The "Project for Closed Loop Interval Ontology" is our exploratory development and architectural design for the integration of semantic ontology and our response to "The Tower of Babel Problem." Major dimensions and implications What is closed loop interval ontology? Assembling process A single line of descent from absolute measurement Unit interval Russell's Paradox

Major dimensions and implications Placeholder | Back We are postulating a master containing and aligning principle, from which we propose all conceptual form is derived, and within which all conceptual form is contained. This structure has a central axis across its descending levels that guides participating collective decision-making towards a common center Foundation for conceptual form -- all form derived from this structure Universal container for conceptual form - all form contained within this structure Integrating (global and local) center-point for convergent decision-making There is an implicit ethics contained within this framework, defined as link between global and local, where global well-being is instantiated and replicated at the local level. This concept is sometimes known as "glocalism" (ie, the global in the local). The well-being of the whole is defined by balance (homeostasis) and that same balance is replicated at the level of local community. The integrity of this connection forms a guiding universal ethic active at all levels (from local to global). Tue, Feb 16, 2021

What is closed loop interval ontology? Sketch | Back We are proposing a new model of the continuum, which we define as a closed loop - a circle - but in this case taking the form of a "Moebius Strip". We are proposing that "an ontology" can be defined within (bounded by) and on (as a strip) a "closed loop". By "ontology", we mean a system or framework of definitions or terms or concepts ("words"), where the distinctions and specifics characterizing those terms are defined within intervals created by distinctions and boundary values. What is a "closed loop?" What is an "interval?" What is an "ontology?" We are exploring the hypothesis that the two "edges" of the strip -- which are actually only one continuous edge -- can define an "interval" across the width of the strip - somewhat as is shown in the animated graphic in the header. In addition to this basic concept of bounded interval (bounded by top edge and bottom edge), we define a range of values nested "between the edges" of this form, which we propose takes the general form of a taxonomy or "hierarchy of abstraction". This hierarchy of nested levels extends from a "top level" (from the top edge), which is an unbounded interval ("the infinite") to a "bottom level" (to the bottom edge) ("the infinitesimal"). This decomposition is analogous to ("isomorphic to") the decomposition of the unit interval into the decimal number system. The exploratory hypothesis is that this simple general form can interpret any special-case instance, and that all taxonomies take this same general form, taking a specialized configuration in service to particular purposes arising in particular contexts. The power of this model emerges from the way we interpret it. The "closed loop" is a container. It's like a cell in a spreadsheet or database table. It has boundaries (or boundary) and something inside those boundaries (or boundary). We understand those boundaries to be defining cascades of nested distinctions, or, at the bottom the cascade (the hierarchy of abstraction), perhaps a single element or instance. How do the purposes defining a taxonomic cascade influence or shape the structure of the cascade? How does the general form interpret any special case instance? Why is this important? Those boundaries can be defined as "boundary values" -- lower and upper limits on some dimensional range. The objective is to understand how it is that "everything is contained within it." This concept is proposed as an absolute bound on the conceivable. Every idea, it is proposed, every concept, every term or category, emerges as a distinction or a "cascaded nest of distinctions" defined within this framework. The entire structure can be understood as a "unit interval" - which we see as a foundation concept for defining the notion of "unit" -- "one of anything". The unit interval can be "decomposed" -- like a taxonomy. A series of levels - like a taxonomy - is shown in the animated graphic. So, this decomposition is similar to (identical to, isomorphic to) the decomposition of decimal numbers -- or any other numbers -- into decimal places and finer and finer (smaller and smaller) measurement distances or units (10ths, 100ths, 1000ths, 10000ths, etc.) The power here emerges when we understand that this entire process can be contained across the limited width of the moebius strip -- the distance between the two edges. One edge of the strip is the open undefined unbounded infinite interval with no end-points -- because it is a circle -- and the other edge is defined as something like the real number line, the finest differentiation possible, and an approach to continuity as a limit. Mon, May 10, 2021

Assembling process Placeholder | Back We want to define precisely how it is that the closed loop (hierarchically-partitioned moebius strip) can be interpreted as a universal foundation for conceptual structure. This is complicated ambition with a lot of moving parts, and the idea is still unproven and intuitive. List these elements here to help guide the coalescence. We need a comprehensive list of these elements, and we have to explain or justify them all individually in a consistent way. This list is a design for project architecture: The pieces that are to be combined The way in which the pieces are to be combined The themes and principles and methods that are brought together in one place The concept of "isomorphic recursion" defined across many simultaneous objects or concepts Terminology: Develop a list of terms and words that are synonyms for "category" -- any dimensionally-bounded unit Develop a list of terms that are synonyms -- or can be defined as synonyms -- for boundary value or "cut" -- the "line that cuts a line" to form a distinction Type of objects that are defined through isomorphic recursion and connected to build a model of reality: Abstract Symbolic - representational Concrete (referent?) or "actual" "What can be done with little boxes"? Everything is defined by a box (a bounded interval) containing some symbolic structure, composed of alphabet or numbers.. Everything is matrix matrix can define everything -- to within some error tolerance Mon, Apr 5, 2021 Closed loop interval ontology Objectives and strategy

