# Experimental Non-Violation of the Bell Inequality
## 6 Relations to Other Approaches
### p-Adic Quantum Theory  
Over the years, there have been attempts to reformulate quantum theory, e.g., by replacing the complex coefficients of Hilbert states with \(p\)-adic numbers (there is a \(p\)-adic correspondence) to the complex numbers. A review of such approaches is given in [21]. One of the motivations for thiswas the idea that space-time may have some fractal structure on the Planck scale. Drawing on this work, [PERSON] [22] proposed a hidden-variable model where the probability distribution \(\rho(\lambda)\) was defined on the \(p\)-adic numbers. Now, essentially because \(-1=1+2+4\ldots\) in 2-adic number theory, this allows for negative probabilities, which, although very esoteric, also arose for different reasons in [PERSON]'s work on relativistic quantisation, and therefore have some basis in quantum physics. [PERSON] argued that the existence of such negative probabilities could negate the Bell Theorem.  
It is important to distance the present work from such \(p\)-adic quantum theory. Here, we do not introduce \(p\)-adic numbers into Hilbert Space or (as a result) into probability distributions. Fundamentally, Invariant Set Theory is a deterministic theory and not a probabilistic theory. A key element of Invariant Set Theory is to reject the notion of the algebraically closed Hilbert Space as a state space for quantum physics, irrespective of whether the Hilbert Space is closed on the reals, the complexes or the \(p\)-adics. The reason for such rejection of this is ultimately so that physics can be described finitely [6]. In Invariant Set Theory, probabilities are defined by elementary frequentism and therefore are simply and intuitively rational numbers on the interval \([0,1]\). In fact, in the work presented in [6], maps on \(p\)-adic integers are defined to describe the deterministic laws that may underpin quantum physics. This is where algebraic closure can be reinstated--not on the Hilbert vectors or associated probabilities.  
Indeed, the metric \(g_{p}\) that plays such a central role in this paper, whilst motivated by the \(p\)-adic metric and the fact that the set of \(p\)-adic integers is homeomorphic to Cantor sets, is not identical to it. \(g_{p}\) has the property that, if \(p\) is large, points which do not lie on \(I_{U}\) are necessarily distant from points that do lie on \(I_{U}\). This reflects the fact that the \(p\)-adic distance between \(p\)-adic numbers that are not \(p\)-adic integers, and \(p\)-adic integers, is \(\geq p\).  
This comparison with \(p\)-adic quantum theory raises an important point. A key motivation for developing Invariant Set Theory (in addition to be able to describe physics finitely) was to make the Bell Theorem understandable. If a particular formalism leads one to trade one type of incomprehensible property (nonlocality) for another (negative probability), then, as far as the author is concerned, there is no compelling reason to adopt such a formalism. As discussed further in the next Section, Invariant Set Theory does not require any esoteric notions to explain the Bell Theorem (once one has embraced the key fact that space-time and state-space can have very different metrics).