Previously, we proposed a consciousness model based on human language and attempted to interpret psychological consciousness, psychological time, and the quantum mechanics concept of Qbism [1-5]. Through this, we believe we have been able to understand the mechanism of consciousness as a physical monism.

In this paper, we focus on the "mind" that we always feel. If consciousness is purely a physical phenomenon, then the mind, as a collective entity, can also be considered a physical entity. However, when we communicate with someone, we always feel that they have a "mind". If the concept of mind is a physical phenomenon, we need to consider the mechanism behind why we feel that someone has a "mind" even though there is no substance as a "spirit". This paper focuses on the mind as an application of the consciousness model proposed by the authors based on human language. The authors have attempted to interpret psychological consciousness, psychological time, and the concept of Qbism in quantum mechanics, and believe that they have gained a certain understanding of the mechanism of consciousness as a physical monism. However, these approaches focus on a physical approach to consciousness and do not directly deal with the concept of the mind. Therefore, in this paper, we propose to use a set-theoretical approach to understand the mind in a physical sense.

If consciousness is assumed to be purely a physical phenomenon, then the mind, as a collective entity, can also be considered a physical entity. On the other hand, when we communicate with someone, we often feel that they have a "mind". If the mind is a physical phenomenon and there is no substance as a "spirit", it seems natural to consider the mechanism behind why we feel that someone has a "mind".

Specifically, by using a set-theoretical approach, the mind can be understood as a set that combines relationships that started from an empty set and have a physical entity. Furthermore, since this approach is known to be useful for understanding abstract concepts, it is expected to be useful as a means of understanding various abstract concepts, not just the concept of the mind.

Thus, this paper proposes to use a set-theoretical approach to understand the concept of the mind in a physical sense.

In recent years, the development of artificial intelligence and machine learning has advanced computer perception and cognition. However, we still cannot fully understand the essence of the human mind. Therefore, research is being conducted to explore the essence of the mind by mathematically representing the concepts of the human mind. Set theory is used as a mathematical model of the mind.

In the past, there was a field called "social psychology," which was first proposed in the 1930s. In this field, relationships between sets were analyzed using sets that included an individual's psychological state as an element.

In his 1983 work, "New Metaphysics," David Lewis proposed modeling consciousness using set-theoretic methods. He attempted to represent the state of consciousness using sets that included an individual's psychological state as an element [6].

Cognitive scientist Douglas Hofstadter attempted to mimic the self-referential structure of the mind in his 1979 work "Gödel, Escher, Bach: An Eternal Golden Braid." He attempted to represent the complexity of the mind using infinitely self-referential sets [7].

In his 1987 work, "The Intentional Stance," Daniel Dennett proposed expressing the phenomenon of consciousness using a formal set that did not have elements. He attempted to understand consciousness by expressing the phenomenon of consciousness stylistically rather than using sets that included an individual's psychological state as an element [8].

John Searle proposed the concept of the "internal nature of consciousness" in the 1970s. He emphasized that consciousness is our inner experience and attempted to represent the phenomenon of consciousness using sets that included the individual's internal experience as an element" [9].

To represent the concept of the mind, a set consisting of an empty set is created, and tangible sets are created by the interaction of these sets. This approach is based on the "empathic consciousness theory", a theory that explores the origin of consciousness by the interaction of individual minds. This research is backed by philosophical discussions, including the origin of the mind. Aristotle, the ancient Greek philosopher, divided the human mind into three parts: intellect, emotion, and desire. In modern philosophy, dualism of mind and body and the discussion of the physical existence of the mind have become prevalent. Against this backdrop, modern mathematicians and philosophers are conducting research to represent and clarify the mind mathematically. However, mathematical models of the mind have raised many debates and concerns, as some argue that the mind cannot be simply expressed as a set. Nevertheless, this research provides interesting insights from various perspectives as one framework to understand the essence of the mind. This study analyzes the mathematical model of the mind using set theory and examines how it is formed based on previous research and philosophical discussions. The study also investigates how this mathematical approach will influence modern philosophy and neuroscience.

Having a mind means that there is some kind of relationship with others, and it is thought that a set with substance is formed by combining the relationships that started from an empty set. This set-theoretic approach is shown to be useful not only for understanding the concept of mind but also for various abstract concepts.

For example, when dealing with abstract concepts such as chord progressions in music, language grammar, and quantum mechanics in physics, it is known that a set-theoretic approach is effective. In particular, in quantum mechanics, mathematical approaches such as set theory and topology are indispensable for dealing with complex concepts called state spaces.

In psychology, set-theoretic approaches are also utilized. For example, psychologist George Miller proposed the concept of the "magical number 7±2" and used set-theoretic approaches to study the capacity of human memory for information that can be memorized at once. Similarly, set-theoretic thinking is sometimes utilized in experiments and data analysis in psychology.

As mentioned above, set-theoretic approaches are useful tools for understanding abstract concepts. In particular, it has been shown that a set-theoretic approach is one effective means of understanding the concept of mind.

