Applications of Non-Standard analysis in Topoi to Mathematical Neurosciences and Artificial Intelligence: Infons, Energons, Receptons (I)

The purpose of this paper is to promote new methods in mathematical modeling inspired by neuroscience—that is consciousness and subconsciousness—with an eye toward artificial intelligence as parts of the global brain. As a mathematical model, we propose topoi and their non-standard enlargements as models, due to the fact that their logic corresponds well to human thinking. For this reason, we built non-standard analysis in a special class of topoi; before now, this existed only in the topos of sets (A. Robinson). Then, we arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi. We also consider the genetic–epigenetic interplay with a mathematical introduction consisting of a study of the Yang–Baxter equations with new mathematical results.


Introduction
This paper contains both known and new results, and it addresses an auditorium of great diversity: neuroscientists, computer scientists, physicians, physicists, biologists, mathematicians, etc. We attempted to write a paper that could be read by different kinds of specialists. The authors and their collaborators belong to various scientific areas: neurosciences, mathematics, medicine, and economics; and are professors, researchers, and students.
The authors hope they have produced a self-contained and readable paper for most categories of scientists who might be interested in the topics. At the same time, the preprint [1] represents an expanded version of this article, from the mathematical point of view, collecting most of what the authors considered as being the necessary mathematical background.
Therefore, in this paper, some already known notions and results are briefly re-explained, but, at the same time, the paper contains many new original contributions. The main original contribution is the idea of considering the non-standard extension of the intuitionistic logic (in topoi) as modeling human thinking, consciousness, and considering together both natural and artificial intelligence, as parts of the global brain, as explained in Sections 2-4. This paper also contributes to the development of the non-standard analysis in set-type exponential topoi, in the aria of pure mathematics, see Theorem 1. Note 1.4. This paper is a review/article one, containing the following topics (the order from the paper does not necessarily coincides with the next one): (1) Presentation of some known results from neurosciences, physics, and cybernetics arranged and interpreted in order to prove the importance of the axiomatics from the next (2) (Sections 2-4); (2) Presentation of the axiomatics IIST (Intuitionistic Internal Set Theory) (Section 7.1); (3) Theoretical introduction of non-standard analysis in topoi (until now this was known only for the topos SET), at the level of general definition (having higher order language)) (Section 6.2); (4) Topoi of SET-type and the construction of a model class for IIST on them, based on non-standard analysis in SET (IIST is then as consistent as ZF(C) is) (Section 5.2 and Theorem 1); (5) Definitions of the notions of infon, energon, recepton, which would use for a simultaneous approach of the natural and artificial neural (quantum) networks, both part of the global brain (Section 7); this approach is based also on a novel study of the global brain (Section 4); and this last-one is based on some novel points of view concerning the human brain, mind, genetics, and epigenetics (Section 2).

Information and Reception: Brain and Mind, Genetics and Epigenetics
The next three Sections try to put in evidence, using known and, by many experts recognized results from biology and physiology, physics, and cybernetics which relates between them the information and its reception with energy. In this section we deal with biology and physiology (see also [2,3,23]).

Information and Entropy
First of all, information is related to organization. Some systems or objects are more complex than others because we perceive them as containing more information. For example, it is obvious that an epithelial tissue contains more information than a cell. Epithelial cells exhibit distinct polarity, having an apical, a lateral and a basal domain. This organization involves more information, as the molecular mechanism responsible for this arrangement is required to create a functional barrier between adjacent cells.
Following Tom Stonier's idea from [23], information is related to entropy, because the latter is a measurement of organization. They are exponentially inverse proportional: as a consequence, if-inside a complex system-the entropy decreases, meaning that matter is more organized, the amount of incorporated information increases. It is really important to see how this new concept-the incorporated information-influences our lives and more specifically the role it plays in the structure of matter. How do atoms know how to bind together and form a more complex structure? How can a simple stem cell differentiate in so many other types of cells?
Information dictates all those processes. Everything contains infons (see Section 7, for their mathematical definition, one of the main purposes of this paper). Let us take, as an example, a sound wave. We can tell for sure that it contains some kind of information, that travels through air until it reaches the human ear. Here, it is transformed into a nervous impulse and travels to the brain, where it will be perceived. A sound wave is a mechanical wave that causes local regions of compression and rarefaction through the medium, deviating the local pressure from the equilibrium pressure. Therefore, sound, which is made of information, changes the structure of matter.
In neuroscience, it is of extreme interest to see how information affects the brain. We, as humans, are born with unconnected neurons. In the first two years, we form synapses while experiencing the world around us. As we grow up, we lose almost 50% of the already formed synapses, but we also form others while interacting with the environment. How does information influence this process? The more we interact with something, the more powerful the synapse becomes. It is like a road through a forest.
At first, there's nothing there, but as more and more people go along it, the grass disappears and a path is formed. The same thing happens in the brain. Information extends one's life. During the first part of our lives, the human organism functions in an anti-entropic way. As it becomes more organized, the entropy decreases, instead of increasing. However, after this period, the entropy of the human organism continues to increase until it reaches a state of equilibrium with the environment, when the organism dies, because no more energy exchange can be done.
By increasing the amount of information one takes in, the entropy of the organism decreases, meaning that it prolongs the process of ageing. Thus, learning new things delays the decay of the brain primarily, but of other organs as a consequence as well. For a much more physico-mathematical approach of such things, see Sections 3 and 7 of this paper.

Information and Reception, Brain and Mind
For further details related to this paragraph, we refer to the books [2,3]. The brain is the main organ where the mind is born, but the brain is far from being the only entity involved in the growth of the human mind. The mind is a by-product of the neural networks from the brain (synapses), in continuous transformations (some synapses disappear while new ones are born), representing the so-called plasticity of the brain. The mind is a by-product of other kind of communications between neurons (and the glial environment), namely the quantum networks connected by the so-called microtubules.
Certainly, these interactions are between the brain, through the sensory organs, moderated by the information from the DNA molecule (genetics), and the environment (epigenetics). The human being has much information in its DNA, but only some of it will be used in a life-long period. Simplifying, there is information that tells us to start trembling when it is cold (or to sweat when it is hot).
However, if we spend a life-time period at the equator we will never experience the trembling (respectively, if we spend a life-time period at the North pole we will never experience sweating). Thus, the information from the around 2 m long DNA molecule is read by the environment epigenetics, the system genetics/epigenetics becoming a kind of (quantum) Turing machine, generating, in particular, the evolution of the mind, which, through plasticity, generates the evolution of the brain, and vice-versa.
We-humans-think using both consciousness (awareness) and subconsciousness. There are precise measurements proving that the brain takes a decision shortly before we became aware of it, so through subconsciousness (or, as another example, the subliminal messages). The logic of consciousness is intuitionistic (we have, mainly, feelings about things, such as about the truth of conjectures in mathematics, as an example), and the reunion between consciousness and subconsciousness is a conservative extension of consciousness (that is, the subconsciousness does not alter our awareness).
Thus, the logic of human thinking is very similar to the logic of infons from Section 7, having, as models, non-standard objects from non-standard enlargements of topoi-see Section 7 again (based on Sections 5 and 6 also). From a mathematical point of view, it is necessary to understand how the (quantum) neural networks from the brain (synapses) are influenced by the genetic-epigenetic (quantum) Turing machine.
In this paper, one consider that the quantum physics approach is better to be based on truth values than on probabilities, the truth values approach being used in topoi (see [8]). Thus, the reunion between consciousness and subconsciousness is modeled here by the non-standard logic in enlarged topoi (using the IIST axiomatics from Section 7, with the theories as infons, and their categories of models as receptons-the details will be given in the forthcoming Section 7).
A theorem of mathematics (for example) is based on the work of many others; therefore, it is a by-product of the global brain. The global brain includes all the human brains and technology, used both for finding information from the outside of the direct human perception, and for connecting the brains (libraries, social networks, etc.)-see Section 4. In this case, the genetic-epigenetic interaction is between the global brain and the environment, generating a kind of global plasticity for the global brain, generating a global mind (for some details, see Section 4). See, also, the previous Section 2.1.

