Some Theorems on the Empty Set as Necessary and Sufficient For the Primary Topological Axioms of Physical Existence

Bounias, M.; Bonaly, A.

doi:10.4006/1.3028742

Cited by 7 publications

(12 citation statements)

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“…However this construction, associated with the application of Morgan's laws to (Ø), allowed the empty set to be attributed an infinite descent of infinite descents and thus to be classified as a member of the hypersets family (Bounias and Bonaly, 1997b).…”

Section: Analysis Of Unordered Versus Ordered Pairs In An Abstract Setmentioning

confidence: 99%

Scanning the structure of ill‐known spaces: Part 1. Founding principles about mathematical constitution of space

Bounias¹,

Krasnoholovets

2003

Kybernetes

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Abstract. Some necessary and sufficient conditions allowing a previously unknown space to be explored through scanning operators are reexamined with respect to measure theory. Some generalized conceptions of distances and dimensionality evaluation are proposed, together with their conditions of validity and range of application to topological spaces. The existence of a Boolean lattice with fractal properties originating from nonwellfounded properties of the empty set is demonstrated. This lattice provides a substrate with both discrete and continuous properties, from which existence of physical universes can be proved, up to the function of conscious perception. Spacetime emerges as an ordered sequence of mappings of closed 3-D Ponicaré sections of a topological 4-space provided by the lattice, and the function of conscious perception is founded on the same properties. Self-evaluation of a system is possible against indecidability barriers through anticipatory mental imaging occurring in biological brain systems; then our embedding universe should be in principle accessible to knowledge. The possibility of existence of spaces with fuzzy dimension or with adjoined parts with decreasing dimensions is raised, together with possible tools for their study.The work presented here provides the introductory foundations supporting a new theory of space whose physical predictions (suppressing the opposition of quantum and relativistic approaches) and experimental proofs are presented in details in Parts 2 and 3 of the study.

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Section: Analysis Of Unordered Versus Ordered Pairs In An Abstract Setmentioning

confidence: 99%

Scanning the structure of ill‐known spaces: Part 1. Founding principles about mathematical constitution of space

Bounias¹,

Krasnoholovets

2003

Kybernetes

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“…The empty set is a non‐well‐founded set or hyperset, whose existence gives rise to existence of abstract mathematical spaces endowed with the power of continuum (Bounias and Bonaly, 1997a).…”

Section: Lemma 11mentioning

confidence: 99%

Universe as a self‐observable, self‐ethical, life‐embedding mathematical system

Bounias¹

2002

Kybernetes

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The non‐well‐founded properties of the empty set provide existence to abstract topological spaces, in which intersections of subspaces with non‐equal dimensions give topologically closed structures. This let space‐time emerge as an ordered sequence of sections upon a defined combination rule, in which interactivity between observable structures is allowed and conditions for conscious perception phenomena are fulfilled. Conditions for evolution and rules for optimal evolution of ecosystems also infer as corollaries. A physical universe and life constitute one single self‐organized and self managed system where life appears as the physical‐like realization of conditions of functionality of the embedding mathematical spaces. The system includes self‐ethical and moral guidelines which should inspire human behavior on Planet Earth.

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“…For the description of observable world, which consists of the systems perceiving both outer objects and an inner self, an apparatus of the set theory was applied (Bounias, Bonaly 1997a). A special type of sets (closed sets) exists upon intersection of topological spaces owning different dimensions.…”

Section: Formal Incorporation Of the Final Cause Into The Description Of Biological Evolutionmentioning

confidence: 99%

“…The closed sets in this approach are similar to the monads of Leibniz (1965) which constitute and observe the Universe. The empty set will correspond to a vacuum that is still not allotted by features (Bounias, Bonaly 1997b). The memory appears as a 'sign-creating activity' (Hegel 1971), linking sets with different dimensions.…”

Section: Formal Incorporation Of the Final Cause Into The Description Of Biological Evolutionmentioning

confidence: 99%

“…Interaction between the whole and the parts can be viewed as an intersection of the sets with different dimensions forming a contradiction in the sense of Russel's paradox (the fixed point) (Bounias, Bonaly 1997a). This intersection may represent a harmony or a disharmony, depending on how parts are observed within a whole observing it.…”

Section: Perfection and Final Causementioning

confidence: 99%

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Biological evolution — a semiotically constrained growth of complexity

Igamberdiev¹

2002

SSS

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Any living system possesses internal embedded description and exists as a superposition of different potential realisations, which are reduced in interaction with the environment. This reduction cannot be recursively deduced from the state in time present, it includes unpredictable choice and needs to be modelled also from the state in time future. Such non-recursive establishment of emerging configuration, after its memorisation via formation of reflective loop (sign-creating activity), becomes the inherited recursive action. It leads to increase of complexity of the embedded description, which constitutes the rules of generative grammar defining possible directions of open evolutionary process. The states in time future can be estimated from the point of their perfection, which represents the final cause in the Aristotelian sense and may possess a selective advantage. The limits of unfolding of the reflective process, such as the golden ratio and the golden wurf are considered as the canons of perfection established in the evolutionary process.

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Some Theorems on the Empty Set as Necessary and Sufficient For the Primary Topological Axioms of Physical Existence

Cited by 7 publications

References 0 publications

Scanning the structure of ill‐known spaces: Part 1. Founding principles about mathematical constitution of space

Scanning the structure of ill‐known spaces: Part 1. Founding principles about mathematical constitution of space

Universe as a self‐observable, self‐ethical, life‐embedding mathematical system

Biological evolution — a semiotically constrained growth of complexity

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