◄ Index 
◄
Main essay
The Einstein-Bohr debate: Can
Bhupinder
Singh Anand[1]
(A
.pdf file of this essay before the current update is available at http://arXiv.org/abs/math.GM/0307104)
If we assume the Thesis that any classical Turing
machine T, which halts, without looping, on every n-ary sequence of natural numbers as input, in a
Turing-computable time t(n), determines a
PA-provable formula, whose standard interpretation is an n-ary arithmetical relation f(x1, ..., xn) that holds if, and only if, T halts, then
we can define
Preamble: The
Einstein-Bohr debate
1. Introduction
1.1 Is our universe pre-destined?
1.2 Is our universe deterministic?
1.3 Is quantum mechanics ‘irreducibly probabilistic”?
1.4 Can we mathematically express a deterministic, yet quantum, universe that is not pre-destined?
1.5 Quantum concepts in a constructive framework of recursive functions and Peano’s Arithmetic
1.6 A possible ambiguity in classical Turing
computability
1.7 Three theorems
2. Defining a deterministic universe
2.1 Defining a quantum universe
3. The Classical Halting Thesis in a quantum universe
4. The Quantum Halting Thesis in a quantum
universe
5.2 Properties with random values in a
deterministic universe
5.3 Probability of an inter-action yielding a given
value
5.4 Superposition
Appendix
1: Defining every partial recursive function as total
Appendix
2: Collapse of the wave function
Preamble: The Einstein-Bohr debate
A study of the limitations, of standard interpretations of classical mathematical logic, suggests that, prima facie, the same foundational issues - logical and mathematical - may be reflected, albeit obliquely, in the dialogue between Einstein and the adherents of the Copenhagen Interpretation of quantum mechanics - spear-headed by Bohr. This essay attempts to address some of these issues.
Bohr’s belief echoes, in a sense, that of Gödel [Go51], and of set-theorists, such as Shelah [Sh91], who hold, Platonically, that the truth of the formal propositions, or even axioms, of a mathematical language, under a given interpretation, need not be intuitive - and may even be unverifiable effectively[2].
Bohr: detailed analysis of atomic phenomena ... is in principle excluded
Thus, Bohr remarks [Bo49] that:
I advocated a point of view conveniently termed ‘complementarity,’ suited to embrace the characteristic features of individuality of quantum phenomena, and at the same time to clarify the peculiar aspects of the observational problem in this field of experience. For this purpose, it is decisive to recognise that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The argument is simply that by the word ‘experiment’ we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.
... in quantum mechanics, we are not dealing with an arbitrary renunciation of a more detailed analysis of atomic phenomena, but with a recognition that such an analysis is in principle excluded. The peculiar individuality of the quantum effects presents us, as regards the comprehension of well-defined evidence, with a novel situation unforeseen in classical physics and irreconcilable with conventional ideas suited for our orientation and adjustment to ordinary experience. It is in this respect that quantum theory has called for a renewed revision of the foundation for the unambiguous use of elementary concepts, as a further step in the development which, since the advent of relativity theory, has been so characteristic of modern science.
Although Bohr appears to express the need for, and appreciation of, intuitively unobjectionable foundations for quantum mechanics, his concerns seem, however, to address[3]:
... only one half of human intellectual endeavour. This half would, first, be the attempt to individually express the state of the static synaptic elements, of the dynamically evolving, neuronic, activity that is taken to represent an individual brain, within a symbolic language. And, second, it would be the subsequent attempt, to individually interpret, and relate, the symbols of a language to an existing state of static synaptic elements, of the dynamically evolving, neuronic, activity that is taken to represent the individual brain.
The other half of human intellectual activity would, then, be that of determining which, of the concepts that are represented by such expressions, can be communicated uniformly in an unambiguous, and effective, manner that is independent of individual interpretations.
As Bohr notes, further [op. cit.], Einstein:
... argues that the quantum-mechanical description is to be considered merely as a means of accounting for the average behaviour of a large number of atomic systems and his attitude to the belief that it should offer an exhaustive description of the individual phenomena is expressed in the following words: “To believe this is logically possible without contradiction; but it is so very contrary to my scientific instinct that I cannot forego the search for a more complete conception.”
Einstein: complete description ... irrespective of any act of observation
In response, Einstein held [Ei49] that:
I am, in fact, firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this [theory] operates with an incomplete description of physical systems. ...
What does not satisfy me in that theory, from the standpoint of principle, is its attitude towards that which appears to me to be the programmatic aim of all physics: the complete description of any (individual) real situation (as it supposedly exists irrespective of any act of observation or substantiation). ...
Now we raise the question: Can this theoretical description be taken as the complete description of the disintegration of a single individual atom? The immediately plausible answer is: No. For one is, first of all, inclined to assume that the individual atom decays at a definite time; however, such a definite time-value is not implied in the description by the Psi-function. If, therefore, the individual atom has a definite disintegration time, then as regards the individual atom its description by means of the Psi-function must be interpreted as an incomplete description. In this case the Psi-function is to be taken as the description, not of a singular system, but of an ideal ensemble of systems. In this case one is driven to the conviction that a complete description of a single system should, after all, be possible, but for such complete description there is no room in the conceptual world of statistical quantum theory. ...