Information, Energy, and Reception
This paragraph could be introduced here, but it can also belong to the next Section 3, and we preferred to put it there (it is mainly related to physics, while this Section is mainly related to biology/physiology). Here, we only say that there are opinions in quantum physics saying that the collapse of a probability wave takes place when we become aware of it (as explained before, through subconsciousness, so the consciousness and subconsciousness should be considered together). In this paper, the waves from classical quantum physics become truth values waves, that is, infons, while the receptons represent the collapse of such kind of truth waves.

Information and Reception: String Theory and Emergent Quantum Mechanics
In this Section, one continue to put in evidence the relation information-energyreception, using known and, by many experts recognized results from celestial mechanics/ physics (see [12,14,16,24]).

Max Tegmark's Multiverses
Certainly, everything happens in our real world, our real universe, or our real multiverse, or etc. This means that the human brain, human being, global brain, etc., represent substructures of the highest structure which must exist. There are discussions on what this highest structure is. We recall here the theory of multiverses belonging to Max Tegmark (see [14]). Namely, he classifies the notions of multiverse in 4 categories. We do not insist too much, because there are scientists which consider it as being speculative, but the interested people might look in the previous reference.
However the quantum computers seems to work in a rank 3 multiverse (quantum), at the macroscopic level such an object being only one, but at nano-, pico-level is a whole network in superposition. First of all, what is an (not the) universe. In these theories an universe is, by definition, the visible universe for some observer (everything here is based on electromagnetic waves, but the things would change if these waves were replaced by the recently found gravitational waves; to detect them one needs some technology, artificial sensors, not the natural ones only; all these belong to global brain, see Section 4). Two observers might see the same universe, or not. Moreover, a visible universe is increasing (with the speed of light) and might unify with another similarly increasing one. Such a family of universes (which might become only one in some future) defines a rank 1 multiverse. On the other hand, the space can, in theory, dilate faster than the speed of light in some arias (this is not forbidden by the general relativity of Einstein), and the observers from different sides will see for ever different universes. M. Tegmark gets in such a way, a family of multiverses of rank one, that is a multiverse of rank 2 (a comment: in physics, a theory is recognized as true, when it was mathematically-or empirically-predicted AND was experimentally confirmed; three Nobel Prize awards confirm this philosophy: the theoretical existence of the Higgs boson, gravitational waves, and black holes were mathematically predicted-by Higgs and others, Einstein, and Penrose, respectively-with decades before being found, but to be found in reality was crucial; indeed, in organic chemistry, for instance, one can imagine a lot of molecules respecting all the known rules, but which do not exist; some of them can be produced in laboratories, but this is another aspect; even the periodic table of Mendeleev was artificially completed with elements produced in laboratories; on the other hand we assumed the Tegmark's axiom for the rank 4 universe, saying that all the non-contradictory theory must have a physical model, see below; however, it is not obligatory that such a model exists in our universe; this is the end of the comment).
Further, there are discussions concerning the interpretations of quantum physics. One interpretation is based on the Copenhagen program, based on the collapse of the waves of probabilities when measured (observed), and another one belongs to Hugh Everett III [25] saying that the probability waves never collapse, but what happens is in fact a bifurcation (multifurcation) of the (rank 2) multiverse depending on all the possible situations (so, Schrodinger's cat is alive in an universe and dead in another one, at the same time). This is the so-called many-worlds interpretation, giving the Tegmark's rank 3 multiverse. Finally, a rank 4 multiverse is represented by pure mathematics, with the axiom saying that any non-contradictory mathematical theory should have real model(s) in the rank 4 multiverse (see also Definitions 6 and 30). This attempts to explain why our universe allows life to exist (all the universal constants should be very, very carefully chosen: the number of Avogadro, the speed of light, the Planck distance or Planck time, etc.). We will restrict ourselves to a rank 3 multiverse, as being the quantum family of our observable universe. Although the quantum logic of John von Neumann is considered as being the quantum logic in the universe (in the sense of the previous definition), the quantum logic in topoi is considered as being the quantum logic in the quantum multiverse (rank 3).
This is the logic that we will use as it is the most appropriate to the human thinking, which is a product of the brain, a substructure of the uni (multi) verse (considered one of the most complex such substructures, along with black holes and dark matter). However, the best theories from physics, namely the Einstein's theory of (general) relativity (modeling the universe at the macro-level), and the quantum physics (modeling the smallest substructures on the uni (multi) verse, at the nano-, pico-, etc. level) are in contradiction in some situations (the quantum entanglement is one of them).
At the same time, the two theories should coexist in black holes, for instance. Thus, a new unifying theory is required, that is a theory unifying the standard model (which unifies the electromagnetic, strong, and electroweak forces), a quantum model, with the gravitation (modeled by the general relativity). There are currently several approaches in this direction: string theory, emergent quantum mechanics, loop gravity, etc. We will restrict ourselves to some comments relative to the first two of them.

Infons and Energons in String Theory
In string theory (see [12]) one consider that any elementary particle is a tiny vibrating chord-string-in a 10-dimensional space, representing the direct product between the Einsteinian 4-dimensional space-time universe (Einstein-Minkowski real 4-fold) and a complex (projective algebraic) Calabi-Yau 3-fold (so a 6-dimensional real manifold). The Calabi-Yau 3-fold is considered because it has very many symmetries, giving the so-called supersymmetric string theory, modeling not only bosons (as the initial string theory did)that is particles with integer spin, but the fermions as well-particles with half integer but not integer spin.
In such a situation, the number of elementary particles (quarks, photons, electrons, etc.) doubled, and one still look for the new elementary particles at CHERN, in Geneva. The 6 additional real dimensions of the universe (multiverse) are supposed to be very compacted, and give only very limited possibilities to the string to vibrate, imposing conditions, as wave length, amplitude, etc. Thus, a string needs energy, in order to vibrate, and information, in order to vibrate in a specific way to give one or another of the elementary particles (or antiparticles).
In [23], the author considers the existence of some hypothetical elementary particles related to information, calling them infons. It is for the first time when the notion appeared Mathematics 2021, 9, 2048 8 of 28 in the literature. However, the present paper is based on the fact that an infon represents only one facet of an elementary particle, the other being represented by the energons, so an elementary particle is like a coin, with two facets: the infon, and the energon. The theory of infons has been already discussed a little in Section 2.
After [23], another theory of infons (propositional calculus) has been built in [11]. The authors came to the conclusion that this theory is a conservative extension of the intuitionistic logic. However, one can show that the axiomatics of non-standard analysis in an environment using intuitionistic (many-valued) logic represents another similar notion of infon which will be used in this paper. Thus, an infon is given here by a non-standard enlargement of a topos of the form SET C op (see Section 5).
In string theory, an infon is a non-standard enlargement of a topos of the form SET C op (see Section 5), hence the theory of infons is the theory associated a category of such non-standard enlargements (so a category of the form CAT, see Remark 3 from Section 5). to while in emergent quantum mechanics an infon remains the theory itself, meaning and the enlargement of the topos of the previous type will be called recepton. In string theory the notion of energon is also necessary, but for now, we move on:

