We consider consciousness attributed to systems in space-time which can be purely physical without biological background and focus on the mathematical understanding of the phenomenon. It is shown that the set theory based on sets in the foundations of mathematics, when switched to set theory based on ZFC models, is a very promising mathematical tool in explaining the brain/mind complex and the emergence of consciousness in natural and artificial systems. We formalise consciousness-supporting systems in physical space-time, but this is localised in open domains of spatial regions and the result of this process is a family of different ZFC models. Random forcing, as in set theory, corresponds precisely to the random influence on the system of external stimuli, and the principles of reflection of set theory explain the conscious internal reaction of the system. We also develop the conscious Turing machines which have their external ZFC environment and the dynamics is encoded in the random forcing changing models of ZFC in which Turing machines with oracles are formulated. The construction is applied to cooperating families of conscious agents which, due to the reflection principle, can be reduced to the implementation of certain concurrent games with different levels of self-reflection.

我们认为意识归因于时空系统，它可以是纯粹物理的，没有生物学背景，并专注于对该现象的数学理解。结果表明，当数学基础中基于集合的集合论转换为基于 ZFC 模型的集合论时，在解释自然和人工系统中的大脑/心智复合体以及意识的出现方面，是一个非常有前途的数学工具。我们在物理时空中形式化了意识支持系统，但这仅限于空间区域的开放域，并且这个过程的结果是一系列不同的 ZFC 模型。集合论中的随机强迫恰好对应于外部刺激对系统的随机影响，集合论的反映原理解释了系统有意识的内部反应。我们还开发了有意识的图灵机，其具有外部 ZFC 环境，并且动态被编码在 ZFC 的随机强迫变化模型中，其中制定了具有预言机的图灵机。该构造应用于有意识主体的合作家族，由于反射原理，可以简化为具有不同自我反射水平的某些并发博弈的实施。