The Formal Layer of {Brain and Mind} and Emerging Consciousness in Physical Systems

Abstract

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.

1 Introduction

It can already be said that a functioning brain, in particular the human brain, is a very complex system that includes many intertwining layers of various structural, physical, biological, chemical, informational, logical, computational and, possibly, some other layers of structures and functions. Needless to say, the layers interplay in dynamic processes that, through miraculous fine tuning, give rise to conscious phenomena, which in turn reveal a highly complex world of structures, meanings, or inner experiences. This entire complex of structures and processes leading to consciousness is called a brain/mind complex. Faced with the complexity of brain function, researchers describe the correlations of various conscious phenomena with the structural behaviour of brain regions. This has led to the observation that the variety of possible dynamic situations, occurring in the physical world, can be grasped by certain, sometimes simple, topological invariants, representing the physical reality. One reason is that the invariants do not depend on the specific geometry of what happens as dynamically changing distances, but restore what essentially underlies the various (continuous) deformations of external data. Such invariant-based data are then accessible internally by a brain and undergo further processing. However, it seems that our conscious perception of the external world does not follow the topological content of data, but rather still distinguish them geometrically, e.g. Tozzi (2022). In any case, relations of this kind are rather a case of a special geometric-topological layer of the functioning brain and in no sense exhaust the whole richness of the problem. A wealth of neural data indicates that mathematics is a natural ordering factor, and researchers have used group theory, categories, homologies, algebraic topology and geometry, sheaf theory, and others to understand the relationship of the structured brain to information perception. However, the approach is in its infancy and remains open to new ideas.

In this paper, we describe how to extract a specific, also inspired by mathematics, layer within the brain/mind complex. The central idea behind the proposal comes from fundamentals of mathematics, more precisely from model theory and set theory. The way toward this comes from observation that any brain/mind complex is deeply nested in physical space-time and much of its behaviour reflects the dynamics in the external space-time. This, the trivial it is, observation open quite fruitful area of studies.

Another basic observation is that space-time is an enormously rigid physical object – indestructible under extreme physical conditions and surviving super-large scales after (or shortly after) the cosmological Big Bang due to its being a carrier of high-energy phenomena (where it can dynamically participate) and eventually ends up in the singularity of black holes. However, much of what we experience as humans is anointed by the presence of physical space-time, or better yet, space + time.

Fixing the research perspective that the brain/mind complex lives in physical space-time, and supplementing it with fundamental insights coming from mathematics, we propose to admit a special procedure that has recently been applied to some physical systems. Namely, every system in space-time can be approached formally, but the formalisation can vary and depends on local regions of space-time. Thus, the maximum depth of formalisation achieved locally in space-time leads to various local descriptions in a uniform constant formal language $\mathcal {L}$, which may vary depending on the region. The last task is the agreement (still in the language $\mathcal {L}$ or its extensions) between local descriptions. The whole procedure follows to some extent the local-to-global way of thinking specific to algebraic geometry and topology, but here it is applied to logical contexts. It also resembles the situation known from quantum field theory, where the inclusion of local gauge symmetries led to the (description of) bosonic fields in space-time.

Thus, we apply the local-to-global procedure to the brain/mind complex from the perspective of model theory. We show that the perspective sheds light on the problem of the emergence of consciousness in structured brains. More precisely, there exists such a formal level where models of Zermelo-Fraenkel set theory (possibly with the axiom of choice – ZFC) bear features usually ascribed to conscious phenomena. This does not mean that any formal model is conscious of something by itself, but rather that the remnant of full-fledged consciousness (f.c.), considered on the layer of formal models, non-trivially imitates the functioning of f.c. In this way, the origin and understanding of consciousness can be deeply traced in universal language of mathematics. Moreover, we consider the formal layer and remnant projected onto it as fundamental underlying components of f.c. and as a formal platform for the analysis of conscious phenomena in the broad perspective of living organisms and artificial systems. This work aims to fill a gap (to some extent) in understanding how consciousness emerges in physical systems in space-time. This method was inspired by the recent suggestion that Turing machines could be considered conscious, using exactly the same “local ZFC” construction. In the case of Turing machines (TM) such an approach is rigid and indicated directly by mathematics, since it is based on the fundamental relationship between Peano arithmetic (PA) and set theory as reflecting the external environment to a TM relation. On the other hand, the TM case is extremely universal, since it concerns, in principle, any ‘efficient’ process in Nature, realised by robots or living organisms. This is an upward approach to consciousness. Here we are developing a rather up-to-bottom way of thinking, starting with entities in space and mathematically grasping emerging consciousness. Both of these approaches must somehow meet, leading to new insights.

Much of the formal content of this work relies on models of a 1st-order axiomatic theory like ZFC and on their forcing extensions. Axioms of ZFC comprise in fact an infinite system (schemata) of formulas expressing formally quite natural properties of sets, see the textbooks like Jech (2006) for the list of axioms formulas of the ZFC system. The language of ZFC is first-order so thus one avoids quantification over subsets of the domain and follows the rules of first-order logic, see also the textbook like above. The structure of the class of models of ZFC is particularly rich. Any model of ZFC if it is a set (and if there exists a set model at all, ZFC is consistent), comprises a set universe M with the relation E which expresses x as an element of y, xEy and all provable theorems of ZFC are still true in the structure (M, E) under a suitable interpretation. If the relation E is the usual relation $\in$, the model is standard. The Mostowski collapse enables one to find a standard model isomorphic to any well-founded model of ZFC. There are plenty of different models of ZFC, in fact infinitely many for any infinite cardinality of the domain. We usually work in the standard transitive models $(M,\in)$ meaning that the domain M is a transitive set or transitive class like V. The models differ also by true formulas which hold true in the models, besides all true ZFC provable sentences there are specific sentences true only in the model. This is the reason for interpreting the individuality of a creature on this or another model of ZFC.

Forcing in set theory, invented by Paul Cohen in 1963, when he showed the independence of the continuum hypothesis (CH, i.e. the statement that the continuum of reals $\mathfrak {c}$ has cardinality $\aleph _1$) from the ZFC axioms and the independence of the axiom of choice from the ZF axioms. Even though the continuum of reals, i.e. $2^{\omega }$, can be defined in every model of ZFC, this is not the absolute concept and from the outside it is represented by different sets in the ‘true’ universe of Von Neumann V (Jech, 2006). From this property it follows peculiar fact that given one model $M_1$ of ZFC one can obtain extended model $M_2$, where the set of reals is a superset containing the set of reals in the model $M_1$. Certainly, we see these extra reals only from the outside the model perspective, but they are particularly important for this paper and for the original Cohen forcing. Namely, Cohen has added so many new reals into the model $M_1$ (where CH holds) that the line of reals $2^{\omega }_{M_1}$ in $M_1$ becomes as numerous as $\aleph _2$ extending the cardinality of $\omega _1$. Thus continuum becomes $\aleph _2>\aleph _1$ in the extended model and CH does not hold any longer in it. So, CH has to be independent on the axioms of ZFC. In this paper we are representing the external stimuli by new reals to the initial model $M_1$ such that they are random with respect to $M_1$ and become tame with respect to the extended model $M_2=M_1[G]$ (G represents a generic ultrafilter corresponding to a new real; such a real is called generic relative to $M_1$). For the new reals to represent true random binary sequences, the forcing has to be not Cohen but random Solovay forcing, but the general idea of extending models of ZFC and adding generic reals remains similar in both cases.

The entire formal procedure in the paper resembles and, in fact, may be considered as a kind of paraphrase method known in philosophy¹ (Woleński, 1985). Indeed, we have here the level of natural language (informal) in which there are formulated the features attributed to a conscious entity or consciousness in general. Next the choice of certain formal language (ZF) plus the theory of models of ZFC with emphasis on forcing, enables to find suitable formal background for the features. However, in this approach we find that two distinct foundations for set theory, universism, distinguishing the unique cumulative universe V, and multiversism, indicating forcing extensions and models of ZF(C) as primary objects (without one extra model), have to both take part and coexist in proper modelling the relation conscious entity/external environment. Another important feature of our formal method is localising into spatial regions, i.e. the models of ZFC and forcings are varying from one local domain to another. We would say that the paraphrase method gained a kind of twist when applied to modelling consciousness, however, we are not focused on the precise relation of our method to paraphrase one in this work. The results obtained are self-explaining and do not depend on the particularities of the relation.

In Sect. 2, we will construct a formal layer and show how certain conscious phenomena are mapped on it. Based on this, we will formulate the requirements that a conscious Turing machine must satisfy. The chosen point of view allows us to solve such questions regarding universal Turing machines with consciousness. In Sect. 3, we will show that a swarm of such conscious systems can cooperate and negotiate, and its functioning and behaviour is well represented by the implementation of certain concurrent and reflective game scenario. In Sect. 4, we will discuss the methodology of our work.

Hence, in this paper, we introduce the TM with consciousness to get some applications of social interaction in multi-agent robotic systems, therefore we show an opportunity of applications of this machine to concurrent games (Abramsky and Mellies, 1999; Gutierrez et al., 2023) and reflective (reflexive) games (Chkhartishvili, 2010; Nelson and Ham, 2012; Novikov and Chkhartishvili, 2004). Artificial social multi-agent systems recently developed in computer science are inspired by some natural elements of intelligent behaviour and cognitive properties that occur in an organised community of various swarms: ants, bees, flocks of sheep (horses), shoals of fish, etc. Using our model, it is possible to develop a general theory of any artificial swarms for modelling the behaviour of groups of robots with elements of social functions and group decisions: (1) for artificial simulation of social and emotional functions of swarms and (2) for programming social and emotional functions of groups of robots.

2 Results

2.1 Formalisation

The formal environment for describing brain-mind and its functions is based on the axiomatic system of Zermelo-Fraenkel set theory with the axiom of choice (ZFC). We consider a brain-mind system as functioning in physical space-time $M^4$ that is formally represented by smooth 4-dimensional Lorentzian noncompact manifold. $M^4$ can be curved, i.e. its Riemannian tensor can be nonvanishing, however at this stage the gravitational degrees of freedom can be safely neglected. Also, the subluminal effects do not contribute significantly at this stage. So, $M^4$ can be flat Minkowski 4-space-time which is $\mathbb {R}^4$ with the global Lorentzian metric and, in the nonrelativistic limit, it can be seen as Riemannian flat $\mathbb {R}^4$ with the Euclidean metric. The smoothness structure on $M^4$ is defined as the maximal atlas of local charts

$$\begin{aligned} \{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in I} \text { where } M=\bigcup _{\alpha \in I}U_{\alpha } \end{aligned}$$

and the maps $\phi _{\alpha }:U_{\alpha }\rightarrow \phi (U_{\alpha })\subseteq \mathbb {R}^4$ are homeomorphisms onto open subsets of $\mathbb {R}^4$ and for all $\alpha ,\beta \in I$ and $U_{\alpha \beta }= U_{\alpha }\cap U_{\beta }\ne \emptyset$ the maps

$$\begin{aligned} \begin{aligned} \phi _{\alpha \beta }:= \phi _{\beta }\circ \phi ^{-1}_{\alpha }:\phi _{\alpha }(U_{\alpha \beta })\rightarrow \phi _{\beta }(U_{\alpha \beta }) \\ (\phi _{\alpha \beta })^{-1}=\phi _{\beta \alpha } \end{aligned} \end{aligned}$$

are all smooth. Then given a smooth atlas $\{(U_{\alpha },\phi _{\alpha })\}_{\alpha \in I}$, we assign models of ZFC, $M_{\alpha }$, to local regions $U_{\alpha }\subset M^4$

$$\begin{aligned} U_{\alpha }\mapsto M_{\alpha }, \alpha \in I. \end{aligned}$$

(1)

It is now possible to extend the relativity principle on smooth manifolds in a way that locally on $M^4$ for any $x\in M^4$ there always exists a coordinate frame $U_x$ such that it is flat, and the gravitational effects can be neglected in $U_x$ and, moreover, the $U_x$ can be chosen flat $R_M^4$ in the ZFC model M, so that $U_x\mapsto M$.