Infons and Receptons in Emergent Quantum Mechanics
In emergent quantum mechanics [16] we consider a different interpretation of the previous notion of infon. Namely, an infon remains the theory itself, meaning a triplet I =(L, A, D), where L is the usual language of the intuitionistic non-standard context (see Sections 6 and 7.1), A is a set of propositions written in the previous language considered as being true (axioms), and D is a set of deduction rules (as modus ponens is), having as models non-standard enlargements of categories (topoi) of exponential type, as in the the previous subsection (i.e., SET C op ) -see Sections 5-7. These categories of models are the receptons (different such theories represent different infons, while different such categories represent different receptons). Moreover, the potential energy is information, so is represented by infons (the information allowing us to know that something can move from one place to another, producing actual energy-as an example, while the actual energy (kinetic, caloric, electromagnetic, etc.) is represented by receptons. The matter is considered a condensed form of (actual) energy (under the action of the strong nuclear force) related to the (actual) energy by the well-known formula E = mc 2 . Both the nuclear fission, which takes place under the action of the electroweak force-from the atomic power plants, and the nuclear fusion, which takes place under the combined action of the strong and electroweak forces-from the Sun, generates loss of mass, producing actual energy from condensed energy, these operations taking place at the level of receptons; the information that this is possible -but not yet effective is potential energy, so infon. Therefore, an energon is either an infon (when virtual, potential), or an recepton (when we deal with actual energy).
Moreover, a stronger argument connecting the (potential) energy with information is represented by the Landauer Principle, quantifying the energy necessary to erase a bit of information (for details see [24]). It means that both energy and information are quantised, and they are somehow inverse proportional (not exponentially, as in the case of entropy, see Section 2, but in the same direction, the relation being given by a precise formula, namely E (min) = k B Tln(2), where k B is the Boltzmann constant, T is the temperature of the environment and ln is the natural logarithm.
Thus, in emergent quantum mechanics everything primordially exists as waves of information = theories in a conservative extension of the intuitionistic logic, all of them being infons. Their associated categories (of models) represent, then, the other facets of the coin, called receptons so, in this setting the two facets are the infon , and the recepton, the category of models of the theory which gives us the chosen infon (which is a non-standard enlargement of a topos of a special type, see Section 7).
We, the human beings, are, at the level of our brains, or to the level of the global brain (which includes both all the networks of the human brains, and the technological discoveries, artificial intelligence included, but also other kind of sensory-type machines) like the radio-sets travelling in a landscape with plenty of electromagnetic waves, and discovering them progressively, function of the increasing quality of the receptor, improved due to the technologies. Thus, we become progressively aware of the surrounding informational waves (theories) improving progressively our consciousness (both at the individual and global level).
At the same time, the subconsciousness works, impacting at every step on our lives, on our level of knowledge. Thus, the receptons become the elementary pseudoparticles of the union between consciousness and subconsciousness, the two facets of the coin being the theory and the associated category (which must be a non-standard topos). In this version of an unification theory, the uni (quantum-multi) verse is emerging, being stepwise discovered by brain and global brain through the measurement (collapse) of the intuitionistic informational waves.
In fact, emergent quantum mechanics is essentially based on information and consciousness (which becomes part of physics), but there are other important facts around, namely the non-determinism (with the meaning that not only the past influences us, but also the future), the discrete space, Gosset Polytope (see Section 5 also, for a topos representing it), E8-crystal, and golden ratio The 4D space-time is pixelated by the projection on R 4 of the Gosset polytope from R 8 with an angle which produce on R 4 two similar 4-dimensional polytopes, such that the quotient of their 4-volumes is equal to Φ. We recall that an energon is either a theory about a potential work (potential energy), so an infon, or a real work (actual energy), which is a recepton.

Cybernetical Principles of the Global Brain
In this Section, we consider a notion of the global brain, using a cybernetical approach, relating again information-reception-energy. The approach is both original and based on ideas coming from [26][27][28][29].

What Is the Global Brain?
The global brain is a bio-technological system composed by physical receptons organized to generate (electromagnetic, gravitational, etc.)-fields of energons, fields which interact to the higher levels of infons and create consciousness. The purpose for humans to create a global consciousness can be achieved by a smart global architecture of a physical human-machine system to implement cybernetic principles of information. As Mihai Drȃgȃnescu proposed in 1990 there is a frontier between physical level of matter and its informational dimension called InfoMatter [29]. At this intersection we identify cybernetic laws which appear in all information systems detected by human sensors (natural and artificial).

Cybernetic Principles of Information in the Global Brain
1. Autonomy of information: information pre-exists at the highest level of consciousness and operates individually (finite) and globally (infinite); 2. Universal connectivity of information manifests on the matrix of all data connections between and inside physical and non-physical phenomena on all dimensions; 3. Free will versus information: human being relates to information in terms of lists of possible actions and non-actions, a basis for the free will; 4. Authorization access to information: at physical level quantum fields interact to create support for information, the access to information on macro level of human beings consists in permissions for entities as humans, robots or other automatic to interchange data according to the free will principle; 5. Exception handling for information ambiguities: natural phenomena of information error consists of contradictions inside information with associated correction choices (multiple choices).
Eric Kandel [30], a recipient of the 2000 Nobel Prize in physiology or medicine for his research on the memory storage in neurons, proved that information is linked inside brain to physical level.

Global Brain Functions Regarding Information
Global brain functions regarding information are derived from data cybernetic principles [31] with the difference that information is a metadata relevant in terms of utility for a recipient (human or robotic).
1. In f ormation aqusition: metadata are identified from perception of existing data inside and outside entities (humans/robots, at the recepton level); 2. In f ormation processing: metadata are created through algorithmic processing of data perceived through sensors (at the infon-energon level); 3. In f ormation Storage: metadata are physically stored in human brain and IT recipients (natural/artificial neurons and machine storage recipients) (infon level); 4. In f ormation sending and receiving: metadata interchange between sender and receiver in request-answer process (recepton level).