Regarding the concept of the mind, it has been discussed in various fields such as psychology and philosophy. For instance, there is a method called introspection to analyze the inner workings of the mind as a model for representing the mind. Philosophically, there exist positions such as dualism, physicalism, and functionalism when considering the mind. Based on these positions, models for representing the mind have been proposed.

On the other hand, the approach based on set theory differs from these models. In set theory, the aim is to capture the essential features of the concept of the mind by representing it as a set with elements. Specifically, by defining the mind as a set rather than something that exists within the inner workings of humans, various psychological phenomena can be explained within the framework of set theory.

For example, in psychology, the states of the mind can be expressed using adjectives such as "sad," "angry," and "happy." To represent these states, sets can be defined with each one having elements corresponding to the concept. For instance, let S1 be the set representing a sad mind with the element "sadness," and let S2 be the set representing an angry mind with the element "anger." By combining these sets, more complex mental states can be expressed. For instance, when sadness and anger coexist, a set can be defined as the intersection of S1 and S2, which is S1∩S2.

According to a set-theoretical approach, the concept of the mind can be thought of as starting from the empty set. That is, an individual's mind initially contains nothing and is represented as the empty set { }. For example, let's say that person A's mind and person B's mind are represented as A = { } and B = { }, respectively.

However, humans are social animals and come to possess a mind through communication with others. Therefore, when A and B meet, they recognize each other's existence and begin to feel that the other has a mind. This sensation is represented mathematically as a set { { } } containing the empty set as an element. That is, the set C consisting of A's mind and B's mind is created by their meeting.

Mathematically, the set C is represented as follows:

C = {A, B} = {{}, {}}      1)

Here, the set C is a set that was born because of A and B meeting and contains the empty set as an element. It can be said that C was able to exist as a set with substance by creating a set containing the empty set as an element.

Furthermore, this example can be extended to consider the case of three or more people. For example, in the case of three people A, B, and C, let's represent each of their minds as A = { }, B = { }, and C = { } and define the set R representing their interactions as follows:

R = { {A, B}, {A, C}, {B, C} }    2)

This set R represents that there are interactions between A and B, A and C, and B and C. The set resulting from this is as follows:

D = {A, B, C} = {{}, {}, {}}                        3)

Here, the set D is a set consisting of the minds of A, B, and C and contains the empty set as an element. That is, even in the case of three or more people, a set with substance is born by creating a set containing the empty set as an element. "It can be proven based on the axioms of set theory whether such a D exists or not. Specifically, the "Axiom of Infinity" guarantees the existence of the power set 2^C, which contains all subsets of C. Here, 2^C refers to the set that includes all subsets of C.

Furthermore, it can be proven based on the "Axiom of Choice" that there exists a subset of 2^C that includes {a, b}. The Axiom of Choice is widely accepted as an important axiom in set theory and plays a crucial role.

This paper examines why we feel that the "mind," which inherently lacks physical substance, exists from the perspective of set theory. The conclusion is that, even though the mind lacks physical substance, one can recognize one's own mind through interactions with others and by feeling that others have minds. In other words, it highlights the recognition of the mind through communication and, conversely, the role of the mind as a tool for communication.

The model of the mind using set theory is a theory that consciousness arises through the interaction of multiple minds using sets that include individual psychological states as elements. This theory captures the complexity of the human mind and provides a theoretical framework for explaining the phenomenon of consciousness. However, this model is still a theoretical construct and current scientific technology cannot fully comprehend the human brain and mind.

This research not only provides a theoretical framework for understanding the phenomenon of the mind, but also suggests directions for future research and development. For example, it suggests that a more precise classification of elements is necessary when representing individual psychological states as sets by gaining a better understanding of the human brain in more detail. Additionally, it suggests that more complex models are required to grasp the essential characteristics of consciousness by expressing the phenomenon of consciousness stylistically.

Furthermore, this research suggests that set theory could be applied to fields such as artificial intelligence and robotics. For example, it may be possible to design social systems where artificial intelligence and robots can interact and possess self-consciousness.

Finally, this research not only challenges the complexity of the human mind, but also emphasizes the importance of scientifically exploring the human psyche. The model of the mind using set theory illustrates the importance of scientific exploration and provides new ways to deeply understand the human psyche."

By developing computational models based on set theory, it is conceivable to simulate various aspects of consciousness and propose testable hypotheses regarding its mechanisms and dynamics. Furthermore, set theory offers a formal language to explore philosophical inquiries about the nature of consciousness, its relationship to the physical brain, and the potential for artificial consciousness. These set theory-based mathematical models should be validated through empirical data and experimental verification to ensure their relevance and accuracy in explaining consciousness phenomena. While this approach provides a structured method for conceptualizing consciousness, it is crucial to recognize its limitations and acknowledge that interdisciplinary collaboration is essential for a comprehensive understanding of this fundamental aspect of human experience.