Explicit mathematics and operational set theory: some ontological comparisons

We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

A new model construction by making a detour via intuitionistic theories I: Operational set theory without choice is Π1-equivalent to KP

We introduce a version of operational set theory, OST−, without a choice operation, which has a machinery for Δ0Δ0 separation based on truth functions and the separation operator, and a new kind of applicative set theory, so-called weak explicit set theory WEST, based on Gödel operations. We show that both the theories and Kripke–Platek set theory KPKP with infinity are pairwise Π1Π1 equivalent. We also show analogous assertions for subtheories with ∈-induction restricted in various ways and for supertheories extended by powerset, beta, limit and Mahlo operations. Whereas the upper bound is given by a refinement of inductive definition in KPKP, the lower bound is by a combination, in a specific way, of realisability, (intuitionistic) forcing and negative interpretations. Thus, despite interpretability between classical theories, we make “a detour via intuitionistic theories”. The combined interpretation, seen as a model construction in the sense of Visser's miniature model theory, is a new way of construction for classical theories and could be said the third kind of model construction ever used which is non-trivial on the logical connective level, after generic extension à la Cohen and Krivine's classical realisability model.

Set theory and metric spaces. Math. 261, autumn, 1950.

Cover title.

Hand receipt covering contents of components of end item (COEI), basic issue items (BII), and additional authorization list (AAL) for radio test set AN/PRM-34 (NSN 6625-01-094-5646).

"2 September 1983."

Hand receipt covering contents of components of end item (COEI), basic issue items (BII), and additional authorization list (AAL) for radiac set AN/PDR-56F (NSN 6665-01-113-9530).

"28 January 1983."

Hand receipt covering contents of Components of End Item (COEI), Basic Issue Items (BII), and Additional Authorization List (AAL) for armament repair shop set (4933-01-082-1663).

"October 1982."

Hand receipt covering contents of Components of End Item (COEI), Basic Issue Items (BII), and Additional Authorization List (AAL) for computer set, field artillery, general (1220-01-082-1646) and computer set, field artillery, missile (1220-01-082-1647).

"May 1983."

Hand receipt manual covering system components of end item (COEI), basic issue items (BII), and additional authorization list (AAL) for CHAPARRAL M48A2 Air Defense Guided Missile System, AN/TSM-95A organizational maintenance shop set, and AN/TSM-96A support maintenance shop set.

"December 1984."

Hand receipt covering contents of components of end item (COEI), basic issue items (BII), and additional authorization list (AAL) for radio set AN/GRC-193A (NSN 5820-01-133-4195).

"14 February 1986."

Hand receipt covering contents of Components of End Item (COEI), Basic Issue Items (BII), and Additional Authorization List (AAL) for radar set AN/TPQ-36, (NSN 5840-01-043-4257).

"10 March 1981."

Hand receipt covering contents of components of end item (COEI), basic issue items (BII), and additional authorization list (AAL) for searchlight set, infrared AN/VSS-3A (NSN 5855-00-177-3529 and NSN 5855-00-405-0404).

"31 March 1981."

Hand receipt covering contents of Components of End Item (COEI), Basic Issue Items (BII), and Additional Authorization List (AAL) for receiving set, radio, AN/TRQ-32(V), including AN/TRQ-32(V)1 NSN 5895-01-167-7655, AN/TRQ-32(V)2 NSN 5895-01-167-7655.

"November 1989."

The theory and practice of bank book-keeping, and joint stock accounts; exemplified and elucidated in a complete set of bank account books, (printed in colors,) arranged in accordance with the principles of double entry ...

Mode of access: Internet.

Achieving completeness: from constructive set theory to large cardinals

This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings. After some introductory chapters in the first part of the thesis where we outline the background used later on, the constructive part contains a categorical formulation of several constructive completeness theorems available in the literature, but presented here in an unified framework. We develop them within a constructive reverse mathematical viewpoint, highlighting the metatheory used in each case and the strength of the corresponding completeness theorems. The classical part of the thesis focuses on infinitary intuitionistic propositional and predicate logic. We consider a propositional axiomatic system with a special distributivity rule that is enough to prove a completeness theorem, and we introduce weakly compact cardinals as the adequate metatheoretical assumption for this development. Finally, we return to the categorical formulation focusing this time on infinitary first-order intuitionistic logic. We propose a first-order system with a special rule, transfinite transitivity, that embodies both distributivity as well as a form of dependent choice, and study the extent to which completeness theorems can be established. We prove completeness using a weakly compact cardinal, and, like in the constructive part, we study disjunction-free fragments as well. The assumption of weak compactness is shown to be essential for the completeness theorems to hold.

Fuzzy Set Theory Approach as the Basis of Analysis of Financial Flows in the Economical Security System

The problems of formalization of the process of matching different management subjects’ functioning characteristics obtained on the financial flows analysis basis is considered. Formal generalizations for gaining economical security system knowledge bases elements are presented. One of feedback directions establishment between knowledge base of the system of economical security and financial flows database analysis is substantiated.