Limits and Out Bounds of the Global Brain
We start with the Figure 1  3.In f ormation Storage: metadata is physically stored in human brain and IT recipients 417 (natural/artificial neurons and machine storage recipients) (Infon Level).

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We start with a picture (it is related to picture 7 from [32]): Gravitation in a single unifying framework (see also [26], [34], [35]  The architecture of global brain generated by machines (artificial intelligence algorithms)-human beings interactions are inspired by Norbert Wiener's cybernetic communication principles [28], and have a possible graphical representation as above (M-Hemisphere, H-Hemisphere, corpus AIH, analogous to the left hemisphere, right hemisphere and corpus callosum of the human brain). Based on Stefan Odobleja's vision [26] on analogy and models of mental psychology, a global brain should attain at least the psychological functions of the human brain.
Therefore: request-answer is a cybernetic model to create bio-technological feedback between global brain entities at the knowledge frontier. In the figure above, the membrane of the global brain is, in fact, an invisible knowledge frontier toward global consciousness, continuously in activity inside the system, and as in Figure 1. Many scientists claim for unified theory in science for the explanations of natural phenomena, one of them is Nassim Harramein [33] with unified field theory (see also Section 3), which connects the standard model and gravitation in a single unifying framework (see also [26,34,35]).
From this viewpoint, the global brain is the instrument to apply and study multiple theories from multiple disciplines: psychology, physiology, informatics, cybernetics, physics, mathematics, and others in one brain framework. Nikola Tesla stated also that we live in the age of electricity [36], where electricity is the meeting point for chemists, physicists, and physiologists, as the electromagnetic energy can be transmitted wireless and its power conducts mankind to progress.
Electricity is a basis for any global system and the global brain can be viewed as an application of Tesla's scientific work. Therefore, Tesla's vision of communication services in a global system have become possible to implement almost a hundred years later due to Internet and machines [37]. Therefore, the previous Figure 1 is a vision of a complex adaptive system composed by artificial machines able to manifest rational, emotional, and social intelligence (levels of IQ, EQ, SQ) and humans with, also given by nature, same characteristics, the communication being done at the level of artificial intelligence based on cybernetic algorithms (certainly, the relatively recent discoveries in the aria of gravitational waves can relate the things with the gravitation/graviton also).
The final scope of global brain is to be a knowledge instrument used by humans to develop more self consciousness in their natural habitat. To preserve the habitat of life with other forms of biological and non-biological entities, humans should design a smart and self adaptive system with ethics and human psychological components. Mihai Golu [38,39] studied the field of neuropsychology and its cybernetic dimension and his work is a startpoint to define global brain to be able to manifest consciousness at a higher level of performance in comparison to nowadays human society manifestations on all aspects of development.
Regarding the mathematical instrument to design and operate inside global brain, the non-standard analysis in topoi is a research startpoint achieved in the following Sections of this work (for previous works see [19,20]). In this setting, we propose a starting definition for infons, energons, and receptons, as working tools used to understand the things better and better.
The things will be partially formalized in a mathematical language, in accordance with the previous characteristics sketched for them, coming from biology and physiology, physics, and cybernetics. The formalism aims to produce a new mathematical approach for the problems of neuroscience, physics, and artificial intelligence. A list with problems from the of neuroscience is contained in the book [40]. These can be also considered a list of problems from physics in emergent quantum mechanics.

Theories versus Categories; Topoi and Mathematical Logic
The results from this Section are mostly known, so it is essentially a review section, representing useful preliminaries for the original mathematical part of this paper consisting in the construction of the non-standard extensions in topoi of the SET-type, from the next Section 6, and for the original definitions from Section 7. Here, we present, in Section 5.1, the interplay between theories and categories in the classical context of the bivalent logic and Zermelo-Fraenkel (Choice) axiomatics, as precursors of the mathematical definitions of the infons and receptons from Section 7.
Later, in Sections 5.2 and 5.3 we will pass to a special kind of categories, named topoi, with their intuitionistic logic detailed in Section 5.3. For more details concerning the topics from this Section, we refer to [4,5,13,41,42]. See also the extended version [1], which have the same structure as this work, but the mathematical notions are presented in much more detail.

Theories versus Categories in ZF(C) Axiomatics as Precursors of the Mathematical Infons and Mathematical Receptons
Definition 1. A class is an arbitrary collection of objects. A set is a class which belongs to another class (one avoid in such a way the well-known Russell's paradox, 5.2 from [1]).

Definition 2.
A language L is the family of all formulae, obtained inductively using conjunctions, disjunctions, negations, implications, and quantifications (existential, universal) of the atomic formulae (represented by equality and membership of variables, sets, t-uples, τ(σ) with τ and σ from before) (see also 5.3 from [1]).
If N is a family of classes stable to ∈ (that is ∀y(y ∈ N ∧ x ∈ y → x ∈ N ), then we denote by L N the restriction of L to N (this means that we replace in the classes with those classes from N only).

Definition 3.
A sentence (or proposition) is a formula having all the variables bounded (quantified), so there is no free variable. A sentence is true or false (bivalent logic, see Definition 7).  1). A model of a theory is a concrete mathematical object on which all the sentences from set A of axioms of a given theory are true. We assume here, by definition, that, in our context, a theory has at least one model (this represents a compatibility with Tegmark's rank 4 multiverse, see Section 3), which is not automatic for the theories of order at least 2 (see Definition 11). However, for the theories of order 1 (i.e., having a first-order language) it is true that such a theory is non-contradictory if it has a model.

Definition 7.
A classical logic (CL) is a theory having the following axioms:

Remark 1.
We recall from the list of ZF the following axioms only (see 5.10-5.14 from [1]): where ϕ is a formula in L M , with the free variables among z, ω 1 , . . . , ω n and having y as a bounded variable. In particular, there are sets of the form {x|ϕ(x)}; note that this axiom allow us to consider such kind of sets only if they are subsets of a bigger set, avoiding in such a way the Russell's paradox. Moreover, this axiom allow us to built the empty set, specifically Definition 8. ∅ = {x ∈ y|x ∈ x ∧ ¬(x ∈ x)} (the empty set). Moreover, the empty set is unique (does not depend on y) by the Axiom (E). U n (T), where U 0 (T) = T and U n+1 (T) = U n (T) ∪ P (U n (T)), ∀n ≥ 0. This is the Universe generated by T (one can consider a bigger universe, where the union is taken over all the ordinals, by transfinite recurrence-the von Neumann Universe-but we do not need it now). It is constructed from the set T, using ZF axiomatics. We remark here that, even when T = ∅, we have 0 = ∅ ∈ U 0 , and, if ω ∈ U n , then S(ω) = ω ∪ {ω} ∈ U (n+1) , so N ⊂ U (∅). Then, take T = N. We find, using the standard constructions of the number fields, that all of them (Z, Q, R, C) are contained in U (N), by the definition of U (N) and the ZF axioms. Continuing, it is easy to see that, in fact, all the necessary mathematical structures can be progressively built starting with the empty set ∅ (whose existence is postulated by ZF), using the ZF(C) axioms, in higher and higher universes as before.

