfinite and standard unifications for physical theories”

Definition 5.2. Suppose one has a nonempty finite set = {C1, . . . ,Cm} of general consequence operators, each defined on a nonempty language Li, 1≤ i≤m. Define the operatorΠCm as follows: for any X ⊂ L1×···×Lm, using the projection pri, 1≤ i≤m, define ΠCm(X)= C1(pr1(X))×···×Cm(prm(X)). Theorem 5.3. The operator ΠCm defined on the subsets of L1 ×···×Lm is a general consequence operator and if, at least, one member of is axiomless, then ΠCm is axiomless. If each member of is finitary and axiomless, then ΠCm is finitary. Proof. (a) Let X ⊂ L1 × ··· × Lm. Then for each i, 1 ≤ i ≤m, pri(X) ⊂ Ci(pri(X)) ⊂ Li. But, X ⊂ pr1(X)× ··· × prm(X) ⊂ C1(pr1(X))× ··· ×Cm(prm(X)) = ΠCm(X) ⊂ L1 × ···×Lm. Suppose that X = ∅. Then∅ =ΠCm(X)= C1(pr1(X))×···×Cm(prm(X))⊂ L1 × ··· × Lm. Hence, ∅ = pri(ΠCm(X)) = Ci(pri(X)), 1 ≤ i ≤ m, implies that Ci(pri (ΠCm(X)))= Ci(Ci(pri(X)))= Ci(pri(X)), 1≤ i≤m. Hence,ΠCm(ΠCm(X))=ΠCm(X). Let X =∅ and assume that no member of is axiomless. Then each pri(X) =∅. But, each Ci(pri(X)) = ∅ implies that ΠCm(X) = ∅. By the previous method, it follows, in this case, that ΠCm(ΠCm(X)) = ΠCm(X). Now suppose that there is some j such that Cj is axiomless. Hence, Cj(prj(X))=∅ implies thatΠCm(X)= C1(pr1(X))×···× Cm(prm(X)) = ∅, which implies that Cj(prj(ΠCm(X))) = ∅. Consequently, C1(pr1 (ΠCm(X)))× ··· ×Cm(prm(ΠCm(X))) = ∅. Thus, ΠCm(ΠCm(X)) = ∅ and axiom (1) holds. Also in the case where at least one member of is axiomless, then ΠCm is axiomless. (b) Let X ⊂ Y ⊂ L1 × ··· × Lm. For each i, 1 ≤ 1 ≤ m, pri(X) ⊂ pri(Y), whether pri(X) is the empty set or not. Hence, Ci(pri(X)) ⊂ Ci(pri(Y)). Therefore, ΠCm(X) =

A hypercontinuous hypersmooth Schwarzschild line element transformation

In this paper, a new derivation for one of the black hole line elements is given since the basic derivation for this line element is flawed mathematically. This derivation postulates a transformation procedure that utilizes a transformation function that is modeled by an ideal nonstandard physical world transformation process that yields a connection between an exterior Schwarzschild line element and distinctly different interior line element. The transformation is an ideal transformation in that in the natural world the transformation is conceived of as occurring at an unknown moment in the evolution of a gravitationally collapsing spherical body with radius greater than but near to the Schwarzsclfild radius. An ideal transformation models this transformation in a manner independent of the objects standard radius. It yields predicted behavior based upon a Newtonian gravitational field prior to the transformation, predicted behavior after the transformation for a field internal to the Schwarzschild surface and predicted behavior with respect to field alteration processes during the transformation.

Nonstandard topological extensions: Addendum

It has recently come to my attention that, independent from my investigations in [1], Professor K.D. Stroyan [2] obtains for a Tychonoff space X and a normal base β a quotient space on * X which is homeomorphic to the Wallman-Frink Hausdorff compactification w ( X , β ). Consequently, if X is Tychonoff, then Corollary 4.1 in [1] implies that Stroyans space is homeomorphic to the space w ( X , β) S which is constructed in Theorem 4.1 in [1]. Stroyans result is obtained by use of a monadic closure operator. I regret not mentioning Professor Stroyans result in my paper [1].