引子    集合  运算   悖论   完备性  完全性、相容性   独立性     形而上    无穷势   全息完全性

逻辑（公理，选择），集合（可分，可加，表示）  、区域（几何，连通，拓扑）  、映射（运算，变换，合成，泛函）、分异（确界，临界，跨界，过界）、统一（层次，维度，尺度，规模）

1. 被主宰的万能之神（全能悖论）

2. 系统本体的依他性（库尔特·哥德尔不完备性定理悖论）

3. 谁给理发师理发（罗素悖论）

4. 随机不可随机选（贝特朗悖论）

5. 连续世界的离散解（芝诺悖论）

6. 因果与序的逻辑对峙（祖父悖论）

7. 模糊需要确界（沙堆悖论）

8. 自我指涉的逆否（苏格拉底悖论）

9. 预测、选择与因果循环（鳄鱼悖论）

10. 同理误判之自掘坟墓（索洛斯的反身性悖论）

11. 认识的束缚是走进自我（柏拉图洞穴悖论）

12. 对叠加行为的时机诠释（薛定谔的猫悖论）

13. 纠缠无可度距离（量子纠缠悖论）

14. 连续可数至不可列（康托尔的连续统假设）

15. 自然很自然（忒修斯之船悖论）

16. 自由意志与选择公理（拉普拉斯妖悖论）

一：From ideology to ideosphere

The transition from "ideology" to "ideosphere" represents an evolution from a single, centralized collection of ideas to a diverse and interactive environment of consciousness. This shift reflects a deepening and broadening of people's understanding of thoughts, cultures, and values.

First, let's examine the concept of "ideology." Ideology refers to a collection of ideas, encompassing one's understanding, cognition, beliefs, viewpoints, concepts, thoughts, and values. The formation of ideology is influenced by various factors, including social, political, economic, and cultural aspects. It possesses a certain degree of stability and continuity, forming a consensus among specific social groups and having a profound impact on people's behavior and thinking.

However, with the continuous development and progress of society, people's understanding of ideology has also deepened and expanded. They have begun to realize that ideology is not a single, isolated system of ideas but is closely linked to the entire society, culture, and environment. At the same time, people also recognize that different social groups and cultural backgrounds will produce different ideologies, which complement and conflict with each other.

Under such circumstances, the concept of "ideosphere" emerged. Ideosphere refers to a diverse and interactive environment of consciousness, encompassing various ideologies, thoughts, and cultural values. In the ideosphere, different ideologies and cultural values interact and influence each other, forming a complex and rich intellectual and cultural landscape. This landscape reflects the diversity and complexity of society and provides people with more choices and possibilities.

The transition from ideology to ideosphere signifies that we no longer view ideological concepts and cultural values as isolated entities. Instead, we examine them within a broader and more diverse social and cultural context. This shift helps us to gain a more comprehensive understanding of human thought and cultural phenomena, and also enables us to better respond to the challenges and opportunities presented by globalization, informatization, and other contemporary trends.

In summary, the transition from ideology to ideosphere represents a deepening and broadening of our cognitive understanding of thought and culture, providing us with a more comprehensive and diverse perspective to understand and address complex and ever-changing social phenomena.

（文化属性之从意识形态到意识生态的文明进程）

二：The paradoxical extension of subspaces

As a partial space with a dimension less than or equal to the full space, a subspace possesses profound logical boundaries that have not fully resolved the intricate interplay between meaning and expression, as well as the imbalance between the content and form of thinking. It involves the process of extending the domain, properties, or structures from one set to a larger set or an entirely new space.In establishing interdisciplinary integration mechanisms, it is necessary to take into account the measurement standards and semantic differences between different fields to ensure accurate transmission and understanding of information.

（小而无内，大而无外之子空间的悖论开拓）

三：The deceptive function of the Stealing Hearts' Trickery

In the field of algorithms and computational science, "deceptive functions" typically refer to functions that are designed to exhibit deceptive properties, aiming to cause algorithms (such as genetic algorithms) to fail, converge to suboptimal solutions, or get trapped in local optima. The purpose of such functions is to test and evaluate the performance of algorithms and to find ways to improve them. In deceptive functions, the relationship between inputs and outputs can be complex, nonlinear, and even unpredictable, posing challenges for algorithms. Through continuous optimization and adaptation, the deceptive strategy is constantly adjusted and optimized to ensure its effectiveness and sustainability.

（本我无我逻辑界限说偷心天机之欺骗函数的构造理论）

四：Intelligent Wisdom: The Frontier of Wisdom in Machine Proof

The wisdom frontier of machine proof manifests itself in various aspects, including the ability to handle complex logical reasoning, process big data and intricate models, integrate with other technologies, and pose challenges to human wisdom and creativity. These aspects collectively constitute the innovative frontier and challenges of the field of machine proof, driving its continuous development and progress. In terms of handling complex logical reasoning, machines need to understand and apply various mathematical logic rules to perform deductive reasoning on premises and arrive at correct conclusions. As mathematics and logic evolve, new logic rules and theorems continue to emerge, and machine proof systems must continuously learn and adapt to these new knowledge to expand their processing capabilities. In practical applications, machine proof systems often need to process large amounts of data and complex models to support various scenarios. In formal verification, the system must handle intricate system models and specifications. To meet these challenges, machine proof systems require efficient data processing and model analysis capabilities to quickly and accurately complete proof tasks. Furthermore, with the advancement of artificial intelligence technology, more and more new technologies have been introduced into the field of machine proof, such as deep learning and reinforcement learning. Enhancing their integration with other technologies enables these new technologies to provide novel ideas and methods for machine proof, enabling it to handle complex problems more intelligently. Additionally, machine proof also needs to integrate with other technologies, such as natural language processing and computer vision, to achieve wider applications. In terms of challenging human wisdom and creativity, although machine proof systems have made significant progress, they still cannot match human experts in certain aspects. For instance, in solving complex mathematical problems, human experts often rely on intuition and creativity to find ingenious solutions, which machine proof systems may not be able to achieve. Therefore, the development of machine proof still requires continuous challenges and surpassing human wisdom and creativity.

（智力边缘之仿生仿真机器学习与证明的智能智慧化）

附记：数学与哲学评估说范畴