Definition 11.
A language L U , associated with the previous Universe U can be defined as in Definition 2. If the quantifiers applies to U 0 (T) = T only, we call the resulting language, denoted by L T , a first order language, if the quantifiers applies to U 1 (T) only, we call the resulting language a second order language, denoted by L U 1 (T) , etc., if the quantifiers applies to U n−1 (T) only, we call the resulting language a n-th order language, etc., in this terminology, L U being called a higher order language.

Definition 12.
The family of all the models (or of a set of models) of a theory, together with the "dictionaries" between them (functions, or other objects, compatible with the structure of the theory) gives a category C T associated to the theory T .

Definition 13
h for which all the compositions are defined; (ii) ∀X ∀Y, X, Y ∈ Ob(C), ∀ f ∈ Hom C (X, Y), the following diagrams are commutative:

Remark 2.
Any category (the models of a theory) is a model of the theory of categories, as are, for instance, the last two examples from Example 2.

Remark 3.
The category associated to the theory of categories is a category of categories, named CAT, that is: Ob (CAT) = categories Hom CAT (C 1 , C 2 ) = the functors (see Definition 14) from C 1 to C 2 , with "•" and 1 C defined canonically. Definition 14. Let C 1 and C 2 be two categories. A functor F : C 1 → C 2 is acting on objects: F(A) ∈ Ob(C 2 ), ∀A ∈ Ob(C 1 ) and on morphisms: F( f ) ∈ Hom C 2 (F(X), F(Y)), ∀ f ∈ Hom C 1 (X, Y), being compatible with compositions and identities, that is:

Remark 4.
The previous definition introduce both the notion of morphism in CAT, and the theory of functors also. Thus, one expects to obtain the category of functors also. Indeed, Definition 15. Let C 1 and C 2 be two categories. We define the category C C 1 2 as follows: Definition 16).
The composition and the identities are defined canonically.
Here we need the following:

Remark 5.
We insisted until now to obtain categories from theories, but the converse also works, through axiomatization. Thus, the theories and the categories are the two faces of the same thing, but they represent different notions. Definition 17. Let C be a category. The dual category of C, denoted by C op is the category having: , ∀X, Y ∈ Ob(C) (reversing the arrows).

Definition 18.
Let us assume further that the reader is familiar with parts of the theory of categories.
If not, we produced the manuscript [1] (Sections 5 and 6) exactly for those readers (from neuroscience, or some physicists, economists) which feel better to have all the necessary things collected in only one place. A much more detailed treatment is presented in [4]. In particular, we assume as known the notions of initial object (usually denoted by 0), final object (usually denoted by 1), monomorphism, epimorphism, isomorphism, subobject of an object, direct product, direct sum, equalizer, co-equalizer, inductive limit, projective limit, pullbacks, pushouts, exponentiation, (usually denoted by X Y , with X, Y objects), subobjects classifier (usually denoted by Ω). The notions product and sum, initial and final, epi and mono, co, etc., and without co, pull and push, inductive, and projective, etc., are dual notions. An example is given by the next g as a functor does). This is the first example of a pair of dual notions (functor vs. cofunctor).

Topos Theory
Now let us pass to the main notions used in this paper: Definition 20. An elementary topos is a category E such that: (T1) E has finite inductive limits; (T2) E has finite projective limits; (T3) E has exponentiation; (T4) E has subobjects classifier (see [1,4]).
Remark 6. It can be shown that the conditions required in the definition of a topos, implies the existence in such categories of the objects having the propriety to be any of the objects from Definition 18 (Therefore, there exists initial and final objects, (finite) direct products and sums, equalizer, coequalizer, pullbacks, pushouts, power object of any two objects, etc.).

Examples of Topoi
(1). The category SET of sets; (2). The category of n-uples of sets with the morphisms defined on components. This can be written as SET n , where n is the discrete category having Ob(n) = {1, 2, . . . , n}, 1, 2, . . . ∈ N as objects and only the identities as morphisms (this category is of type from Definition 15); (3). SET C op is also a topos. The previous example is a particular case, when C = n.

Definition 21.
A topos of the form (3) will be called exponential topos. If the topos from (3) is obtained for a category C having a finite number of objects and morphisms, we will call it a SET-type topos.

Remark 7.
In Section 6, we will construct non-standard extensions for SET-type topoi, as finitegeneralization of the non-standard analysis in the topos SET, introduced in [9], see also [1,6]. This will be an original mathematical contribution. The general case of the exponential topos will be considered in Part II of this work.

Oriented Graphs as Topoi of SET-Type
An oriented graph − → G consists in a finite number of points A = {A 1 , A 2 , . . . , A m } (the word point, is only formally used, the elements of A are not necessarily geometric entities), some (or all) of them being joined with some others (or all) of them by oriented segments (arrows); we note that the elements of A can be categories, and the arrows can be functors (for example).
Let us suppose here, additionally, that for each i, j, i, j ∈ {1, . . . , n} there is at most one arrow, either from A i to A j or from A j to A i , that the path of two consecutive arrows can be replaced with an arrow, and implicitly assume the existence of an unique arrow from A i to itself, for any i as before, namely the identity arrow (which has no orientation). This graph can be organized as a category of type n with some additional morphisms: add to n a morphism from i to j, i, j ∈ n if there is an arrow from A j to A i in − → G . Let us denote this new category by n− → G . Then, the SET-type topos SET n− → G op can replace the given graph. In fact one can work with a category of n-uples of sets, (X 1 , . . . , X n ) requesting additionally that X i ⊂ X j , whenever there is an arrow from A i to A j in − → G . This approach can be used to study various kind of graphs, in particular graph neural (quantum) networks of various kinds (optical, convolutional, etc.) with the quantum aspects modeled in topoi.
Such a development is related to deep learning, artificial intelligence, quantum computers (starting with the quantum Turing machine). Moreover, mathematical game theory is also related to graph theory, and some economical applications can be obtained, for instance related to the new aria of neuroeconomics. There are also connections between graphs and the evolutionary games (with applications in the study of climate change, evolutionary biology, etc.).

Geometric Topoi
One can use the previous topos-interpretation of the oriented graphs to build geometric topoi. Let us consider the following oriented tetrahedron, as in Figure 2: from i to j, i, j ∈ n iff there is an arrow from A j to A i in G . Let's denote this new category 661 by n− → G . Then the SET-type topos SET n− → G op can replace the given graph. In fact one can 662 work with a category of n-uples of sets, (X 1 , . . . , X n ) requesting additionally that X i ⊂ X j 663 whenever there is an arrow from A i to A j in − → G . This approach can be used to study various 664 kind of graphs, in particular graph neural (quantum) networks of various kinds (optical,   Let us denote by − → T the oriented graph given by the previous tetrahedron, and by n− → T the discrete category n at which we added the additional morphisms 2 → 1, 3 → 1, 4 → 1, 3 → 2, 4 → 2, 4 → 3. Then, the previous tetrahedron can be recognized in the topos SET n− → T op or, alternatively, in the topos whose objects are the quadruplets and having as morphisms quadruplets of functions compatible with restrictions. In such a way, one can represents in topoi any polygon, polyhedron, and polytope from R n , n ≥ 2.
The 12 vertices of the three squares are the middle of the edges of a cube having edges of length 4, the center in O, and the faces parallel with the planes of coordinates. Thus, this polyhedron is determined by a cube and a (regular) tetrahedron. It can be represented as a topos using the method from Section 5.2.3. However, even the Gosset polytope from R 8 can be represented in such a way once we give orientations to its edges. Moreover, the group of symmetries of the Gosset polytope is a group which can be found in the structure of the (−2) curves from a degenerate del Pezzo surface of degree 1. We will deal with this subject in a forthcoming paper, based on [43,44].