Remark 1

One can consider the model $M_{\alpha }$ as a more formal (although local) stage of $M^4$ at $U_{\alpha }$ than just $M^4$ in V (the universe of sets). The entire family $M^4, \{U_{\alpha },M_{\alpha } \}_{\alpha \in I}$ is a more formal than $M^4, \{U_{\alpha } \}_{\alpha \in I}$ description of the smooth manifold $M^4$. Note that given $M_{\alpha }=M=V$, then both presentations are the same.

$M^4$ for which it holds $\forall _{\alpha \in I}M_{\alpha }=M=V$ is called ZFC-homogeneous space-time. This is the usual description of the smooth $M^4$ entirely in the universe of sets V.

Remark 2

The assignment (1) has appeared in the context of quantum mechanics, where $M_{\alpha }=M=V^B$ – the Boolean-valued model of ZFC for B, the measure Boolean algebra (the dimension of the Hilbert space of states, $\dim {(\mathcal {H})}=\infty$) (Król and Asselmeyer-Maluga, 2020; Król et al., 2017). In the context of smoothness structure in cosmological scales (1) has been also considered, with $M_{\alpha }$ given by models of set theory in certain categories (smooth toposes) (Asselmeyer-Maluga and Król, 2019).

A system S in (ZFC- inhomogeneous) space-time $M^4$ can be considered as a ZFC-inhomogeneous system, which means that the functioning or structure of S requires at least two different models $M_1,M_2$ of ZFC in the description of S.

Example 1

A quantum system S on the infinite dimensional Hilbert space with the lattice of projections $\mathbb {L}$ such that the lattice determines different models of ZFC, $\{M[\mathcal {U}_i] :i\in J\}$. The models are random forcing extensions of certain transitive model M (Król and Asselmeyer-Maluga, 2020). This is in fact the direct conclusion of the Boolean valued model $V^B$ assigned to the quantum system S, as in the Remark 2, because there holds the relation: $V^B/\mathcal {U}_i=M[\mathcal {U}_i]$ for each $i\in J$ (Król and Asselmeyer-Maluga, 2020; Król et al., 2017).

2.2 ZFC-Inhomogeneous Systems and Emerging Pre-consciousness

We merge the ZFC-inhomogeneous system with one given in the Example 1, however, we do not assume any quantum origins of S. We claim that such S is capable to generate and mimic conscious phenomena. In fact we are translating the features usually ascribe to conscious behaviour onto corresponding formal behaviour when on the level of models of ZFC. First, we take a closer look at the ZFC-inhomogeneous system S and understand its allowed space of actions from the formal point of view.

Thus, the ZFC-inhomogeneous system S, is understood as a set of allowed states or a set with certain algebraic structure, like the lattice $\mathbb {L}$ in Example 1, which is augmented by functions $f,\hat{f}$. It is defined as follows.

Definition 1

Let S be certain algebraic structure. The ZFC-inhomogeneous system $(S,f,\hat{f})$ comprises of

• i) a spatial domain $D\subset \mathbb {R}^3$, D homeomorphic with $\mathbb {R}^3$;

• ii) a cover by subregions $D_i,i=1,2,3\dots$ of D, $D=\bigcup _{i}D_i, D_i\subset \mathbb {R}^3$, each $D_i$ is homeomorphic with $\mathbb {R}^3, i=1,2,3\dots$;

• iii) an assignment f defined on certain class of substructures $\{S_i\hookrightarrow S :i=1,2,3,\dots \}$ with values in $\{D_i, i=1,2,\dots \}$, fulfilling

• 1. for any $s\in S$ there exists $S_i$ such that $s\in S_i$;

• 2. $f(S_i)=D_i,i=1,2,\dots$;

• iv) an assignment $\hat{f}:S_i\rightarrow M_i,i=1,2,\dots$ where $\{M_i\}$ is the family of transitive standard models of ZFC such that

• 1. there exists a pair $(i,j), i\ne j$ such that $\hat{f}(S_i)=M_i, \hat{f}(S_j)=M_j$ and $M_i\ne M_j$;

• 2. $D_i\in M_i,i=1,2,\dots$.

There follow direct consequences from this definition

• C1. $i\ge 2$ so there are at least two subsystems $S_i$ and two spatial subregions $D_i$ (from iv1. above).

• C2. $D_i$ and $D_j$ becomes $R_{M_i}^3$ in $M_i$ and $R_{M_j}^3$ in $M_j$ (from iv2. above).

• C3. The definition above represents static spatial picture of the system in 4-space-time, i.e. in certain $t=t_0$ instant of time.

In what follows as a rule we are using S instead of the triple $(S,f,\hat{f})$, if the meaning is clear from the context.

To understand the dynamics of S we should learn more about models $M_i$ and their relation. Again we follow Example 1. Given arbitrary formal theory (in the first order language $\mathcal {L}$) like ZFC, the models of it are such construed that all provably true formulas of ZFC remain true in any model $M_i$. However, there remain additionally statements which are true in a model but are not ZFC -provable. These $M_i$-theorems indicate the model $M_i$. In the broad logical context, we can speak about theories of a model $M_i$, $Th(M_i)$, comprising all theorems true in $M_i$. Thus, the identity of a model as model should be found among these true formulas of $Th(M_i)$ which are not ZFC-provable. This is the indication that certain formal situation can distinguish, or identify, models among other models (of the same theory). We will make use of it when interpreting conscious phenomena. Let us note here that if an ith creature $C_i$, or maybe its content of consciousness, were somehow connected with a model of ZFC $M_i$, $C_i$ would be seeing the ZFC world ‘individually,’ i.e. distinguishable to others visions through $M_j, i\ne j$. So, even though the ‘objective’ (provable) ZFC world would be the same for all creatures $\{C_i :i=1,2,\dots \}$, they see it individually and differently. What is more such possibility follows from, and is written in, the formal setup of ZFC.

However, there is much more to be learnt about dynamics of the visions (contents of consciousness) from merely studying models of ZFC. Note that there emerges an internal vs. external relation of $C_i$ and the world (given by the universe of sets V of ZFC). Internal to $C_i$ is what can be inferred (about sets at this stage) within the model $M_i$, while external to $C_i$ is what can be proved in ZFC but also this externality to $C_i$ should contain the internal truths of other creatures $C_j, i\ne j$, i.e. provable in $M_j$s. However, these truths are not verified in general as (provable) truths by $C_i$.

Now, what is the dynamics of a conscious creature in such a world? This is the point, where one needs different models of ZFC – the dynamics is written in a variety of models and, in particular, in the ability to build extensions of a given model $M_i$. For instance, natural set theoretic operations (forcing), governing the changes of models, embody the entire dynamic process of reaction on random external stimuli of various kinds and the changes that follow the (conscious content of a) creature itself.

Hence, we consider a creature $C_i$ as represented by a general ZFC-inhomogeneous system $S^{(i)}$ that assigns different (standard, transitive) models of ZFC to regions of space. This representation can be recapitulated by specifying the following conditions

• A1. To each creature $C_i$, $i\in I$, there is assigned a family of models $M_{\{i \}} :=\{M_{j} :j\in \{i\}\}$ of ZFC.

• A2. The model $M_j$ for $j\in \{i\}$ is determined as individual rather than “up to isomorphism.” Let us consider a local creature $C_i$ for $i\in I$ as given by a single model $M_i$ (corresponding to a local spatial domain $U_i$). The system $S_i$ is thus reduced to a single $M_i$ and the local domain $U_i$. The isomorphic copy $M_k$ of $M_i$ represents different system $S_k$ of another local creature $C_k$. This simply follows from the content of what is accessible to the local creature $C_i$ (“what $C_i$ believes in” or ‘knows’): this content indicates that $k\ne i$ (as local creatures) even though $M_i\equiv M_k$. This follows from the fact that first-order formulas expressing what $C_i$ knows about sets can not express the fact that $M_i,M_k$ are models of ZFC, nor they are isomorphic or non-isomorphic models.

• A3. From the above it follows that the local creature $C_i$ is confined by its deductions inside $C_i$. This is a formal counterpart of the “knowing the identity of itself (of $C_i$ by $C_i$).”

It follows from Definition 1 [iv)1., 2.] that any creature $C_i$, as ZFC-inhomogeneous system, “identifies itself” and “is identified” with the instance of the model $M_i$ internally rather than with an external universal ZFC point of view. Thus, even though the same (up to isomorphism) model is assigned to another creature, as in A1. above, this is a different ‘model’ from the $C_i$’s perspective (A2.). The formal perspective expressed in A2. on the individuality relies on the inability to prove the isomorphism $M_i\equiv M_j$ in $M_i$ or the facts that $M_{i,j}$ are models of ZFC (A3.). All ZFC provable theorems become now trapped internally in the model $M_i$ and in the individualised instance of $M_i$: what one proves in $M_i$ is not generally accessible in different $M_j$. There also follows the distinction between internal to $M_i$ ZFC world and external to it in other models as well in the universe of sets V. We will develop these important points more formally in the next subsection.

