The concept of strong and weak virtual reality

Abstract

We approach the virtual reality phenomenon by studying its relationship to set theory. This approach offers a characterization of virtual reality in set theoretic terms, and we investigate the case where this is done using the wellfoundedness property. Our hypothesis is that non-wellfounded sets (so-called hypersets) give rise to a different quality of virtual reality than do familiar wellfounded sets. To elaborate this hypothesis, we describe virtual reality through Sommerhoff’s categories of first- and second-order self-awareness; introduced as necessary conditions for consciousness in terms of higher cognitive functions. We then propose a representation of first- and second-order self-awareness through sets, and assume that these sets, which we call events, originally form a collection of wellfounded sets. Strong virtual reality characterizes virtual reality environments which have the limited capacity to create only events associated with wellfounded sets. In contrast, the logically weaker and more general concept of weak virtual reality characterizes collections of virtual reality mediated events altogether forming an entirety larger than any collection of wellfounded sets. By giving reference to Aczel’s hyperset theory we indicate that this definition is not empty because hypersets encompass wellfounded sets already. Moreover, we argue that weak virtual reality could be realized in human history through continued progress in computer technology. Finally, within a more general framework, we use Baltag’s structural theory of sets (STS) to show that within this hyperset theory Sommerhoff’s first- and second-order self-awareness as well as both concepts of virtual reality admit a consistent mathematical representation. To illustrate our ideas, several examples and heuristic arguments are discussed.

Appendix

We employ a generalization of ZFC⁻ + AFA set theory and follow the original work of Baltag (1999), in which a structural concept of sets is introduced, and we briefly outline some of the elementary ideas in STS. A structural understanding of sets is in a sense dual to the classical iterative (i.e., synthetic) concept of set. While in the latter sets are built from some previously given objects in successive stages (the iterative concept of set), the former presupposes that a priori a set is a unified totality that reveals its abstract membership structure only step by step through the process of structural unfolding. This stepwise discovery of the set structure is generated by imposing questions (which Baltag calls analytical experiments) to the initial object; the answers to these questions are the stages of structural unfolding. Thus at each stage we have a partial description of the object considered. The unfolding process can be defined by recursion on the ordinals: for every ordinal α and every set a, the unfolding of rank α is the set a ^α, given by

$$ \begin{aligned} a^{\alpha +1}=&\{b^{\alpha}:b \in a\}\\ a^\lambda=&\langle a^\alpha\rangle_{\alpha < \lambda}, \hbox{for limit ordinals}, \lambda \end{aligned} $$

Surely, this definition is meaningful for all wellfounded sets, but for a larger universe it is inappropriate in general since ∈-recursion is equivalent to the Axiom of Foundation.

To arrive a definition of structural unfolding for more general objects, i.e., systems or classes, Baltag argues that at every ordinal stage we can only have a partial description of a system. Here, an essential concept is observational equivalence between systems—a generalization of the bisimulation concept for systems (Aczel, 1988; Devlin, 1991). As a equivalalence relation, bisimulation is not applicable when large systems are considered, i.e., when those systems fail to be graphs because their collection of edges/nodes become proper classes. In such cases observational equivalence is given by modal equivalence, and not by the usual definition of bisimulation, and it turns out that with infinitary modal logic observational equivalence between systems can be defined to incorporate even large systems.

In STS, a modal theory th(a) for every set or class a is given through the so-called satisfaction axioms, and before we quote these axioms we may first introduce the underlying modal language.

• 1. Negation. Given a possible description φ and an object a, we construct a new description $\neg \varphi$, to capture the information that φ does not describe a.

• 2. Conjunction. Given a set Φ of descriptions of the object a, we accumulate all descriptions in Φ by forming their conjunction $\bigwedge\Phi$.

• 3. Unfolding. Given a description φ of some member (or members)of a set a, we unfold the set by constructing a description ⋄φ, which captures the information that a has some member described by φ.