Remark 10.
Given a topos E , it is possible to define, for any X ∈ Ob(E ), operations on its subobjects, which are similar, in some sense, with the union, intersection, complementary, empty set, total set from the case SET (and are exactly these operations for SET). In a general topos, for any a, b ∈ Sub E (X) ( = the family of subobjects of X) one defines: union a ∪ b (using the existence of the equalizers and direct sums), intersection a ∩ b (using the existence of pullbacks), implication a → b (using the existence of the exponentiations), minimal object = 0 ∩ X, and maximal object = 1 ∩ X. In the case of SET, the usual operations between sets gives a structure of Boolean algebra. In a general topos, the previous operations gives a structure of a Heyting algebra. We obtain an analogue of the power set of X. We suppose all these things as known. For details, see [4,42] (the constructions are included in 6.37-6.42 from [1] also).

Remark 11.
For the topos SET, the subobjects classifier is Ω = {0, 1} (via the usual characteristic function), and, in CL and ZF(C) axiomatics, the number of truth values is the cardinal of Hom SET (1, Ω)(here 1=any singleton), that is 2. For a general topos, on Hom SET (1, Ω) one can define logic operators ∨, ∧, →, ⊥, using the so called Ω-axiom appearing in the definition of the subobjects classifier. Then, (Ω, ∨, ∧, →, ⊥, ) is a Heyting algebra, in general. When it is a Boole algebra, one says that the topos E is a Boolean topos. We again assume these things as known. Details, again in [4,42] (the constructions are also included in 6.43, 6.44 from [1]). Moreover, one can give to Ω a structure of MHA also. See [45,46] (referred in [4], p. 457). See also 6.51-6.53 from [1].

Remark 12.
Fix a language L, and, for a given topos E having Ω as subobjects classifier, we assign to any atomic sentence a from L (see Definitions 2 and 3) an element V(a) from Hom E (1, Ω). Then, extend this association V to all the sentences from L, using the logical connectors for L, and, correspondingly, the operators from Remark 11 in Hom E (1, Ω) (V(p ∨ q) = V(p) ∨ V(q), with the ∨ from the left being the logical connector, and ∨ from the right being the connector from Remark 11, similarly for the other operations ∧, etc.). This extends the known Boolean valuations. We remark that card(Hom E (1, Ω)) represents the number of truth values in E (possible infinite). They are called the global sections of Ω.

Non-Standard Analysis in SET-Type Topoi
Let us begin this Section by shortly recalling (in Section 6.1) the notion of non-standard enlargement of a standard universe, denoted by U ⊂ * U , where U = U (T) is as in Definition 10 (with its mathematics), as introduced by Abraham Robinson in [9]. This works in the category (topos) SET. Modern details can be found in [6]. See also [47,48]. Most of the results from [6] are also presented in 7.20-7.45 from Section 7 of [1]. Then, in Section 6.2, we extend the things to SET-type topoi, introducing for the first time non-standard extensions for such kind of topoi. These are original results from pure mathematical point of view.

Non-Standard Enlargement in SET
universes. Let us denote by * U := * (U ) ⊆ U and by * a the image of a, * a := * (a), ∀a ∈ U . For a sentence ϕ in L U , let us denote by * ϕ the sentence in L U obtained by keeping unchanged the primary relations ∈ and =, as well as the logical quantifiers ∨, ∧, ¬, →, ∀, ∃, and replacing all the constant symbols a ∈ U by * a ∈ U . One requests that * satisfies the axioms: S, T : An L U -sentence ϕ is true ⇔ * ϕ is true as L U -sentence; the ⇒ is called transfer (T), and ⇐ is called standardization (S); • I (idealization): If R ∈ U is any finitely satisfiable binary relation (i.e., ∀n∀x 1 , . . . , x n ∃y(yRx 1 ∧ . . . ∧ yRx n )), then * R is globally satisfiable (i.e., ∃y ∀x(y * Rx)). We should remark that the xs and y run inside U , while y runs through U ( * x = x, ∀x ∈ domR, y ∈ U , y ∈ U ). We note that the order of the variables is not important, because, if R = R(x, y) is a binary relation, so is R(y, x) = R −1 .
In the previous situation, one call the extension U * → U , a non-standard extension of U . We note that the inclusion U (T) ⊂ U ( * T), ( * T ∈ U ) is a non-standard extension ([6], Enlargement Theorem, p. 185).
Remark 13. 1. If U = U (T) and U = U (T ) are universes, with T ⊂ T , then take T such that T ⊂ T , T ⊂ T and let U = U (T ). Then L := L U = L U |U and L := L U = L U |U (notations from Definition 11). We call ZFC + CL + I + S + T := IST (internal set theory), with CL from Definition 7, ZFC from Remark 1, and I,S,T from before. The triplet (L = (L U , L U ), IST, D = {MP, G, P}) (see Example 1) represents the Theory of non-standard extensions. We point out that this theory has a higher order logic, that is, it is possible to quantify over any entity.
2. If U has ZFC then * U has IST as axiomatics. Moreover, a sentence ϕ in L U conserves its truth value (0 or 1) when passing to * ϕ in * U . We say that the extension U * → * U is conservative. Moreover, U is not contradictory if * U is (because we can construct models for * U starting from U with ZFC, see [6], or look at [1] 7.38-7.40).

Remark 14.
One can define the sets * N, * Z, * Q, * R, * C of the non-standard number sets. In particular, in * R one obtains the notions of infinitesimal, unlimited, infinite, etc., numbers. We also can obtain further properties of non-standard numbers. See The union and intersection of any internal collection of sets are internal: if {A i |i ∈ I} ∈ * U and A i internal for any i, then i∈I A i and i∈I A i are internal; • If a 1 , . . . , a m are internal, then both {a 1 , . . . , a m } and (a 1 , . . . , a m ) are internal; Example 6. 1. N is an external set in U . 2. * (A) := { * a|a ∈ A}, A infinite is external (note that * (A) = * A).

Non-Standard Analysis in SET-Type Topoi
We start now to achieve our main objective from the mathematical point of view, namely to build a non-standard analysis (NSA) in topoi, extending the previous theory from the classical setting (in the topos SET) to more general topoi. We will be only interested in building NSA in the exponential topoi of the form SET C op (such kind of topoi were also used in quantum physics). However, for the moment we will give the general definitions, but we will exemplify the ideas on the SET-type topoi (see Definition 21).

Remark 15.
Let E be a topos. In Remark 11, we recalled the existence of all the logical operations in the space of sections Hom E (1, Ω), such that this space become a MH A (see Definition 23). Hence, we can interpret the notion of theory (in the sense of the Definition 6) in any topos.