Hence, this so far static picture is augmented further by forcing relation towards a dynamic one. Forcing is one of the most powerful technique known in formal set theory with many facets and variants and with still open important fundamental problems (e.g. Jech (2006)). It has been invented by Paul Cohen as a technique to show the independence of the continuum hypothesis (CH) from ZFC axioms and the reformulation of the proof of the independence of the axiom of choice (AC) from the ZF axioms (Jech, 2006). Here we emphasise on forcing as a way of the extension of a given standard, transitive model of ZFC, say $M_i$, to another standard transitive model $M_i[G]$ where G is the symbol for the so-called generic filter. This G exists in $M_i[G]$, but it is not any member of $M_i$ for the so-called nontrivial forcings extensions. For some class of forcings, like Cohen or random, the extensions $M_i\rightarrow M_i[G]$ parallel the corresponding extensions of the lines of real numbers $R_M\rightarrow R_{M[G]}$ that leads to the internal to the models descriptions of regions $D^3_M\subset R_M^3$, $D^3_{M[G]}\subset R_{M[G]}^3$, since it holds

$$\begin{aligned} R_M\subset R_{M[G]}\subset \mathbb {R} \text {, so that }R_M^3\subset R_{M[G]}^3\subset \mathbb {R}^3, \end{aligned}$$

however, these subsets relations are not accessible internally to models. This leads to the proposal that (still on the formal ground of the approach), the extensions might reflect both, the reaction of the creature on the external to it stimuli, i.e. changes caused by these stimuli in the mind/brain complex of the creature, and also the recognition of the stimuli as externally originated. This last feature means that the appearance of some stimuli has to be independent of the internal constitution (provability or knowledge) of the creature about what can happen externally. Or in other words, among stimuli there have to be random ones as experienced by a creature. If all stimuli were deducible or predictable by $C_i$ than the system: the creature acting and experiencing in the world and the world itself, would reduce to just the internal world of $C_i$, i.e. $M_i$. This irreducibility is one of basic characteristics of a conscious entity in the world. Another is related with a degree of intentionality characterising certain conscious acts.² This we will assign to a forcing relation as possibly ‘seen’ or construed internally to M but still containing an independent external component.

2.3 Interpreting Consciousness

In this subsection, we will show that features usually ascribed to conscious creatures are actually well-represented by the formal counterparts of systems like S. This is not the fact that the system is aware but rather that S is a formal carrier of the remnant of the fact that in physical space-time (or rather space-and-time) there are functioning aware creatures. This very fact has its impact on effective set theory of the resulting system: $S+ \text { space-and-time}$.

Let s be a set, formally it is $s\in V$, where V is a universe of sets. What would it mean that a creature $C_i$ be aware (on the level of set theory) of s? It could be just aware the existence of s as the element of the external to $C_i$ world W or internal to it, $W(C_i)$ ($W(C_i)$ is the internal set theory world of $C_i$). This aware noticing might have its expression in a formal statement like $s\in W$. Moreover, it is also the possibility that $C_i$ grasps (is aware of) the existence of $s\in W$ as well $s\notin W(C_i)$, but also some internal trace of s, $\hat{s}$, is represented in the internal world $W(C_i)$ to $C_i$. At this stage, we do not stipulate on the full knowledge of what s is (what are its properties). It is enough to demand that some s be in the external world, internal or both and that $s\in W, s\in W(C_i)$ can be intentionally approached by $C_i$ or such s can act on $C_i$ irrespective it comes from the external W or the internal $W(C_i)$. We will see that this scenario can be naturally represented by the formal counterparts of models of ZFC.

Let V be the universe of sets, $V\models \text {ZFC }$, which formally represents the world in which (with relation to which) there are functioning, also conscious, creatures $C_i,i\in I$. Let $M_i$ be the model of ZFC assigned to the system S representing $C_i$ (Definition 1). Note that saying that $C_i$ knows that, e.g., $\forall _x\phi (x)$, means in set theory that $\forall _x\phi (x)$ is true in $M_i$ ($\phi (x)$ is a formula in the language of ZF which is not necessary ZFC-provable). This observation motivates the search for a more complete representation of conscious phenomena based on models of ZFC. First, we will single out certain basic elements or moments of “being conscious” (pre-conscious phenomena) that should have its formal counterparts.

Let us define the following dictionary translating the pre-conscious phenomena into the formal counterparts.

Definition 2

(Dictionary)

• i. The awareness of the identity of itself “$C_i$ knows it is $C_i$”$:=$ $\exists _{M_i} M_i\models \text { ZFC }$ and $M_i$ is present in $S^{(i)}$ of $C_i$, the internal truths to the model $M_i$ determine what $C_i$ knows about sets in general (e.g. in V). Thus, the natural to $C_i$ is the internal world $W(C_i)$ depending on $M_i$, irrespectively at this moment whether the external W exists or not.

• ii. “$C_i$ knows it is distinct from the external world W”$:=$ $\exists _{M_i} M_i\models \text { ZFC } \wedge M_i\ne V \wedge \exists G\in V, G\notin M_i\wedge M_i[G]\ne V$.

  • a) there exist random (independent on inferences in $M_i$) external V-defined events acting on $C_i$;

  • b) there are (ubiquitous) dynamical reactions of the external events on $C_i$ which modify $C_i$.

• iii. “Random external to $C_i$ events, $G_j,j\in K$, modify $C_i$” $:=$ forcing extensions $M_i[G_j]$ replace $M_i$ such that $M_i[G_j]$ represent $C_i$ after modification and $G_j\notin M_i$ and $G_j\in M_i[G_j]$. Randomness of $G_j, j\in K$ relies on the independence from the ZFC axioms.

• iv. “$C_i$ is aware of s coming from the external W” $:=$ $\exists _{M_i} s\notin M_i\wedge M_i\ne V\wedge M_i\subset V$;

• v. “$C_i$ is aware of $\hat{s}$” $:=$ $\exists _{\tilde{M}_i}\tilde{M_i}\models \text { ZFC} \wedge \tilde{M}_i \text { is a ground for } M_i\wedge \hat{s}\notin \tilde{M}_i \wedge \hat{s}\in M_i$.

• vi. “$C_i$ intentionally aims at external s (internal $\hat{s}$)” this is represented by the reflection principle in set theory (see below) for the finite fragments of ZFC.

Remark 3

The reason for ii. that $C_i$ knows it is different than V is the existence of nontrivial forcing extensions $M_i[G]$. This is also the reason that $N=M_i[G]\ne V$ since there exists N[H] and so on. This is also crucial point in the multiverse approach to foundations of ZFC (see section 2.4).

We consider this definition as expressing what formally would mean that a creature $C_i$ is aware of sets as formal objects. This serves as the preliminary conditions directing next more complete understanding. It seems that the approach locates itself at the deep formal level and is quite distant to real consciousness of, say living entities. However, we will see that the approach (suitably enhanced as below) is in fact quite universal and similar constructions appear in the context of Turing machines allowing for consciousness. Moreover, the formalisation leads to the set theoretic remnant of true conscious behaviour of organisms which is the collection of mathematical preconditions underlying consciousness in various systems (including living organisms). In other words, the fact that there are conscious creatures functioning in physical space-and-time might have back-reaction on the deeper set-theoretic and logical layer which might have been imprinted its existence on the formal layer. This is the reason we are investigating these formal imprints. From the other side the layer appears to be quite universal among effectively functioning organisms in nature (see Subsecs. 2.4 and 2.5 and also Król and Schumann, A.: Turing machines as computing conscious machines, (2023)).

In what follows we are applying the formal tools developed so far into the realistic case of conscious systems including human beings and, thus, reduce the distance between the formal considerations and true consciousness. Hence, below (following Van Gulick (2022)) we will interpret several features usually ascribed to the conscious behaviour by some set theoretic counterparts. The list is not exhausted, but rather it can be extended by other features not included here, however, this is the ability of conscious creatures to possess the features, but not the necessity to maintain them permanently. The important thing is that we are identifying the formal core that enables a conscious behaviour and activates the features.

Thus, we are first citing a characteristics of conscious entities from Van Gulick (2022), and then building the formal model supporting them.

Awareness of something (transitive consciousness). In addition to describing creatures as conscious [ ... ] there are also related senses in which creatures are described as being conscious of various things. The distinction is sometimes marked as that between transitive and intransitive notions of consciousness, with the former involving some object at which consciousness is directed (Rosenthal, 1986).

Again, it seems plausible that given any conscious system it should be able to be found in a state of being aware of something, i.e. it can eventually be aware of something without the necessity to be so permanently.

As a consequence, we define the act of becoming aware of a set or set-like property P in a way that $C_i$ becomes aware of P. This can be done twofolds, first goes ‘ideal’ or sharp notion of consciousness of s. Let $M_i$ be a model assigned to the algebraic structure $S^{(i)}$ of $C_i$, then

• 1. Awareness of the external P (“$C_i$ sees P” or “$C_i$ becomes aware of P”) $:=$ $M_i[G]\vdash \exists _{S\in M_i[G]} P\in S$;

• 2. Awareness of the internal P ($C_i$ sees P or $C_i$ becomes aware of P) $:=\exists _{s\in N[H]} N[H]\vdash P\in s$ where $N[H]=M_i$ (N is ground for $M_i$).

There is also a phantom awareness of something, where there emerges the internal image of s, but ZFC is finitistically truncated. We consider this as the basic kind of consciousness, since to be aware of an object P means that such a creature makes the internal copy of P that is always on the subjective side of the creature. Such a phantom awareness of something is the base for intentional acts and conscious acts in general (see the analysis further in this subsection)

• 1P. Phantom awareness of the full external P (“$C_i$ sees phantom P,” $P_{Fin}$ or “$C_i$ becomes aware of phantom P,” $P_{Fin}$) $:=$ $M_i[G]\vdash \exists _{S\in M_i[G]} P\in S$ and $P_{Fin}\in M_{i,Fin}$ (the model of a finite fragment of ZFC – $\text {ZFC}^*$ – see below);

• 2P. Phantom awareness of an internal P (“$C_i$ sees P as phantom,” $P_{Fin}$ or “$C_i$ becomes aware of $P_{Fin}$”) $:=\exists _{s\in N[H]} N[H]\vdash P\in s$ where $N[H]=M_i$ (N is ground for $M_i$) where $P_{Fin}$ is now in $N_{i,Fin}$ the model of a finite fragment $\text {ZFC}^*$ in N.

Remark 4

Taking the external perspective, i.e. allowing for the reference to models of ZFC (or even quantification over models) the condition 1. can be expressed as: $\exists _{t_0<t_1} P\notin M_i$ at the moment $t_0$ and $P\in M_i[G]$ at the moment $t_1$.

2. can be expressed as: $\exists _{t_0<t_1} P\in M_i$ at the moment $t_0$ and

$$\begin{aligned} (\exists _{N\text { a model of ZFC}}N\text { is a ground for }M_i)\wedge (P\notin N) \wedge (P\in N[H]=M_i) \end{aligned}$$

at the moment $t_1$).

Remark 5

The stimuli acting on a creature $C_i$ leads to the random forcing extension $M_i[r]$ and this is unconscious (random) process. The creation of the “phantom r” in $M_i[r]$, which is r in the model of a finite ZFC part of the stimuli r, allows for the subsequent conscious seeing r (seeing phantom r). So unconscious random action results in conscious seeing ZFC-finite part of r and this is what the creature is aware of. Similarly for the internal stimuli r: as fully random in N[r] (with respect to a ground model N) this is unconscious process which results in the conscious seeing of r as phantom in $N_{i,Fin}$. This conscious grasping of phantom r in each situation above can be caused by either of the intentional act of the creature or by random (external) stimuli action on the creature.

We are going to explain and explore further the process of taking the finite fragment of ZFC models by reacting creatures, which becomes the base for sensing the internal or external worlds.

Sentience. It may be conscious in the generic sense of simply being a sentient creature, one capable of sensing and responding to its world (Armstrong 1981). Being conscious in this sense may admit of degrees, and just what sort of sensory capacities are sufficient may not be sharply defined. Are fish conscious in the relevant respect? And what of shrimp or bees? (Van Gulick, 2022).

