A Note on the Reality of Incomputable Real Numbers and Its Systemic Significance
Abstract
:1. Introduction
“Mathematicians, however, freely fantasize with infiniteprecision real numbers. Nevertheless within pure math the notion of a real number is extremely problematic.”
2. Computational Processes and Emergent Computation
“What I have called the Babylonian idea is to say, ‘I happen to know this, and I happen to know that, and maybe I know that; and I work everything out from there. Tomorrow I may forget that this is true, but remember that something else is true, so I can reconstruct it all again. I am never quite sure of where I am supposed to begin or where I am supposed to end. I just remember enough all the time so that as the memory fades and some of the pieces fall out I can put the thing back together again every day’”([7], p. 45).
“In physics we need the Babylonian method, and not the Euclidian or Greek method”(Ibid., p. 47).
3. Arbitrary Discretization
“Experimental physicists know how difficult accurate measurements are. No physical quantity has ever been measured with more than 15 or so digits of accuracy”[3].
The reality of IRNs lies in the representative modeling roles of some of their properties, such as their infiniteness, uniqueness, singularities, and noncomputability, which represent and correspond to theoretical incomplete properties of processes such as emergence and quantumlike phenomena, whereas theoretical representation, partly differentiating from the classical representation, can be applied to model very complicated transient dynamics between phases occurring when classical and quantum aspects mix.
The research on IRNs should then not be considered an abstract exercise of mathematicians but as research in a field considered to possess properties representing and corresponding to those detectable in real, even if extreme, phenomena, such as emergence and quantum phenomena.
3.1. Equivalences and Symmetries
3.2. Discretization, Multiple Modeling, and NonPredictability
 The infiniteness of IRNs corresponds to the theoretical incompleteness of several phenomena and processes mentioned above, such as the processes of emergence and the phenomena of continuous balance between equivalences and interchangeabilities, e.g., ergodic, and quantum.
 The noncomputability of subsets of real numbers, such as irrational algebraic and transcendent, i.e., IRNs, corresponds to their nonzippability into complete analytical formulae or exhaustive single models and their combinations of complexity, such as processes of emergence, phenomena that continuously acquire, lose, and recover multiple coherences, e.g., collective behaviors.
3.3. Computability
 Nondeterministic computation when, from a given input and state, the abstract machine may jump to several different possible states;
 Hypercomputation of the socalled Ω Chaitin constant or the halting probability, consisting of a real number expressing the probability that a randomly constructed program will halt depending on the program encoding used and its length [17,53,54,55]. The importance of the Chaitin constant lies in the fact that various problems in number theory are equivalent to solving the halting problem for special programs, such as searching for counterexamples and halting if one is found. For instance, this is the case for the socalled Goldbach conjecture, which states that every even integer number greater than 2 can be always expressed as the sum of two primes [56].
 Inductive Turing machines, which perform a list of defined instructions depending on the initial states and acquire a series of successive states by applying inductive reasoning and being dependent on environment phenomenology [57].
 Quantum computers, which are based on the possibility of being in superposed states. Whereas a bit can only have two states, i.e., 0 or 1, a qubit state is a linear superposition of the basis states that is described by probability amplitudes. Multiple qubits can exhibit quantum entanglement. The Quantum Universal Turing Machine (QUTM) takes advantage of the superposition principle and the entanglement among qubits [58,59,60,61,62,63], raising the research issue of whether or not it is a Turing machine.
 Qualitative analysis, which involves determining and elaborating qualitative properties of a phenomenon in a way that allows the identification or exclusion of subsequently acquired properties, such as convergence, that deal with numerical sequences, analytic properties of a function without calculating it, and pattern and multiple dynamic coherence recognition. We believe this is a very promising field of research and one that can be used to introduce new, unimaginable approaches.
 The infiniteness of IRNs corresponds to the necessary but unpredictable collapse of any equivalence, reaching arbitrary points of difference at suitable levels of description, e.g., the quantum description level;
 The incomputability of IRNs corresponds to the nonzippability of complexity into a single analytical formula or exhaustive single model [69].
4. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Computable (availability of an algorithm that computes in finite time and with arbitrary precision) Real Numbers  Incomputable (unavailability of an algorithm that computes in finite time and with arbitrary precision) Real Numbers (IRNs) 
Generic cases include rational nonperiodic, limited, algebraic decimal numbers, whereas specific cases include perfect roots.  Include algebraic, irrational, trigonometric (based on Euler’s formula), transcendent numbers, some points of convergence, and some special numbers, such as the Chaitin omega number. 
Effective Computability  Incomputability  Emergent Computation 

We differentiate between symbolic computational processes and their effective computability. In the last case, the stepbystep computation process is formally available and is Turingcomputable, i.e., it ends always and in a finite amount of time.  We differentiate between symbolic computational processes and their noneffective computability. In the last case, although the stepbystep computation process is formally available, it is not Turingcomputable; for instance, it does not end in a limited amount of time. In such a case, we do not use results. Rather, it is possible to use approximations or computational symbolic representations for possible groupings and simplifications, such as numbers with exponents and numerical fractions, simplifications that are impossible when using approximate values that are indeed only calculated at the end.  The computation cannot be analytically completely represented, and acquired properties cannot be anticipated by considering the formal computation process.The computational processing is nonexplicit, not analytically represented, and called subsymbolic, even if the program being performed is an explicit, computable algorithm, such as ANN. Systemically, properties are acquired during ongoing networked computations. 
The role of equivalence in maintaining coherence in collective systems 
Within collective systems of interacting elements, equivalence is intended to be a reason for their unpredictable microscopic behavior, incomplete representations, and regardless of the source stability, the robustness of the collective behavior, which can be represented in different though equivalent ways. 
Equivalence relates to the interchangeability of roles assumed by interacting elements. There are countless (nonequivalent) ways of maintaining the same equivalent level of coherence even after going through temporary local incoherencies that are recovered in a number of ways. 
For instance, there are several variable correlations among temporary, local, and different nonequivalently correlated communities, e.g., groups of people that make up the nightlife, rather than only establishing the same longrange correlation, e.g., flocks. 
A model represents a system that is logically closed when 
(a) There is an almost complete formal, explicit (symbolic, not subsymbolic) description of the relations between the state variables of the model available. 
(b) There is an almost complete and explicit, i.e., analytically describable, description of the interaction between the system and its environment available. 
(c) The knowledge of values related to the previous two points allows the deduction of all (assumed to be finite and limited numbers) possible states and properties that the system can take together with its structural characteristics. 
A system is classified as logically open when there is a violation of at least one of the three points above. Logical openness is a specification of the theoretical incompleteness. 
Turing computability 
A function is said to be Turingcomputable if all the function’s values can be computed with a Turing machine. 
We may specify such generic definition. 
Formally, a Turing machine is specified as a quadruple T = (Q, Σ, s, δ) where 

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Minati, G. A Note on the Reality of Incomputable Real Numbers and Its Systemic Significance. Systems 2021, 9, 44. https://doi.org/10.3390/systems9020044
Minati G. A Note on the Reality of Incomputable Real Numbers and Its Systemic Significance. Systems. 2021; 9(2):44. https://doi.org/10.3390/systems9020044
Chicago/Turabian StyleMinati, Gianfranco. 2021. "A Note on the Reality of Incomputable Real Numbers and Its Systemic Significance" Systems 9, no. 2: 44. https://doi.org/10.3390/systems9020044