One may not merely ask: “Does a definite time instant for the transformation of a single atom exist?” but rather: “Is it, within the framework of our theoretical total construction, reasonable to posit the existence of a definite point of time for the transformation of a single atom?”. One may not even ask what this assertion means. One can only ask whether such a proposition, within the framework of the chosen conceptual system - with a view to its ability to grasp theoretically what is empirically given - is reasonable or not. ...
Roughly stated the conclusion is this: Within the framework of statistical quantum theory there is no such thing as a complete description of the individual system. More cautiously it might be put as follows: The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems. In that case the whole “egg-walking” performed in order to avoid the “physically real” becomes superfluous. There exists, however, a simple psychological reason for the fact that this most nearly obvious interpretation is being shunned. For if the statistical quantum theory does not pretend to describe the individual system (and its development in time) completely, it appears unavoidable to look elsewhere for a complete description of the individual system; in doing so it would be clear from the very beginning that the elements of such a description are not contained within the conceptual scheme of the statistical quantum theory. With this one would admit that, in principle, this scheme could not serve as the basis of theoretical physics. Assuming the success of efforts to accomplish a complete physical description, the statistical quantum theory would, within the framework of future physics, take an approximately analogous position to the statistical mechanics within the framework of classical mechanics. I am rather firmly convinced that the development of theoretical physics will be of this type; but the path will be lengthy and difficult.
Psi-functions may be formal representations of hidden, non-algorithmic, functions
So, a reasonable view would be that Einstein’s objections were not so much against a probabilistic interpretation of quantum mechanics - since it is unarguably effective as a scientific theory - but against the absence, first, of suitable intuitive interpretations of such probabilities; and, second, against the denial of a need for intuitively unobjectionable axiomatic foundations for the theory, without which its formal assertions cannot be treated as being capable of unambiguous, and effective, communication, under any interpretation, by an intelligent life-form.
Now, a thesis of these essays[4] is that the acceptance, of non-standard interpretations of Peano Arithmetic, and, implicitly, of counter-intuitive interpretations of quantum mechanics, are, both, aesthetically unappealing, consequences of Tarski’s failure to define mathematical truth effectively; this would, reasonably, prevent the postulation of unique values for the outcome of gedanken experiments that are based on standard interpretations of classical mathematics.
If we eliminate this lacuna, and define mathematical truth effectively - by, say, extending Tarski’s definitions, and Church’s thesis, suitably - we can, indeed, arrive at constructive interpretations, of Peano Arithmetic, which are intuitive, isomorphic, and verifiably complete. Hence, it is not unreasonable to conjecture that intuitive, isomorphic, interpretations of quantum mechanical concepts may also follow, in which the functions and relations that are represented by the Psi-function are effectively computable / decidable - but only non-algorithmically.
A feature of such, non-algorithmic, functions / relations is that, first, they cannot be introduced as primitive, or defined, symbols into any recursively definable axiomatic theory without inviting inconsistency; and, second, that, although they are Turing uncomputable / undecidable, there is always some effective method for determining their value for any given set of values of their free variables - which would correspond to a measurement, or collapse of the Psi-function, for that particular set of values.
A consequence of the first is that such, non-algorithmic, functions / relations can only be represented in a recursively definable axiomatic theory through their values, and that, given any finite set of such values, based on a sequence of measurements, there are denumerable arithmetic functions / relations that could generate the measured set in the theory. The question, thus, as to which particular non-algorithmic function / relation gave rise to a particular set of values, and so prediction of the value at a subsequent measurement, cannot, therefore, be determined uniquely within the theory, although probabilities associated with a particular determination may be possible in the interpretation.
Such functions and relations could, thus, effectively be treated as the “hidden functions” of quantum mechanics. Psi-functions, like the Gödel-Beta functions that can represent a recursive function within a Peano Arithmetic (cf. [Me64], p131, Propositions 3.21-3.23), may simply, then, be formal manifestations of such “hidden functions”.
Are our
current theories of physics consistent with the concept of a universe that is completely
deterministic, yet not pre-destined?
In
other words, can the initial conditions and all physical laws at any instant,
say, for instance, at the time of a projected Big Bang, be knowable completely
in a manner that is consistent with our current theories of physics?
1.1 Is our universe pre-destined?
As Ian
Stewart remarks ([St97], p6), the pre-quantum,
classical concept was that our universe is both deterministic and pre-destined,
a view expressed “eloquently” by Pierre Simon de Laplace, one of the leading
mathematicians of the 18th century, in his “Philosophical Essays
on Probabilities”:
An intellect[5] which at any given moment knew all the forces that
animate Nature and the mutual positions of the beings that comprise it, if this
intellect were vast enough to submit its data to analysis, could condense into
a single formula[6] the movement of the greatest bodies of the universe
and that of the lightest atom: for such an intellect nothing could be
uncertain; and the future just like the past would be present before its eyes.