In the ZFC terminology of this paper the sentience of a conscious creature is the capacity for “forcing extensions” as the reaction on external (and internal) stimuli and the possibility to see the extensions as both internal and external to M. This is the reflection property of ZFC namely any finite fragment of ZFC axioms is set-modelled in any M and thus is conceived as the part of internal M. However, the entire ZFC universe M[r] reflects also fully random external part of the stimuli.

To be more specific, let $\mathbb {A}$ be the set of formulas in the ZF language, M a model of ZFC and V the universe of sets as usual. We say that M reflects $\mathbb {A}$ iff

$$\begin{aligned} \forall _{a\in \mathbb {A}}M\models a \iff V\models a. \end{aligned}$$

Let $\text {ZFC}^*$ be a finite fragment of ZFC. From the Reflection Principle it follows Jech (2006, p.168)

Lemma 1

For any finite fragment $\text {ZFC}^*\subsetneq \text {ZFC}$ there exists a countable transitive model M in V such that M reflects $\text {ZFC}^*$.

Lemma 2

For any model M of ZFC and a finite fragment $\text {ZFC}^*\subsetneq \text {ZFC}$ there exists a countable set N in M such that N reflects $\text {ZFC}^*$.

Consequently, the internal part of the stimuli is described by the reflection principle as the set-models of finite fragments $\text {ZFC}^*$ in the model M, while the unpredictable random external component corresponds to the random r which leads to M[r]. The following general picture of the “structured stimuli” arises. The internal component comprises of many set-models of various $\text {ZFC}^*$ finite fragments. The fragments contain different sets of ZFC axioms (finite) and this choices of finite number of axioms is the intentional component of the stimuli/consciousness complex. The third component of the complex relies on the external random forcing extension M[r] (or its ground model shrink) and employs entire infinite set of the ZFC axioms.

Remark 6

The choice of the finite fragments $\text {ZFC}^*$ of ZFC is a very rich family of different realms in general which varies inbetween its members substantially. One example is given by the axiom of extensionality which can, or need not be included into the finite set of axioms. In the first case we always have transitive models of the finite fragments of ZFC in a given M, while in the second the transitivity is excluded (Jech, 2006). Intentionally a conscious entity can be inclined to the external world (internal world) as if it is non-extensional (even though, formally the world is extensional). In such a case the corresponding model of $\text {ZFC}^*$ without the axiom of extensionality is created in M.

Thus being a sentience creature would mean the ability of building the internal set-models in M of finite fragments of ZFC (which do not agree with ZFC itself) as the response to the external stimuli (usually represented by random forcing). Sensing the space (orientation, position) is due to different models $M_i$ of ZFC subordinated to different local frames $U_i$ of the space. Each $M_i$ is capable of building internal finite responses (set-models of $\text {ZFC}^*$) to stimuli, with a degree of intentionality. As a consequence, the internal finite axioms models in $M_i$ are augmented by information about spatial localisation that is decoded in varying models $M_i$. Additionally, on the overlapping spatial domains like $U_i\cap U_j$ the model $M_i$ ($M_j$) can be stimulated by $M_j$ ($M_i$) and responds (also with a degree of intentionality) by building internal $\text {ZFC}^*$ models. This is the general scheme for intertwinning the spatial orientation with sensing external/internal world.

There are models of ZFC with capacity of taking their forcing extensions or grounds and this also modifies the possible inferences relative to models. To be sentience means that the creature has assigned the ZFC-inhomogeneous dynamical system S as its formal layer with the entire capacity for building internal models locally in space. It follows that the spatial extension of a sensitive entity is an important determinant of its behaviour. Following the Stanford Encyclopedia of Philosophy Archive remarks regarding conscious organisms let us quote

Wakefulness. One might further require that the organism actually be exercising such a capacity rather than merely having the ability or disposition to do so. Thus one might count it as conscious only if it were awake and normally alert. In that sense organisms would not count as conscious when asleep or in any of the deeper levels of coma. Again boundaries may be blurry, and intermediate cases may be involved. For example, is one conscious in the relevant sense when dreaming, hypnotized or in a fugue state? (Van Gulick, 2022).

In our model of consciousness wakefulness can find its representation as follows. A local model $M_i$ being acted upon an external stimuli leads to the internal state ${M}_{i,Fin}$ which is the model of a finite fragment $\text {ZFC}^*$. This model of $\text {ZFC}^*$ has a set in $M_i$ as its domain, and this set is considered as excited domain by an external stimuli $r_i$. The external view of the excitement is the set in $M_i$, while the internal meaning is the model of $\text {ZFC}^*$ based on this domain. Wakefulness relies on keeping the model ${M}_{i,Fin}$ (the internal meaning of the excitement) as fixed for the (incoming) stimuli. This ‘awareness’ corresponds to the excited domain which, when relaxed, disappears (at this stage we do not formulate any direct correspondence between $\{\text {excited domain of } \,{M}_{i,Fin}\} \in M_i$ and excited domains in the brain neural system, however, this is left as certain possibility). The lack of the wakefulness thus corresponds to undetermination of the internal state, i.e. without fixed ${M}_{i,Fin}$ as a reference model.

Let us quote again from the Stanford Encyclopedia the commentary regarding self-consciousness

Self-conscious. ... yet more demanding sense might define conscious creatures as those that are not only aware but also aware that they are aware, thus treating creature consciousness as a form of self-consciousness (Carruthers 2000). The self-awareness requirement might get interpreted in a variety of ways, and which creatures would qualify as conscious in the relevant sense will vary accordingly. If it is taken to involve explicit conceptual self-awareness, many non-human animals and even young children might fail to qualify, but if only more rudimentary implicit forms of self-awareness are required then a wide range of nonlinguistic creatures might count as self-conscious (Van Gulick, 2022).

In the proposed formalism we define self-consciousness again in terms of variety of ZFC models, $M_i$, and its finite fragments $\text {ZFC}^*$ models, $M_{i,Fin}$. Model theory yields the possibility of building models inside models and so on. Augmenting the process by awareness or self-reference it seems that this is the model theory as a mathematical way towards including self-consciousness and we follow this possibility.

Remark 7

There is an important activity in foundation of set theory which resonates with the attempt here, it is the reformulation of set theory in a way that its interest becomes studying models as main objectives rather than sets. The practice and development of set theory within last decades clearly shows the tremendous effort in building new models and studying the relations between them rather than studying just sets as sets (Antos, 2022; Hamkins, 2012). This is the change of the reference frame, or departure point in foundations of set theory and mathematics.

Let us now apply the tools for the case of self-consciousness. The system S is ZFC-inhomogeneous (according to Def. 1) and it comprises of variety of models $\{M_i\}$ augmented by the dynamics decoded in forcing relations between the models. The random stimulus can act upon S by the external environment or by an internal excitation. In both cases there is the fully random stimulus r which enters the game as the random extension of certain $M_i$ or ground $N_i$ in the internal case. The mind/brain complex of the entity with the subordinated system S makes a finite fragment of ZFC model copy of the model M[r] and the stimulus r (N[r] and r correspondingly). This finite fragments of ZFC models are fully recognised by S (Lemmas 1 and 2) and S becomes aware of the entire situation (of the finite fragment model grasping r) by its internal means of modelling the external and internal worlds. Moreover, the action of the stimulus r on this specific $M_i$ (not on any other) contains information about the space localisation of r. There are, however, possible many other seeings of this internal $r_{i,Fin}$ in different models $M_j$ in S (at a specific moment of time t). This is simply because of that that any $M_j$ can become the model for any finite fragment model of ZFC like $M_{i,Fin}$, giving rise to its image $M_{ij,Fin}$ in $M_j$ along with the image $r_{ij}$ of $r_i$. There is the partial order of such images and internal finite fragments models, however, for any finite number of models there exists finite fragment model, $\overline{M}_i$, which models all these models. We can take the minimal such model $\overline{M}_i$. In this way we have a kind of awareness of the entire excitement structure within S pictured in $\overline{M}_i$. This is what corresponds to formal origins of self-consciousness in S subordinated to the creature C. Thus, “dynamically react on changing world with being aware of himself and the external world” gains its new formal meaning as above.

There is the connected to this self-consciousness picture “what it is like” criterion of Thomas Nagel which can be addressed coherently here

What it is like. Thomas Nagel’s (1974) famous “what it is like” criterion aims to capture another and perhaps more subjective notion of being a conscious organism. According to Nagel, a being is conscious just if there is “something that it is like” to be that creature, i.e., some subjective way the world seems or appears from the creature’s mental or experiential point of view. In Nagel’s example, bats are conscious because there is something that it is like for a bat to experience its world through its echo-locatory senses, even though we humans from our human point of view can not emphatically understand what such a mode of consciousness is like from the bat’s own point of view (Van Gulick, 2022).

In our approach “what it is like” can be reformulated just by observing that the internal image of the world is encoded into the finite fragment models or into the minimal $\overline{M}_i$ and they can be models for different finite fragments depending on the creature $C_i$. Thus, subjective images of the external world can be substantially different for different creatures (see Remark 6). The sensory apparatuses specific to each creature have their significant influence on seeing the world in this or another way. In this approach the subjective relation to the world, or subjective picture of it, is a function of the senses apparatus and conversely.

So far, we have not sought to explain why consciousness appeared in nature, but rather to look for clues in mathematics that make the existence of consciousness ‘natural.’ Consequently, we are interested in mathematics that models conscious phenomena as correctly as possible at an early stage of their emergence, and, thus, serves as their precondition. In the following two subsections, we are interested in the universality issues behind the emergence of consciousness, and we will see that the preconditions found are not random, but rather reflect the general attitude of nature favoring the systems operating in it towards the acquisition of consciousness under certain natural conditions.

2.4 Emerging Consciousness and Set Theory Multiverses

Does any kind of consciousness assigned to living organisms or (future) robots, actually factorises through the formal layer recovered in the previous subsections? Or is any conscious phenomena in nature or laboratory assigned to its formal level by which it emerges? These are particularly important issues of universal character which we want to approach here. One clue is that randomness considered sufficiently formally refers to the layer of (random) forcing extensions, hence speaking about random stimuli, when formal, leads to the constructions in this paper. Another indication is the way how we formalise in general and apply it to physical systems in space-time. We consider the conscious systems as necessary dwelling space-time such that the formal level of it is related to the formal level of the system. Is this relation strict in any sense? The way to answering it is in formalisation. Any conscious system is acting in, or oriented on, the ever present space and time. Saying that one formalises a theory in set theory means that constructions and reasoning within the theory are expressible in the language of ZF. In the case of the differentiable manifolds, their theory is usually formalisable in the part of ZFC i.e. $\Sigma _3$ formulas actually suffice. This in particular means that the entire formalisation (if needed) might be performed in the universe V of sets. However, here we follow a different possibility, also present in the foundations of set theory. Namely, any single universe does not suffice and this is rather a variety of models of ZFC, some being forcing extensions of $M_i$, another grounds of it, so that there emerges the dynamics of changes of what $M_i$ ($C_i$) actually ‘knows’ about sets. This picture is augmented by the internal finite fragments models $\text {ZFC}^*$ reflecting what $C_i$ is aware of. Hence, certain class of formalisation, “localised on space-time regions,” has been assigned to the conscious systems filling the domains of space-time such that different spatially separated parts of the system are grasped formally by different models $M_i,i\in I$ of ZFC (Definition 1). If there were a single model, say V, the conscious phenomena would not have been interpretable.