The language generated by these three rules is called infinitary modal logic, which allows infinite logical conjunctions. With $\bigvee$ and □ as the duals (obtained by substituting $\wedge \mapsto \vee$ and $\diamond \mapsto \square$) to $\bigwedge$ and $\diamond$, respectively, we introduce the following operators:

$$ \begin{aligned} \diamond\Phi=:&\{\diamond \varphi: \varphi \in \Phi\},\\ \square\Phi=:&\{\square\varphi : \varphi \in \Phi\},\\ \varphi \wedge\psi=:& \bigwedge\{\varphi, \psi\},\\ \varphi\vee\psi =:&\bigvee\{\varphi, \psi\},\\ \bigtriangleup\Phi=:& \bigwedge\diamond\Phi \wedge \square \bigvee\Phi. \end{aligned} $$

The satisfaction axioms presume the existence of a class a Sat, each element of Sat is a pair of a set a and a modal sentence φ. Writing a⊧φ for (a, φ) ∈Sat, these axioms read as

$$ \begin{aligned} \hbox{(SA1)}& \quad a \models \neg \varphi \hbox{if} a \not\models \varphi\\ \hbox{(SA2)}& \quad a \models \bigwedge \Phi \hbox{if} a \models \varphi \hbox{ for all } \varphi \in \Phi\\ \hbox{(SA3)}&\quad a \models \diamond\varphi \hbox{if} a^{\prime} \models \varphi \hbox{ for some } a^\prime\in a \end{aligned} $$

with this setting the notion of unfolding of a set a admits now an expression through modal sentences φ^α _a defined for any cardinal α as

$$ \begin{aligned} \varphi^{\alpha+1}_a=:&\triangle \{\varphi^\alpha_b : b \in a\},\\ \varphi^\lambda_a =:&\bigwedge\{\varphi_a^\beta:\beta < \alpha\} \hbox{ for limit cardinals}, \lambda. \end{aligned} $$

Unfoldings of rank α are maximal from an informational point of view as they gather all the information that is available at stage α about a set and its members. In formal language this statement reads as the proposition: b ⊧φ^α _a iff b ^α = a ^α. This explains the notion of observationally equivalent: two sets, classes or systems are said to be observationally equivalent if they satisfy the same infinitary modal sentences, i.e., if they are modally equivalent. In STS, the existence of sets is guaranteed by the bijection th(·) between maximal weakly consistent theories and the sets (This correspondence is a direct consequence of the Super-Antifoundation Axiom in STS.). Weakly consistent theories are those theories in which all sub-collections of descriptions that are satisfied by a set are closed under infinitary conjunctions. It follows that non-wellfounded sets or classes are exactly those which do not admit satisfaction by any finite conjunction in infinitary modal logic.

On this background, we now make the observation that STS—constructed with infinitary modal logic— provides a tool for restating Sommerhoff’s first- and second-order self-awareness in set theoretic and logical terms. First, a universal set U exists in STS being observationally equivalent to the universal class U _c  = {x: x is a set}. We want to use U as the collection of all events E ₁ in weak virtual reality. This is possible because any universe ^* V ₁ for ZFC⁻ + AFA set theory is observationally equivalent to the set U (Baltag, 1999). Thus a weak virtual reality {R ₁, U, S } can be defined within STS because any arbitrary event E ₁(P,S) ∈U is associated with its modal description φ ∈th(U), i.e., E ₁(P,S) ⊧ φ.

Any event E ₁(P,S)∈U is then understood as the image of $P\subseteq W_1$ under the map d: = th _S ○ th ⁻¹ _S , i.e., E ₁(P,S) = d(P) = th ⁻¹ _S (th _S (P)), with th _S being the operator mapping classes $P\subseteq W_1$ onto their modal theories and th ⁻¹ _S is the inverse map; both maps exist in STS (Baltag, 1999), and together they define the denotation function d: = th ○ th ⁻¹ with d: R ₁ → U. First-order self-awareness are mathematically represented by the denotation d, and second- order self-awareness is established through the unfolding rule, i.e., rule (3), as ⋄φ now encodes the information that U has a member described by φ. But ⋄ φ as a sentence is a member of th _S (U), because th _S (U) already contains the collection {⋄ φ: φ is a consistent modal sentence} (Baltag, 1999). Then th ⁻¹ _S (⋄ φ) must be a member of U, and we identify it as th ⁻¹ _S (⋄ φ) =  E ^*₁ (P,S). This logical statement represents Sommerhoff’s second-order self-awareness as introduced in Definition 2.1, because E ^*₁ (P,S) indeed satisfies the modal sentence stating that U has E ₁(P,S) as part of its current state. Finally, since rule (1) and rule (2) are satisfied, they recognize the fact that modal descriptions of events can be logically conjoined to create valid descriptions of events, Proposition 5.1 follows.

The special case of strong virtual reality is obtained by using a denotation d which maps classes in R ₁ onto wellfounded sets. These sets altogether comprise the totality of events in strong virtual reality V ₁. In that case the corresponding theory is th _S (V ₁)⊂ th _S (U).