Pseudo-Superstructures as Nonstandard Universes

An Axiomatic Presentation of The Nonstandard Methods in Mathematics

A nonstandard set theory ∗ZFC is proposed that axiomatizes the nonstandard embedding ∗. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. ∗ZFC is a conservative extension of ZFC. Introduction. For practitioners, the most popular foundational approach to nonstandard methods is the so-called super...

Better Nonstandard Universes with Applications

Linearly Stratified Models for the Foundations of Nonstandard Mathematics

Assuming the existence of an inaccessible cardinal, transitive full models of the whole set theory, equipped with a linearly-valued rank function, are constructed. Such models generalize superstructures and provide a global framework for nonstandard mathematics. MSClassification: 03H05, 03C20.

Size-Sorting and Pattern Formation of Nanoparticle-Loaded Micellar Superstructures in Biconcave Thin Films

Biconcave thin water layers represent a template to induce organization of supramolecular structures into ordered monolayers. Here we show how micelles form extensive micrometer-sized pseudo-2D superstructures that reveal size-sorting and geometric pattern formation, as shown by cryo-transmission electron microscopy (cryoTEM). Electron-rich gold particles inside the micelles facilitate direct v...

The Strength of the Isomorphism Property

In §1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second–order types in model theory. In §2, several applications are given. One of the applications answers a question of D. Ross in [R] about infinite Loeb measure spaces.