Let us now turn to the first case above, i.e. randomness. In mathematics, there have been developed domain of algorithmic randomness to express properly (and effectively) the meaning assigned to a binary infinite sequence $\sigma \in 2^{\omega }$ to be random. There is the definition by Martin-Löf of 1-random $\sigma$ which roughly says that such $\sigma$ has to avoid all arbitrarily small Lebesgue measure subsets of $\mathbb {R}\simeq 2^{\omega }$, which are generated in the arithmetic class $\Sigma _1^0$ (this sigma notation expresses the complications of formulas in the language of PA – see, e.g., Soare (2016)). This 1-randomness is naturally extended over n-randomness (avoiding $\Sigma _n^0$ ‘small’ subsets) and over $\omega$-randomness, though they are not that universal as 1-randomness of $\sigma$s. One especially important point is that forcings, both Cohen and random, become present in recovering formal content of the relation of randomness and Turing computability. First, they appear as the Peano arithmetic miniaturisations of the full set theory forcings, but their set theory counterparts are shown to be involved as limiting cases in the description as well. We do not write down here the exact definitions of various concepts, since they are not in use in the further argumentation and their rough meaning is sufficient. The interested reader is referred to such textbooks as Soare (2016) or Downey and Hirschfeldt (2010) (see also Król et al. (2023)). Thus, various levels of randomness and Turing uncomputability classes when seen sufficiently formally, are necessary affecting each to the other and coexist nontrivially with Cohen and random genericities in set theory. This is where formalisation of randomness recovers Cohen and random forcings extensions as in set theory.

To see the robustness of the relation of the formal level of the system S and this of space-time we apply the formalisation to smooth models of space-time (smooth manifolds) that gives the direct result. Let $M^n$ be a n-dimensional generalization of space-time, i.e. a smooth n-dimensional manifold allowing for Lorentzian, hence Euclidean as well, structures. By a nontrivial formalisation subordinated to an atlas $\{ U_i\}_{i\in I}$ on a smooth manifold $M^n$ we understand the set of the pairs $\{(a,b):a= U_i, b=M_i \}$ (where $M_i$ are standard transitive models of ZFC and not all $M_i,i\in I$ are isomorphic) which result in the internal local frames $\{ U_i\in M_i:i\in I\}$.

Lemma 3

Every nontrivial formalisation subordinated to an atlas on $M^n$ gives rise to a ZFC-inhomogeneous system S.

This follows directly from the fact that the formalisation constitutes identically a system and the condition of its nontriviality thus making the system ZFC-inhomogeneous.

As a consequence, the formalisation alone performed over space-time leads to the ZFC-inhomogeneous system S which is also the condition for supporting conscious phenomena within S. We also saw that the formalisation applied to random phenomena in space-time leads to genericity, hence to the nontrivial forcing extensions of models of ZFC. Thus, we recover main components of the model supporting consciousness in this work (the system S, models of ZFC and forcing extensions) via formalisations, ‘localized’ in space-time regions. One can find yet another, geometrical, interpretation of S and space-time. Namely, the depth of the system can be measured by the number of iterated forcing extensions (grounds) starting from a model $M_i$ and this depth is in a sense orthogonal to the manifold $M^n$.

Remark 8

The construction above resembles the appearance of local gauge symmetries in space-time known from gravitational and particle physics. New kind of local in space-time symmetries emerges (gauge symmetries) which are orthogonal to the horizontal tangent space to $M^n$ and the connection coefficients, living in the vertical part of such fibre bundle, correspond to the new interactions of particles (gauge fields).

Following the remark’s above perspective we see that the horizontal direction to $M^n$ corresponds to random forcing extensions which reflect the action of the random stimuli over the creature’s system S. If there were a single global model, say M, there would be just “zeroth global cross section” of the forcing bundle, i.e. there would be no inhomogeneous ZFC-system, and there would be no room for self-reflecting effects underlying the conscious behaviour. Similarly, the higher global cross-sections of the forcing bundle (meaning there is the same model M[r] for every local frame in $\mathcal{O}(M^n)$ and in this model M[r] the construction of $M^n$ is performed) would obstruct the existence of the self-reflections and the dynamical reactions of S on the random stimuli. That is why the nontrivial horizontal structure of the bundle is the precondition for the interpretability of conscious phenomena in the model. This bundle analogy can be developed further, however, this is just the geometrical-like perspective helping understanding which does not influence much the entire model. It can, however, suggest that the appearance of consciousness in nature is not that distant, as one could think, from what has been already found in the physical theories of fundamental interactions.

The variable models of set theory, underlying the presented here approach to conscious systems in space-time, have yet another structural aspect which leads to the multiverse approach (mV) in foundations of set theory and mathematics. mVs appear as the analogue of the structural symmetry group for the locally trivial fibre bundles over $M^n$ generalized by the varying models of ZFC over local coordinate patches of $M^n$. This is the repetition of the above construction, however, now we try to understand the invariance of the structure. Let one of the models assigned to the local open neighbourhood of $M^n$, $U_i\in \mathcal{O}(M^n)$, be $M_i$. As it was stated before, the models along the nontrivial cross sections over $M^n$ undergo the jumps $M_i\rightarrow M_j$ which on the nonempty intersection $U_i\cap U_j$ corresponds to the forcing extension, $M_j=M_i[G]$, or taking a ground model, $M_j=N$ where $N[G_j]=M_i$. Consequently, the most general symmetry structure is taking all forcing extensions or grounds for $M_i$. Such a structure determines what is called in set theory the mV of the model $M_i$. There emerges the external invariant layer of the creature $C_i$ which is the pair $(M^n,\text {mV}(M_i))$. The word ‘external’ reflects this fact that the content of the consciousness of $C_i$ is based on the internal models of finite fragments of ZFC which are not present in the external perspective (but such models can always appear, in principle, in any $M_i$). As a result, the variable models represent a deep universal property of consciousness emerging in systems ‘living’ in some regions of space-time.

From the point of view of internal models of ZFC, i.e. standard transitive models, N, containing all ordinals, the following problem has arisen (Hamkins, 2005; Reitz, 2007). Does it exist any N such that we can reach V by a forcing extension of N, i.e. whether there exists generic G such that $N[G]=V$? The answer is in general terms a bit indirect but is fixed by the following assumption. The non-existence of any N, a model of ZFC, such that its forcing extension N[G] would be V is called the ground axiom (GA) discussed in set theory. If GA is true it has consequences to conscious creatures. In particular, GA shows the separation of the realm based on mV of $M_i$ and V itself and the separation corresponds to the external world, where the random stimuli are originated and are going to react on the creature $C_i$, and the state-space of the allowed models which $C_i$ can admit. The external world $W(C_i)$ described by V is never reached by any forcing extension of any $M_i$ of $C_i$, but still any finite fragment $\text {ZFC}^*$ from the $W(C_i)$ is well-modelled inside $M_i$ and any of its forcing extension. We conclude that a system S to be consciously reacting on the external stimuli is spanned on multiverse paradigm, while the external world $W(C_i)$ is on the universe V. In the next subsection, we will see that this is especially important also when studying conscious Turing machines.

2.5 How Universal can the Model be?

So far we have approached the universality of the systems with consciousness in space-time by means of formalisation which would be localised in spatio-temporal domains or subordinated to the locally interacting system. Then Lemma 3 indicates that the formal layer of S and of space-time smooth manifold, in fact, trivially agree. In this subsection, we discuss the interpretation of conscious phenomena by formal constructions of ZFC in the context of Turing machines (TMs), since TMs are presumably able to represent universally what is effective and computable in Nature, as predicted by the famous Turing-Church thesis, e.g. Soare (2016). The expectation is to find yet another universality feature characterising ZFC constructions.

One might wonder whether the ZFC’s point of view limits too much the understanding of consciousness in general, or in living beings or robots in particular. A successful transfer of the paper’s ideas to the TM context, on the basis of the universality of the Turing-Church thesis, would show the broad applicability of these constructs. We have recently studied this problem in ref. Król and Schumann, A.: Turing machines as computing conscious machines, (2023) and analysed the general conditions under which TM could become a conscious computing machine. The ZFC constructs from the present work have been applied in the cited work as necessarily characterising TMs with consciousness.

Let us briefly overview these results. Again we do not present here full definitions since they are not essential for understanding of what follows. Interested reader can find all details regarding TMs and computability in the textbooks like Soare (2016). Special role is devoted to TMs with oracles (Soare, 2016) (see also Król and Schumann, A.: Turing machines as computing conscious machines, (2023)). In general, TM is a formal construct showing how any computational process realised in nature, hence effective, can be represented or reduced to the functioning of a simple machine which is TM (simple but equipped with an infinite tape). Such a machine can be defined and described, in principle, by Peano arithmetic augmented by the existence of an infinite countable set of order of $\omega$ of cells of the tape (PA is bi-interpretable with ZF without the axiom of infinity by taking its negation “every set is finite” Kaye and Wong (2007)). We see that in a sense TM refers to ZFC as its natural external environment. This is the crucial point for understanding TMs as creatures allowing for conscious reactions for the external random stimuli.

We follow the analysis in ref. Król and Schumann, A.: Turing machines as computing conscious machines, (2023) where TM is to be seen dynamically and, in particular, able to assimilate external random stimuli leading to better “understanding of the world” by TM. To show this kind of statement the following strategy has been employed. First, the list of features typically assigned to conscious behaviour is built. The point is that one should carefully distinguish the conscious behaviour which is not merely algorithmically determined which would not be just deduced as the result of computing over the incoming data, from those which are fully Turing computable. If there is no such domain of uncomputable events the alleged conscious TM would be just computable TM. Moreover, one requires that these unpredictable stimuli affect the growth (or the shrinkage) of the creature (and corresponding TM) by the assimilation of such random external factors that would result in the extension of the creature’s capability by learning. Second, the definition of TM with consciousness is given. Third, one proves that such defined TM respects all the points in the list and that is why it can be considered as conscious TM. Such process, indeed, has been performed and successfully led to certain results (see ref. Król and Schumann, A.: Turing machines as computing conscious machines, (2023)).

More precisely the initial step in defining the conscious TM is by formulating TM in a model M of ZFC. This fixes the external environment to TM as ZFC but the cumulative universe V of sets is the proper model for the environment while for TM it is the model M. Moreover to include the dynamics in the reactions of TM to the external stimuli and enabling for learning driven changes, one refers to forcing extension of M, M[r]. Now a generic random real number r represents a random stimuli and this r is suitably assimilated by M[r]. To represent properly the learning skills of TM one places a generic real r in the oracle for TM.