Definition 26.
Let E be a topos, and X ∈ Ob(E ). Then, we can associate to any two morphisms α ∈ Hom E (1, X) and β ∈ Hom E (1, X) their direct product (α, β) ∈ Hom E (1, X × X) (1 is (the) final object, so 1 × 1 = 1). Then, a binary relation R on X is a set of pairs (α, β) as before. As per usual, we write αRβ instead of (α, β) ∈ R. We say that a binary relation on X in finitely satisfiable if for any positive integer n and α 1 , . . . , α n ∈ Hom E (1, X) there is β ∈ Hom E (1, X), such that α 1 Rβ, . . . , α n Rβ. Definition 27. Let E be a topos. An enlargement of E is a monomorphism * : E → * E in a category CAT containing both categories E and * E , satisfying: is any valuation (see Remark 12) (here P is the set of sentences in the language L E , 1 is the final object is E , and Ω is the subobjects classifier from E ), then * V( * p) = V(p) for any p ∈ P (here * V : * P → * Hom * E (1, * Ω) is the non-standard image of the original map V, 1 = * 1 is the final object, and * Ω is the subobjects classifier from * E ); • 3. For any X ∈ Ob(E ) and any finitely satisfiable relation R on X there is g ∈ * Hom * E (1, * X), such that α * R g for any α ∈ Hom E (1, X) (we say that any finitely satisfiable relation is globally satisfiable) (if R ⊆ Hom E (1, X × X) then * R ⊆ * Hom * E (1, * X × * X)).

Definition 28.
The inequality * V( * p) ≤ V(p) from 2. is called standardization (S), the converse inequality is called transfer (T), and the propriety 3. is called idealization (I). They generalize the similar proprieties from CL appearing in Section 6.1, and can be formalized in the appropriate language. ∀x((∀y ∈ x ∧ φ(y)) → φ(x)) → ∀xφ(x), for every formula φ . Note that this in setting the double negation, de Morgan's relations, and tertium non datur, do not hold.

Definition 30.
An intuitionistic theory is an usual theory T = (L, A, D) (see Definition 6) in an intuitionistic context, so assuming that A contains the axioms IL and IZF from before, and that D = {MP} (see Example 1). As in Definition 6, we assume, by definition, that a theory has at least one model otherwise it is not considered a theory in our sense).

Definition 31.
Using the same notations as in Remark 13, but with IZF from Definition 29 for intuitionistic set theory, we denote by IIST = IL + IZF + I + S + T with IL from Definition 22 and I, S, T from Definition 28. As in Remark 13 we define the theory of intuitionistic non-standard extensions, replacing IST by IIST and taking only D = {MP} from Example 1. The propositional calculus of this theory is a conservative extension of the intuitionistic logic (that is by T and S one conserve the truth values).

Infons, Receptons, Energons
Here are presented the essential definitions for which this work was done, namely:

Definition 32.
A generalized infon is a Theory I = (L, A, D) in the context of IIST. An infon is an entity of the previous form, whose category of models form a non-standard enlargements of a topos of the form SET C op , where C is any category (see Section 5 also) and a preliminary defined infon is the infon whose associated category is a non-standard enlargement of a topos of SET-type (see Definition 21).
Correspondingly, Definition 33. A generalized recepton is the category of models of a generalized infon, an recepton any non-standard enlargement of a topos of exponential type SET C op , where C is any category (see Section 5 also) and a preliminary defined recepton is any non-standard enlargement of a topos of SET-type (see Definition 21).
We recall in the context that the logic from topoi is intuitionistic, and its non-standard enlargement is a conservative extension (see Sections 5 and 6), so the infon and the recepton are compatible entities, representing the two facets of the same entity.
An energon is also generalized, normal and preliminary, as before. It is partially infon (when dealing with theories related to potential energy) and partially recepton (when dealing with the kinetic, caloric, electromagnetic actual energies), see also Section 3. (2). The Nicolas Bourbaki group of prominent mathematicians from France considered that the fundamental structures in mathematics are (both in the language of theories and of categories) the algebraic structures, topological structures, and order structures, all the others (like the differential ones, for example) being combinations between them.
Thus, in order to apply mathematics in various arias, one needs to consider collections of such structures, communicating between them. This means that we need to consider various theories and categories together. In the case of categories, the structures are collected in categories, and the communication between them is done with the help of the functors. This means that, in fact, one needs to work with categories of type CAT (see Remark 3), and, as a consequence, with appropriate infons and receptons.
This means that an important family of receptons will have the form SET CAT op and similarly for the corresponding theories. For example, in multiversal quantum physics several mathematical structures appearing in the usual quantum physics were put together in such kind of topoi, such as Hilbert spaces with compact self adjoint operators, von Neumann algebras, and others (see [7]). However, when studying the human and global brain, is necessary to enlarge the things to the non-standard environment, due to the role played by the subconsciousness (see [1,19]).

The Yang-Baxter Equations
In forthcoming Part II of this work, we will present, in more detail, the connections between DNA, quantum computers, and the sequence knots, braid groups, and Yang-Baxter equations (YBE). Here, we begin with an introduction to the Yang-Baxter equation. After that we will present related results and applications of this equation.
Let V be a vector space over k. Let I = I V : V → V be the identity map of the space V. We denote by τ : Now, we will give the main notations for introducing the Yang-Baxter equation. (1) We also require that the map R is invertible.

Remark 17.
An important observation is that if R satisfies (1) then both R • τ and τ • R satisfy the QYBE (the quantum Yang-Baxter equation): Remark 18. The Equations (1) and (2) are equivalent.

Remark 19.
There is a similar terminology for the set-theoretical Yang-Baxter equation. In this case V is replaced by a set X and the tensor product by the Cartesian product. We will explain this definition in the next examples below.
Remark 20. Let X be a set containing three (bivalent) logical sentences p, q, r (i.e., p, q, r ∈ X). We can choose X as rich as we wish for the moment. Later, we will try to find the smallest X which fits for our theory.
It follows that One way to check that (3) holds is to make a table with values for p, q, r.

Remark 21.
Another interesting solution to the set-theoretical Yang-Baxter equation is the following. Let R : X × X → X × X, be defined by R(p, q) = (p → q, p). Again, one way to check the above statement is to make a table with values for p, q, r.
It is now the moment to discuss about X. What can be said about X in general (what is the smallest X for which R is well-defined)? One could consider a set X containing more than three logical sentences. What is the interpretation of the set-theoretical Yang-Baxter solutions in this case?
For example, if B is a Boolean topos (see Remark 11) the pairs from Remark 21 gives solutions for the YBE (1) from the global sections of subobjects classifier Ω (i.e., Hom B (1, Ω)) of B, while the pairs from Remark 20 gives solutions from the global sections of subobjects classifier Ω (i.e., Hom E (1, Ω)) for both equations YBE and QYBE ( (1) and (2)) and for any topos E .
We can go a step further and consider an algebra of type (2,2), (A, * , •), and call the operations * and • YB conjugated if R(a, b) = (a * b, a • b) satisfies the set-theoretical Yang-Baxter equation. We propose the study of algebras with two operations of type (2,2) which are YB conjugated. (Distributive lattices, groups and self-distributive structures fall into this category.)