A single line of descent from absolute measurement Placeholder | Back We want to define an aligning axis as a guide to interpretation across levels ranging from global to local, from "infinite" to "infinitesimal" It's a guide to interpretation, and is a guide to convergence towards common center ** Wed, Feb 17, 2021 Closed loop interval ontology One Integral Tree - The Intuitive History

Unit interval Placeholder | Back the top level in the strip – the undifferentiated “one” at the top of the hierarchy of layers, defined in a recursive way like a “holon”, such that every nested interval is also a unit interval Sun, Apr 4, 2021 Reference In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1). However, the notation I is most commonly reserved for the closed interval [0,1]. URL https://en.wikipedia.org/wiki/Unit_interval Closed loop interval ontology Closed Loop Ensemble Core vocabulary

Russell's Paradox Placeholder | Back Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction. Russell’s paradox Bertrand Russell (1872-1970) was involved in an ambitious project to rewrite all the truths of mathematics in the language of sets. In fact, what he was trying to do was show that all of mathematics could be derived as the logical consequences of some basic principles using sets. At this time (around 1900), it was generally believed that any property of objects could define a set. For example, the property “x is a natural number between four and seven” defines the set {4, 5, 6, 7}. We could also write this set as {x| x is a natural number between 4 and 7}. What was believed was precisely that for any property “P” that you can think of, it made sense to talk about the set {x| x has property P}. Certainly, sets that consist of numbers make sense this way. But when you allow any objects in your sets, you can run into trouble. Russell was the first one to notice this. Russell’s insight was the following. First, it is possible for a set to be an element of itself. (Remember that elements are the objects which make up the set, e.g. the number 4 is an element of the set {4, 5, 6, 7}). An example of a set which is an element of itself is {x| x is a set and x has at least one element}. This set contains itself, because it is a set with at least one element. Using this knowledge, Russell defined a special set, which we’ll call “R”. R is the set {x| x is a set and x is not an element of itself}. Russell then asked: is R an element of the set R? Let’s think about this question. If R is an element of R, that exactly means that it is an element of itself. Which means that it can’t possibly be in R - by definition R is the collection of all sets which are not elements of themselves. Since this option is impossible, we must agree that R is not an element of R. But in that case, R is not an element of itself, so by definition it belongs to the collection of sets which are not elements of themselves. Uh-oh! [It is worth pausing a minute here and reassessing the situation on your own. Do you believe that R is an element of R or not? Neither? What’s the problem?] Mathematicians and logicians thought for a while that the problem with the set R was going to undermine their whole project to do all mathematics in terms of set theory. Fortunately, they eventually came up with a technical solution that changes the way sets are constructed (slightly) and prevents R from being considered a set (whew!). This technical solution is why you sometimes see a set being described in the form {x ? X| x has property P}, where “X” is supposed to denote some other set, or “the universe” (whatever that means!) We won’t worry about that issue in our class, as most of our sets will be sets of numbers and these are always safe. So did Russell succeed? Yes and no. The program to do all mathematics in terms of set theory was the birth of modern day logic, a rich and exciting area of mathematics. But ultimately, the logician Kurt G¨odel showed that the original goal of the project – to deduce all mathematics from some axioms (rules) concerning sets – was itself mathematically impossible! For more details on this story, I recommend the excellent book Logicomix by Doxiades and Papadimitriou. You may be delighted to know that a) it’s a graphic novel and b) the Reg library has a copy. Sun, Mar 28, 2021 Reference In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901,[1][2] showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered in 1899 by Ernst Zermelo[3] but he did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other members of the University of Göttingen. At the end of the 1890s Cantor himself had already realized that his definition would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter.[4] According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically: {\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}{\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R In 1908, two ways of avoiding the paradox were proposed: Russell's type theory and the Zermelo set theory. Zermelo's axioms went well beyond Gottlob Frege's axioms of extensionality and unlimited set abstraction; as the first constructed axiomatic set theory, it evolved into the now-standard Zermelo–Fraenkel set theory (ZFC). The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed, while Russell altered the logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be first-order logic.[5] URL https://en.wikipedia.org/wiki/Russell%27s_paradox Closed loop interval ontology Project under development What is a universal container? Set