Remark 9

Oracle A for TM is an arbitrary subset $A\subset \omega$ (it can be Turing uncomputable) which is coded as 0, 1 digits of its characteristic function on the additional (infinite) to be read only tape (see (Soare, 2016) for details and further information). Such an extended TM becomes oracle TM and we write o-TM or $\text {TM}^A$.

Thus, the definition of the conscious TM is o-TM defined in the model M such that the oracle contains the random generic r (representing the random stimuli from the external environment) and TM is now defined in the extended model M[r] (after assimilation of the stimuli r). In this way TM can compute, based also on random (Turing uncomputable) data and TM is now formulated in M[r] as $\text {TM}^r$. The scope of its predictablity power is hence extended or changed – in the extended model there are sentences to be true which are not necessary true in M (of course all theorems of ZFC remain to be equally true in both models).

The remarkable feature of the analysis is that the tools applied to TMs here reflect and follow closely the tools developed in this paper in the broader context of conscious creatures which were also based on models of ZFC, their forcing extensions or taking grounds, and their relation to PA. To find out more complete relation of TMs and ZFC-inhomogeneous systems in space-time, we follow the rules below. Let $M_i,i=1,\cdots , N$ and $M_{ij}$ be CTM of ZFC.

• 1. S spreads over a spatial region $U\subset \mathbb {R}^3$; let $U=\bigcup _{i=1}^NU_i$ be U with its local open cover and with a family of ZFC models $\{M_i :i=1,2\cdots N \}$ such that $U_i\in M_i$ for each i.

• 2. Let $r_i$ be a random stimulus which is acting upon S at $x_i\in U_i\in M_i$ such that $M_i\rightarrow M_i[r_i]$.

• 3. Given another local interaction of S with the external stimulus $r_j\in M_j[r_j]$ on $U_j, i\ne j$ at $x_j\in U_j\in M_j$, then on the intersection $U_i\cap U_j$ there are assigned two models $M_i\ne M_j$.

• 4. It can happen one of the two possible situations: $M_i$ is $M_j[s]$ (or $M_j$ is $M_i[s]$) or $M_i, M_j$ are not any forcing extension one of the other.

• 4a. In the forcing correlated case above and supposing s be random generic real, one can consider $M_i$ and Turing machine $\text {TM}_i$ subordinated to $U_i$ and the change of the coordinates map $f_i:U_i\rightarrow U_j$ as governed by the random forcing extension $M_i\rightarrow M_j=M_i[s].$

• 4ba. For the forcing uncorrelated models $M_i\ne M_j, i\ne j$ it can still happen that there exists a hyper model $M_{ij}$ extending both $M_i$ and $M_j$ as its submodels with the same ordinals. Then we assign the $\text {TM}_{ij}$ TM on the intersection $U_i\cap U_j$.

• 4bb. Or it can happen that the hyper model $M'_{ij}$ requires adding extra ordinals and we assign $\text {TM}'_{ij}$ on the intersection $U_i\cap U_j$.

Remark 10

We have the family $\{\text {TM}_i :i=1,2,\cdots , N\}$ of TMs augmented eventually by $\{\text {TM}_{ij} :i,j=1,2,\cdots ,N \}$ for the intersections that means that each $\text {TM}_i$ is oracle $\text {TM}_i^{r_i}$ which is formulated in $M_i$ ($\text {TM}_{ij}^s$ in $M_{ij}$). Consequently, S undergoes local random stimulations which is captured locally by TMs with consciousness. The case 4ba. expresses the situation that two random extensions of $M_i$ can be compatible in a sense of Hamkins and 4bb. that they are incompatible in the sense that there does not exist a hyper model of ZFC for two forcing extensions $M_i[r_1],M_i[r_2]$ with the same ordinals as in $M_i[r_1],M_i[r_2]$.

The family $\{\text {TM}_i :i=1,2,\cdots , N\}$ as in the remark above is called the covering family of internal TMs for the conscious system S in space-time. Note that even though the full forcing extensions can be compatible (4ba) or incompatible (4bb), any finite fragment of ZFC is still modelled as the excited set in any model $M_i$. As we saw before, this reflection principle of set theory lays behind the conscious grasping of the internal as well external world where the randomness of both is reflected in the forcing random extensions of full models. There emerges the following characteristic of the reaction of the conscious TM on random phenomena in terms of computability. Let a conscious TM as before be an oracle o-TM in the model $M_i$. Then

Proposition 1

If a conscious TM assimilates external random stimuli, then the extended TM computes certain previously uncomputable reals.

This follows from the fact that arithmetic random real r is arithmetic weak random and, hence, Turing uncomputable in higher degrees n (e.g. Downey and Hirschfeldt (2010)). The set forcing is the extension of the arithmetic case for models of ZFC and when r becomes the oracle for TM computations are now performed on the base of such r extending the model $M_i$.

We have seen that the full case 4ba. deals with compatible (concurring) extensions while 4bb. with incompatible ones which suggests the possibility that one can grasp also the case of many cooperating conscious systems in space-time by multiverse language. This is the topic of the next section.

3 Applications: Concurrent and Reflective Games in Multiverses

Adding social functions to multi-agent systems takes their utility aspects to a qualitatively new level. There are two recent approaches to analysing multi-agent systems with social functions from the point of view of game theory: concurrent games (Abramsky and Mellies, 1999; Gutierrez et al., 2023) and reflective games (Chkhartishvili, 2010; Nelson and Ham, 2012; Novikov and Chkhartishvili, 2004). Using the agent paradigm of both approaches, we can find some practical applications, based on our mathematical model of TMs with consciousness.

3.1 Concurrent Games of Turing Machines with Consciousness

Now, we can assume that there are several conscious creatures $C_i$, $i = 0, 1, \dots$, called agents, who can act concurrently and interact with each other. Let us take the set of all agents $Agt = \{C_1, C_2, C_3, \dots \}$, and the set of all stimuli $Sgn = \{r_{ij}\}$, where in $r_{ij}$ the index i means an i-th stimulus for agent $C_j$. Then the extensions $M_j[r_{ij}]$ replace $M_j$ such that $M_j[r_{ij}]$ represent $C_j$ after modification. This $r_{ij}$ can denote, for example, the route that the agent $C_j$ plans to take. However, the planning strategy by $C_i$ or to be aware by $C_i$ of the eventual collision in future with another agent’s $C_j$ plans, require to be explained. This is where the internal content of the consciousness given by models of the finite fragments $\text {ZFC}^*$ becomes crucial. These ‘finite’ models are always modelled internally in any M by sets in it. Thus, even though building a model of ZFC internally in another model M is out of reach, as are forcing extensions, these ‘finite’ models can be always build internally to M. Consequently, the agent $C_i$ can model both the external environment including plans of other agents, and $C_i$’s internal images of them.

In particular, agents can deal with a small, finite number of properties like what $C_i$ knows about the states of a game at the moment t, and will increase its knowledge about certain sentence (property) which reflects the state of the game in the near future (after performing a step in the game) and which will be (considered as) true for $C_i$. This future sentence can be independent on the initial few properties $C_i$ started with. Such independence is indeed possible (see e.g. Paterek et al. (2010)) and reflects the forcing relation $\Vdash$ reduced to the small number of properties. Below we are using the forcing symbol $\Vdash$ in this finite sense, i.e. $r\Vdash \phi$ iff $\phi$ has been independent on the finite set of properties available to $C_i$ at t and becomes available for $C_i$ at the stage r. This corresponds to the extension of finite fragments models containing the finite number of the properties and we are using the same symbol M for such models (as for full ZFC models). Then the forcing relation $\Vdash$ expresses the extension of M by adding independent sentence $\phi$ even though the true forcing over full models would be trivial in many such cases. Hence, internalisation to the $C_i$ knowledge at various moments of time makes also forcing relation to be truncated to the finite set of properties.

Remark 11

Given the finite number of internal finite fragments of ZFC models they are all representable in a single model, like $M_i$, as the excited sets and this leads to the ability to consider planning strategies (paths) of variety of agents as they all are in a single model. Still intentionally by $C_i$ they could correspond to possibly full forcing extensions of models from the outside reflecting random phenomena in the ‘real’ world.

In general, the following situation is possible when two or more agents $C_{j_m}$ and $C_{j_n}$ are planning paths $r_{ij_{m}}$ and $r_{ij_{n}}$ between which a collision necessarily occurs, that is, the agents $C_{j_m}$ and $C_{j_n}$ will certainly collide at some time point t, passing their planned paths $r_{ij_{m}}$ and $r_{ij_{n}}$. For example, the agents may compete for the same resource, and then a collision may occur between them at time t. This collision will make it impossible to implement the plan. Then we have the following possible cases:

• $C_{j_m}$ is planning $r_{ij_m}$ and $C_{j_n}$ is planning $r_{ij_n}$, but only $M_{j_m}[r_{ij_m}]$ takes place because of the collision at t;

• $C_{j_m}$ is planning $r_{ij_m}$ and $C_{j_n}$ is planning $r_{ij_n}$, but only $M_{j_n}[r_{ij_n}]$ takes place because of the collision at t;

• $C_{j_m}$ is planning $r_{ij_m}$ and $C_{j_n}$ is planning $r_{ij_n}$, but neither $M_{j_m}[r_{ij_m}]$ nor $M_{j_n}[r_{ij_n}]$ has a place because of the collision at t.

To avoid collisions, agents must not only plan their own path, but also evaluate the paths of other agents. We suppose that they do it in a (finite) concurrent game that is a tuple (Kuznetsov et al., 2023)

$$\begin{aligned} \mathcal {G} = (Player_k, States, Act_{m\le k}, N^d_{k}, Mov^{m\le k}, Tab^{m\le k}, (\preceq _a)_{a\in Player_k}), \end{aligned}$$

where

• $Player_k = \{a_1, \dots , a_k\} \subseteq Agt$ is a finite set of players presented by those who analyse other agents.

• $States = \{r^t_{ij}:i = 1,2,\dots ; j = 1, \dots , k; t = 0,1, \dots \} \subseteq Sgn$ is a set of states presented by the ith path $r^t_{ij}$: $[t,t+1] \mapsto \mathbb {R}^3$ passed by the agents $a_j$ at the time step from t to $t+1$. Two paths $r^t_{1j}$ and $r^{t+1}_{2j}$ of the same agent $a_j$ can be concatenated and the result of concatenation $r^t_{1j} r^{t+1}_{2j}$: $[t, t+2] \mapsto \mathbb {R}^3$ will be also the path passed by the agent $a_i$ at the time interval $[t, t+2]$. Any composed path $r^t_{1j} r^{t+1}_{2j}\dots r^{t+n}_{(n+1)j}$ will be denoted by $r^{t+n}_{j}\in States^n$ at the time interval $[t, t+n+1]$.

• $Act_{m\le k}$ is a non-empty set of actions, $m \in (0, k] \subset \mathbb N$ is said to be a radius of actions in which a current state of m players is taken into account. This m can be different at each time step, but it cannot be bigger than k. These are precisely the neighbours whose reactions must be taken into account in a new decision to pave the way forward. This set $Act_{m\le k}$ represents decisions which an agent can make to avoid a collision with his or her neighbours, and m can be interpreted as the depth of the agent’s memory about previous states of m agents or as the horizon of the agent’s movement planning with m neighbours. The action is a map from $States^m$ to States. An output of action is called a move.