Remark 22.
We now consider other equations for a k-linear map R : V ⊗ B V → V ⊗ B V: (4) and (5) then R is a common solution for (1) and (2). If a k-linear map R : (1) and (2) then R is a solution for (6). If a k-linear map R : V ⊗ B V → V ⊗ B V verifies (1) or (2) then R is a solution for (7).
Proof. We only prove the first claim: The other claims follow in a similar manner.
We will write the above results in the following manner: (1) ∧ (2) → (6); (1) ∨ (2) → (7). As a direct application of this theorem, one can check which of the functions presented in this section are common solutions for the braid condition and the quantum Yang-Baxter equation.
We are now preparing the framework for the next theorem.
For an equation p (for example, R 12 • R 23 etc. = 0), we can associate ¬p (for example, R 12 • R 23 etc. = 0). Now we can consider the equation the equation always true (p ∨ ¬p) and by ⊥ = ¬( ) the equation with no solutions.
We now can state a theorem with applications in the understanding the theory of the Yang-Baxter equation.
Theorem 3. The set of equations of operators of type R : , has a natural distributive lattice structure. In this lattice, one can define a partial order, a minimum ⊥, a maximum , and operations ¬ and →.
The Yang-Baxter equation represents some kind of compatibility condition in logic. Let us consider three logical sentences p; q; r, and let us suppose that if all of them are true, then the conclusion A could be drawn, and if p; q; r are all false, then the conclusion C can be drawn.
Modeling this situation by the map R, defined by (p; q) -> ( p or q; p and q), helps us to comprise our analysis. The Yang-Baxter equation guarantees that the order in which we start this analysis is not important (see [56]). How to obtain solutions for set-theoretic Yang-Baxter equations in a context of Boole or Heyting Algebras is also indicated in [52,57,58], or even in [59].

Conclusions and Further Considerations
To conclude, in this paper we obtain three kinds of contributions. One is related to pure mathematics and consist in constructing the non-standard analysis in SET-type topoi (Theorem 1); another one is related to foundations of mathematics; arguments from physics, neuroscience, cybernetics are used in order to justify the importance of introduction of the IIST axiomatics and to provide models for it (based on Theorem 1 again); and, last but not least, the introduction and justification of the infons, energons, receptons: how they should look like and which is their usefulness (Sections 2-4). We should also refer to non-mathematical original contributions: building a cybernetic model of the global brain (Section 4), inspired from original interpretations of the human brain (Section 2).
We will continue our work with part II, where we will build non-standard analysis in general exponential topoi, with applications to infons and receptons in their natural form. Then, these notions should be used in concrete problems of physics, neuroscience, cybernetics, artificial intelligence, economics, etc. In particular, we will study the (quantum) neural networks and (quantum) Turing machines in the context of topoi, related to the interplay of brain and mind (genetic and epigenetic) and in connection with the global brain. Braid groups will be also considered in the context. Example 8. We end this paper noting that the theory from Section 7 (ZF from Section 5 in the classical case) represents an infon, containing the empty set. From it (so, from nothingness), one built all the mathematics, by receiving the theory through various receptons, parts of the global brain. By Tegmark's axiom for multiverses of rank 4 (see Section 3.1) all the physical entities progressively appear, through consciousness (and subconsciousness).

Remark 23.
We might put the problem in which way the information from infon is received by recepton. We propose as a model of recepting the (quantum) neural networks, as defined in [60], modeled in the IIST axiomatics, in a non-standard extension of a special case of temporal topoi (of exponential type) as in [61]. These are not SET-type topoi, so we have to develop non-standard analysis in such topoi also, in the second part of the paper. The dendrites are connected to infons, and the axons to receptons.
In our brains, the neural networks (synapses) help the concrete understanding of the existing entities, and in the global brain the artificial neural networks also plays an important role. Due to the work of R. Penrose [62][63][64] and S. Hameroff, we must consider the quantum phenomena from the brain, at least partially, although there are discussions on their existence (see [65]); however, it seems that some phosphorus compounds extend the period of superposition in the worm and humid environment from the brain, with possible applications to quantum computers, which might work at the room temperature also, not necessarily close to 0 degree Kelvin).

Remark 24.
In the literature there are two more notions belonging to the family of words related to consciousness, namely superconsciousness and unconsciousness.
(A) The word superconsciousness has several meanigs: religious, related to meditation etc. (in general, belonging to the spiritual world). We are not interested now in these meanings, the definition which we use is that superconsciousness is the consciousness of the global brain, topic already considered in Section 4. We give three classes of examples of informations comming to us through superconsciousness, namely: (i) the Human Genome Project (aiming to make a map of the information from a human ADN: genomics): https://www.genome.gov/human-genome-project; (ii) the Human Brain Project (aiming to make a map of the neurons and synapses from a brain: connectomics): https://www.humanbrainproject.eu/en/, and https://www.epfl.ch/research/ domains/bluebrain/; (iii)to make the map of the known univers: https://www.nature.com/articles/d41586-021-014 66-1.
In all these examples there is no scientist (or human being) to know all the information, but the specialists can see the maps (which are not yet ready, it is work in progress in all three cases). In order to produce these maps, networks of supercomputers are necessary (situated, preferably, in cloud, so with unknown locations, to avoid the hacking). So, the information comes vrom many scientists from many arias of research, and uses high technology, most probably without the help of quantum computers the works cannot be done. All these statements related with many human brains and technology are parts of the global brain. So, we become aware of these things with the help of global brain.
(B) The word unconsciousness has also several meanings, including in a medical sense. Here we consider it as denoting that information which exists, both in our brain (in neurons, synapses) and in ADN, but we become-maybe-aware of it only through the interplay between genetics and epigenetics (see Section 2). It can be mathematically modeled using both (quantum) neural networks and (quantum) Turing mashines. It can be specific both to a human being, and to the global brain (archetypes, cultural customs, etc.). For further information, see also [66,67].
Author Contributions: The authors worked on the subjects from this paper collaborating in the multidisciplinary scientific seminar Mathematical Modeling: Brain and Global Brain at the "Simion Stoilow" Institute of Mathematics of the Romanian Academy (IMAR). Parts of this paper were presented at the international Conferences entitled: "The 14-th International Workshop on Differential Geometry and its Applications", (https://www.upg-ploiesti.ro/ro/14-th-international-workshopdifferential-geometry-and-its-applications-petroleum-gas-university), and "Geometry and Physics", https://events.theory.nipne.ro/gap/index.php/conference, held at IMAR, where the authors also met. Conceptualization, Methodology, and Supervision, O.P.; Section 2 is entirely due to C.E.M.; Section 4 is entirely due to I.R.B.; Section 8 is entirely due to F.F.N.; Sections 3, 6 and 7 are entirely due to O.P.; Section 5 is mostly a review section; Sections 1 and 9 (which are not formal Sections) are due to I.