• $N_{k}^d =\{a_j\in Player_k:\text {the distance of }a_j \text { to }a_m\le d\}$ is a neighbourhood for the agent $a_m$, where $j, m \in (0, k]\subset \mathbb N$ and d is a radius with its centre in $a_m$ for including other agents to the neighbourhood.

• $Mov^{m\le k}$: $States^m \times Player_k \mapsto 2^{Act_m^{Player_k}}{\setminus } \{\emptyset \}$ is a mapping indicating the available sets of actions to a given player in a given set of states. A move $\textbf{m}_{a_j}^m$ is available if and only if $M_m[r^{t+n}_{m}]$ takes place for all $m \in (0, k]\subset \mathbb N$ at time $[t, t+n+1]$.

• $Tab^{m\le k}$: $States^m \times Act^{Player_k}_m \mapsto States$ is the transition table which associates, with a given set of states and a given move of the players, the set of states resulting from that move.

• For each $a \in Player_k$, $\preceq _a$ is a preorder (refexive and transitive relation) over $States^m$, called the preference relation of player a, indicating an appropriate social function for the agents; for each $\pi$, $\pi ' \in States^m$, by $\pi \preceq _a \pi '$ we denote that $\pi '$ is at least as good as $\pi$ for a and when it is not $\pi \preceq _a \pi '$, we say that a prefers $\pi$ over $\pi '$.

Members from $Act^{Player_k}_{m\le k}$ can be understood as a set of propositions of the players for each $m \in (0, k]\subset \mathbb N$, which can be true or false. The propositions of $Act^{\{a_j\}}_m$ are true for $a_j\in Player_k$ (where $j \in (0, m]$) if they have models $M_j[r^{t+n}_{j}]$. Let the set of propositions for the player $a_j$ be denoted by $Prop_j$. A concurrent game $\mathcal G$ of players looking for a path is a model $\mathcal M$ such that $\mathcal M = (\mathcal G, \phi _j)$, where $\phi _j$: $States \mapsto 2^{Prop_j}$.

Let us consider the game

$$\begin{aligned} \mathcal {G}' = (Player_5, States, Act_{m\le 5}, N_5, Mov^{m\le 5}, Tab^{m\le 5}, (\preceq _a)_{a\in Player_5}) \end{aligned}$$

with no more than 5 players $a_1$, ..., $a_5$ at each decision step for each player. Then $States_{t+n} = \{r^{t+n}_{j}:j = 1,2,\dots , 5; t = 0,1, \dots \}$ is a set of all possible game states at the time interval $[t, t+n+1]$. Let $\textbf{s}_0 = (r_1^0, \dots , r_5^0) \in (States_0)^5$. It is true before the game. This means that $M[r_k^0]$ takes place for all $k \in [1, \dots , 5] \subset \mathbb N$. The next states $\textbf{s}_{i}\in (States_i)^5$ for $i = 1, 2, \dots$ are defined through the game. Without loss of generality, let us regard the localisation of all five players at the time interval $[t, t+n+1]$ as depicted in Fig. 1. We see that $N_1=\{a_1, a_2, a_3\}$, $N_2 = \{a_1, a_2\}$, $N_3 = \{a_1, a_3\}$, $N_4 =\{a_4\}$, $N_5 =\{a_5\}$, according to the radius $d = 3.5$. From this it follows that the players $a_4$ and $a_5$ ignore the behaviour of all other agents at this time step, the player $a_1$ analyses the behaviour of both $a_2$ and $a_3$, while the agent $a_2$ analyses only the behaviour of $a_1$, and $a_3$ analyses only the behaviour of $a_1$. In this case $\textbf{s}_{t+n} = (\max (r_1^{t+n}, r_2^{t+n})\wedge \min (r_1^{t+n}, r_3^{t+n}), r_1^{t+n} \Rightarrow r_2^{t+n}, \max (r_1^{t+n}, r_3^{t+n}), r_4^{t+n}, r_5^{t+n})\in (States_{t+n})^5$. Let us assume that this move is available. This means that we have new individual models after forcing: $M_1[\max (r_1^{t+n}, r_2^{t+n})\wedge \min (r_1^{t+n},r_3^{t+n})]$; $M_2[r_1^{t+n} \Rightarrow r_2^{t+n}]$; $M_3[\max (r_1^{t+n}, r_3^{t+n})]$; $M_4[r_4^{t+n}]$; $M_5[r_5^{t+n}]$, and they are contained in the game model $\mathcal {M}' = (\mathcal G', \phi _j)$, where $\phi _j$: $States \mapsto 2^{Prop_j}$ for $j=1,\dots , 5$. Then this $\textbf{s}_{t+n}$ belongs to $Tab^{5}$ at the time interval $[t, t+n+1]$. At this step, only three agents coordinate their behaviour in order to avoid collision: $a_1, a_2, a_3$, and for this they use not their own trajectories, but Boolean compositions of their trajectories and trajectories of their neighbours.

Fig. 1

The different localisation of the five players $a_1$, ..., $a_5$. As a result, the neighbourhood $N_1$ for the agent $a_1$ contains three members: $a_1$, $a_2$, $a_3$. The neighbourhood $N_4$ for the agent $a_4$ contains only $a_4$ and the neighbourhood $N_5$ for the agent $a_5$ contains only $a_5$. Hence, the agents $a_4$ and $a_5$ ignore all other players in their decisions at the time step $t+n+1$, while the player $a_1$ should analyse the behaviour of the players $a_2$ and $a_3$. We assume that $a_1$ constructs a maximisation of the routes $r_1^{t+n}$ and $r_2^{t+n}$ and a minimisation of the routes $r_1^{t+n}$ and $r_3^{t+n}$ in his or her new decision

3.2 Self-Reflection in Concurrent Games of Turing Machines with Consciousness

To avoid any possible collisions, the players of the game $\mathcal {G}$ in their decisions may construct a Boolean superposition of the trajectories belonging to their neighbour players. In this way we can define a knowledge operator:

Definition 3

(Knowledge operator for own trajectories) Let $Th(r^{t}_{j})\in Prop^t_j$ be a proposition of the player $a_j$ at the time step t, describing $r^{t}_{j}$. Then the agent $a_j$ knows $r^{t}_{j}$ at t, symbolically: $K_j r^{t}_{j}$, if and only if $Th(r^{t}_{j})$ is true in $\mathcal {M}$, that is, $M_j [r^{t}_{j}]\models Th(r^{t}_{j})$.

In our game $\mathcal {G}$, we assume that the players can know the trajectories of their neighbours. From this it follows that a player $a_j$ can try to reconstruct reasoning of his or her neighbour, based on the own reasoning and the past trajectories of his or her neighbours. Let $\textbf{B}^t_j$ be a Boolean superposition of a trajectory $r^{t}_{j}$ of $a_j$ with trajectories of his or her neighbours at t (we remember that the number of neighbours can vary for different t). Then we can extend Definition 3 as follows:

Definition 4

(Knowledge operator for different trajectories) Let $Th(\textbf{B}^t_j)$ be a proposition of the player $a_j$ at the time step t in respect to the behaviour of his or her neighbours. Then the agent $a_j$ knows $\textbf{B}^t_j$ at t, symbolically: $K_j \textbf{B}^t_j$, if and only if $M_j [\textbf{B}^t_j]\models Th(\textbf{B}^t_j)$.

Let $A_t$ be a new state in the game $\mathcal G$ at t. Then the agent $a_j$ knows it, i.e. $K_j A_t$ if and only if $A_t$ is $r^{t}_{j}$ or $A_t$ is $\textbf{B}^t_j$ at t and $Th(A_t)$ is true in $\mathcal M$, according to Definitions 3, 4:

$$\begin{aligned} K_j A_t = \{r :A_t \subseteq \textbf{A}_j(r) = \{r :r \,\, \Vdash \,\,Th(A_t)\}\}. \end{aligned}$$

From this understanding of knowledge operator, we can obtain its following properties:

$$\begin{aligned} (K_{j} A_t\cap K_{j} B_t)\Rightarrow K_{j} (A_t\cap B_t). \end{aligned}$$

(2)

Indeed, if $K_{j} A_t\cap K_{j} B_t$ is not empty, then there exists $r\in K_{j} A_t\cap K_{j} B_t$ such that $r \,\,\Vdash \,\,Th(A_t) \wedge Th(B_t)$. Then $K_{j} (A_t\cap B_t)$ is not empty. On the other hand, let $K_{j} (A_t\cap B_t) = \{r :(A_t\cap B_t) \subseteq \textbf{A}_j(r) = \{r :r \,\,\Vdash \,\,Th(A_t \wedge B_t)\}\}$ be non-empty. It does not mean that there exists $r\in \textbf{B}_j(r)\ne \textbf{A}_j(r)$ such that $r \,\,\Vdash \,\,Th(A_t \wedge B_t)$.

$$\begin{aligned} K_{j} (A_t\cup B_t)\Rightarrow (K_{j} A_t\cup K_{j} B_t). \end{aligned}$$

(3)

The expression $K_{j} (A_t\cup B_t)$ means that there exists $r\in K_{j} (A_t\cup B_t)$ such that $r \,\,\Vdash \,\,Th(A_t \vee B_t)$, that is, for each ‘condition’ $q \le r$ there exists $p \le q$ such that $r \,\,\Vdash \,\,Th(A_t)$ or $r \,\,\Vdash \,\, Th(B_t)$. Then $K_{j} A_t\cup K_{j} B_t$ is not empty. Furthermore:

$$\begin{aligned} K_{j} (A_t\cup B_t)=(K_{j} A_t\cap K_{j} B_t). \end{aligned}$$

(4)

Indeed, $K_{j} (A_t\cup B_t)$ consists of all $r\in K_{j} (A_t\cup B_t)$ such that $r \,\,\Vdash \,\,Th(A_t \vee B_t)$. And it also contains $A_t\cup B_t$ as its subset. It is exactly the set, including all $r\in K_{j} (A_t)\cap K_{j}(B_t)$ such that $r \,\,\Vdash \,\,Th(A_t)$ and $r \,\,\Vdash \,\, Th(B_t)$.

Other properties are evident:

$$\begin{aligned}{} & {} A_t\subseteq B_t\Rightarrow K_{j} A_t\subseteq K_{j} B_t; \end{aligned}$$

(5)

$$\begin{aligned}{} & {} A_t\subseteq K_{j} A_t. \end{aligned}$$

(6)

$$\begin{aligned}{} & {} K_{j} K_{j} A_t =K_{j} A_t. \end{aligned}$$

(7)

3.3 High Levels of Reflection

We can define self-reflection of higher levels. Let $A_t$ be a game state that is known by the agent $a_j$, that is, $A_t\subseteq K_j A_t$. Let $\textbf{B}^t_j(A_t)$ be a Boolean superposition with $A_t$ made by $a_j$ with trajectories of his or her neighbours at t. If this $\textbf{B}^t_j(A_t)$ is true for $a_j$ at t, then $\textbf{B}^t_j(A_t)\subseteq K_j A_t$. Assume that we have a bimatrix game with only two players $a_j$ and $a_i$. Then

$$\begin{aligned} \textbf{B}^{t, 0}_j(A_t) = \textbf{B}^t_j(A_t); \textbf{B}^{t, 0}_i(A_t) = \textbf{B}^t_i(A_t) \\ \textbf{B}^{t, 1}_j(A_t) = \textbf{B}^t_j(\textbf{B}^t_i(A_t)); \textbf{B}^{t, 1}_i(A_t) = \textbf{B}^t_i(\textbf{B}^t_j(A_t)) \\ \textbf{B}^{t, 2}_j(A_t) = \textbf{B}^t_j(\textbf{B}^t_i(\textbf{B}^t_j(A_t))); \textbf{B}^{t, 2}_i(A_t) = \textbf{B}^t_i(\textbf{B}^t_j(\textbf{B}^t_i(A_t))) \\ \dots \\ \textbf{B}^{t, 2n}_j(A_t) = \underbrace{\textbf{B}^t_j(\dots \textbf{B}^t_i}_{2n}(\textbf{B}^t_j(A_t))\dots ); \textbf{B}^{t, 2n}_i(A_t) = \underbrace{\textbf{B}^t_i(\dots \textbf{B}^t_j}_{2n}(\textbf{B}^t_i(A_t))\dots ) \end{aligned}$$

Then we can define the n-th level of reflection for each $n =0,1, 2, \dots$:

Definition 5

(n-th level of reflection) The agent $a_j$ knows $A_t$ at the n-th level of reflection, symbolically: $K_j^n (A_t)$, if and only if $\textbf{B}^{t, n}_j(A_t)\subseteq K_j^n (A_t)$.

From this definition it follows that $A_t\subseteq \dots \subseteq K_j^nA_t\subseteq K_i^{n+1}A_t$ and $A_t\subseteq \dots \subseteq K_i^nA_t\subseteq K_j^{n+1}A_t$. Therefore $K_i^{n+1}A_t\cap K_j^{n+1}A_t \ne \emptyset$. Indeed $A_t \subseteq K_i^{n+1}A_t\cap K_j^{n+1}A_t$.

Hence, we can interpret $K_jA_t$ (accordingly, $K_iA_t$) as the agent $a_j$’s (accordingly, the agent $a_i$’s) cognitive or emotional estimations of states of affairs $A_t$ with a subsequent perlocutionary effect of these estimations on agent $a_j$ (accordingly, on $a_i$) at t. So, $K_jA_t$ means “$a_j$ + performantive verb + that $A_t$” (e.g. “$a_j$ thinks that $A_t$”) and the agent $a_j$ follows this statement in his or her behaviour at t. Further, let $K_jK_iA_t$ (accordingly, $K_iK_jA_t$) mean the agent $a_j$’s performative estimation of perlocutionary effect $K_iA_t$ (accordingly, the agent $a_i$’s estimation of perlocutionary effect $K_jA_t$) and $K_jK_iA_t$ (accordingly, $K_iK_jA_t$) is a new state for both agents $a_i$, $a_j$. For instance, $K_jK_iA_t$ means “$a_j$ knows that $a_i$ hates $A_t$ at t” and this statement determines the behavioural strategies of both agents: $a_i$ hates $A_t$ and $a_j$ takes into account this perlocution of $a_i$. Thus, the state of affairs $A_t$ can be interpreted differently by agents $a_i, a_j$: $K_iA_t$ (“$A_t$ from $a_i$’s viewpoint”), $K_jA_t$ (“$A_t$ from $a_j$’s viewpoint”). But images which are generated by one of the players can be foreseen by others, too. As a result, there are performative effects $K_jK_iA_t$ (“$K_iA_t$ from $a_j$’s viewpoint”) and $K_iK_jA_t$ (“$K_jA_t$ from $a_i$’s viewpoint”), etc.

We can generalise the situation of bimatrix game with higher levels of reflection up to the case of game with many players $a_1, \dots , a_k$. Then for each agent $a_j$, $j \in (0, k] \subset \mathbb N$, who knows $A_t$, there is a reflection level n, such that $K_j^n A_t$ takes place (Schumann, 2014; Mints et al., 2020).

We can suppose the following possibilities of the reflexive game:

• 1. both agents $a_i$ and $a_j$ are at the n-th level of reflection about the same state of affairs $A_t$: $K_i^{n}A_t$ and $K_j^{n}A$ simultaneously take place;

• 2. we have $K_j^{n+1}A_t = K_j^{n}A_t$ and $K_j^{n}A_t \subseteq K_i^{n+1}A_t$; this means that the agent $a_i$ stays at the $(n+1)$-th level of reflection about the same state of affairs $A_t$, but the agent $a_j$ stays at the n-th level of reflection;

• 3. we have $K_i^{n+1}A_t = K_i^{n}A_t$ and $K_i^{n}A_t \subseteq K_j^{n+1}A_t$; this means that the agent $a_j$ stays at the $(n+1)$-th level of reflection about the same state of affairs $A_t$, but the agent $a_i$ stays at the n-th level of reflection.

Hence, self-reflection within reflective games appears as a self-adaptation of mobile agents. That is what means that the agents possess social functions. The divisions of agents into different social roles (to cooperate, compete or counteract) will be a result of self-learning of this system, based on different knowledge operators with different levels of self-reflection. This allows for organising the network systems like swarms.

4 Methods

There exists an approach in foundations of set theory that considers models of ZFC as basic objects of studies (e.g. Hamkins (2012); Antos (2022)). Models are primordial constructs not sets and their elements. Such an approach directly leads to the foundations based on varying models together with their forcing extensions and taking grounds, i.e. on multiverses. In this paper, we apply multiverses to the analysis of the emergence of consciousness in various systems occupying some domains in space-time. The readiness of a system to perpetually react on external or internal random stimuli and perform learning processes based on this is reflected by readiness of any model M to be extended or contracted like in a multiverse.

Thus, given the foundations of mathematics, based on the models of ZFC replacing sets we apply this to conscious phenomena and draw universal conclusions. In particular, we are able to understand the emergence of consciousness in various systems in space-time in mathematical terms. The same method based on models of ZFC allows for working out the counterpart for conscious Turing machine and obtaining the universality results in the domain of conscious systems. When truncating to finite fragments of ZFC and their models (which the content of conscious phenomena is based on) one can consider the cooperation within the ‘swarms’ of conscious creatures in terms of concurrent games, where decisions of $C_i$ about subsequent steps of the strategy are made internally. The truncation to the finite fragments allows us to consider forcings as trivial which reflects what a creature $C_i$ knows approximately about full random extensions (stimuli). Let us note that the similar methods based on models of ZFC and forcings have been applied already to certain physical systems, e.g. Benioff (1976); Asselmeyer-Maluga and Król (2019); Król and Asselmeyer-Maluga (2020); Król and Klimasara (2020).

5 Conclusions and Future Work

Obviously this work is not in any sense the first attempt, especially in philosophical inquires, to formalize consciousness. However, the way how we do that, via forcing and formal set theory, is probably the first such proposal. Anyway it would be highly instructive and desirable from philosophy point of view, to formulate the predictions about the relation ‘external world – conscious acts’ based on our work. These predictions would allow the reader to navigate through existing approaches in the context of the relation and locate among them the proposal. Namely does the external world determine or even create conscious acts which merely reflect the world? or, maybe, do the acts create the world around us and up to what degree? The discussion of such topics can be made arbitrary long and involved, let us see what the presented scenario is saying about them. Firstly, there is a counterpart for the external/internal which are named as cumulative universe V of sets of von Neumann for the external (ZFC) vs. models of ZFC for the internal, but this distinction is just the formal construction which eventually leads to something essential about external/internal in the context of consciousness. Secondly, the external generates stimuli acting over the models of ZFC. Thirdly, the stimuli can be fully random depending on the models of ZFC. Next, the models assimilate the fully random stimuli as forcing extensions which result in the extended models of ZFC where the stimuli are not random any longer but rather are generic real numbers. Further, the internal context of this scenario is built via mapping the full random forcing extensions onto the finite fragment models of ZFC. Saying all this above we can draw the conclusions about the conscious acts and the external world. Namely, on the one hand the external world reacts on the internal (formal random stimuli) and enforces the reflection of the world into the internal domain where conscious acts dominate, however, this reflection is highly creative not just conservative. This last means, on the other hand, that conscious acts create new formal environment (extended models of ZFC) which is referred to the external world. This extended model is not determined by the axiomatic ZFC and V itself. Next step, the truncation to finite fragment model of ZFC and storing as the corresponding conscious content is also undetermined by the external ZFC. From these reasons any conscious act, focusing on the external stimulus, is creative rather than just reflecting the external objective state. This can be further developed and specified in the context of phenomenological components of a conscious act,³ e.g. Yoshimi (2007); Krysztofiak (2020) We find the approach suitable for addressing and understanding the structure of conscious acts which has been one of the major interest in phenomenology from the beginning. Forcing and extensions of ZFC models constitute a universal structure underlying the conscious behaviour of entities in space. The notion of transcedence has its natural meaning in the language of forcing extensions (s is not in a model M but is available in M[G]) and the intentionality of conscious acts is naturally expressed (see the Results section). We do not decide here whether a phenomenological reduction could meet non-trivially this universal formal level, but such possibility seems highly intriguing and worth pursuing. One last comment regarding a way how we mathematise conscious act comparing to applying the language of dynamical systems (Yoshimi, 2007). The use of a formal theory perspective and the perspective of its models as simultaneous formal departing point for the formalisation process, do not allow for the unique outside ‘objective’ point of view based entirely on the universe V or the multiverse $\text {m}V$. This kind of inhomogeneous foundations for the mathematising conscious phenomena is not present elsewhere in other formal approaches where such perspective is always possible, but here this inability is the necessary condition for the proper representation of consciousness.

Our work relates also more practical problems. Currently, the most promising AI task is to design the group behaviour of robots. For example, the organisation of intelligent, hierarchical network systems, acting as ad hoc transport networks, where vehicles (cars, planes, etc.) behave as autonomous agents together with each other, as well as with stationary agents of urban infrastructure, installed on city buildings. Another example is the Internet of Things. But there is no general mathematical theory defining the possibilities and limits of this group behaviour of robots. Turing machines give a general idea of computation and computability and we intend to extend Turing machines to some systems with a mathematically defined form of consciousness. It will be a general mathematical language to describe a system with several units where they share some reciprocal behaviours such as cooperation, coordination and negotiation and have the advantage of distributed and concurrent problem solving. For example, there are different models of artificial swarms used in robotics: artificial bees, artificial ants. But we may present an artificial swarm mathematically in general terms as a Turing machine with consciousness. Hence, for networked systems, the implementation of our mathematical language may offer the possibility of programming complex dynamic systems, such as transport systems, as a self-organising multi-agent system with social functions.