Such a
notion of a pre-destined universe evolved as “the revolution in scientific
thought that culminated in Newton led to a vision of the universe as some
gigantic mechanism ... In such a vision, a machine is above all predictable” ([St97],
p6).
1.2 Is our universe deterministic?
As
Stewart observes further ([St97], p329), the post-quantum belief that our universe may be
deterministic in a yet unknown, but fundamental, way (which may not necessarily
be pre-destined) is reflected in Einstein’s well-known remark, in the following
excerpt from a letter to Max Born:
You believe in the God who plays dice, and I in
complete law and order in a world which objectively exists, and which I, in a
wildly speculative way, am trying to capture. I firmly believe, but I
hope that someone will discover a more realistic way, or rather a more tangible
basis than it has been my lot to do. Even the great initial success of the
quantum theory does not make me believe in the fundamental dice game ... .
1.3 Is quantum mechanics ‘irreducibly probabilistic’?
Stewart
notes, further ([St97], p330), the prevalent
view that, despite Einstein’s predilections, the universe, or at least our
present quantum mechanical description of it, is of an “irreducibly
probabilistic character”. He then suggests that we may need to seriously
consider the “... possibility of changing the theoretical framework of physics
altogether, replacing quantum uncertainty by deterministic chaos, as Einstein
would have liked”.
1.4 Can we mathematically express a deterministic, yet quantum, universe that is not pre-destined?
Clearly, the extent to which we can address this issue meaningfully will depend on how precisely we can express the above concept, of a deterministic, yet quantum, universe that is not pre-destined, within a constructive[7], and intuitionistically unobjectionable language.
In this essay, we broadly address the above issue of whether such
expression can be made mathematically for a Big Bang type of universe U that,
we assume, creates at most denumerable particles at the instant of creation,
all super-posed upon each other at a single “point”, where the states, and
inter-actions, of the particles are dynamic and follow deterministic laws, and
where the future can, reasonably, be interpreted as irreducibly
probabilistic.
Specifically,
we consider the following:
(i) Can we
constructively define intuitive concepts such as “particle”, “deterministic”,
“quantum”, “knowable”, “pre-destined”, and, most significantly, “
(ii) Is there a
mathematical description of the state of a given particle P at a given
inter-action[8] t, such
that, given any property x of P,
there is always an effective method by which we can determine the value y of the property x uniquely?
(iii) Can such a description
allow for the possibility that, given any arbitrary set of properties of a
given particle P, there is no effective method such that we can always
determine the precise values of the given properties simultaneously at any
inter-action t?
(iv) Can such a
description allow for the possibility that the sequence of values of some
property x of a given
particle P, measured at discrete inter-actions t1, t2, ...,
may be Li-unpredictable?
Preliminary Definition 1: A class of natural numbers is Li-predictable[9] if, and only if, it is a well-defined mathematical object[10] (such as, for instance, any well-defined set determined by a consistent set theory[11]).
We assume that, in this case, Li is aware of any member of the class directly, without depending on any effective method for computing it.
Preliminary Definition 2: A sequence of natural numbers is Li-unpredictable[12] if, and only if, it does not define a mathematical object.
We assume that, in
this case, Li is not aware of the value of
the i’th
term directly for any given i, but must depend on an effective method for computing it[13].
(v) Can
we assume that fundamental particles have a distinct identity throughout their
“lifetime” in such a description?
(vi) Can two
distinctly different particles “occupy” the same “space” simultaneously in such
a description?
1.5 Quantum concepts in a constructive framework of recursive functions and Peano’s Arithmetic
The
underlying thesis of this essay is that a constructive, non-classical,
mathematical language for quantum mechanical descriptions, which addresses
these questions appropriately, can, indeed, be built around asymmetric ([An02c],
§2.1] recursive number-theoretic
relations. These relations, characteristically, are not the standard
interpretations of any of their formal representations in any standard first
order Arithmetic that is based on Dedekind’s formulation of the Peano Axioms
for the natural numbers [An02c].
1.6 A possible ambiguity in classical Turing computability
However,
in order to address these questions appropriately, we argue that there is a
need to make the concept of classical Turing-computability[14] less ambiguous. This ambiguity is highlighted when we
consider the two meta-theses:
(a) Classical Halting Thesis: Every PA-formula[15], that is a true[16] n-ary arithmetical[17] relation f(x1, ..., xn) under the standard interpretation[18], determines a classical Turing machine T that halts on every n-ary sequence k1, ..., kn of natural numbers as input if, and only if, f(k1, ..., kn) holds.
(b) Quantum Halting Thesis: Any classical Turing machine T that halts on every n-ary sequence of natural numbers as input in a Turing-computable time t(n) is equivalent to a PA-provable formula, whose standard interpretation is an n-ary arithmetical relation f(x1, ..., xn) such that, for any sequence k1, ..., kn of natural numbers, f(k1, ..., kn) holds if, and only if, T halts.
Clearly, the two meta-theses are mutually inconsistent, since (a) would determine a classical Turing machine T that halts on every n-ary sequence k1, ..., kn of natural numbers as input even for a PA-unprovable formula that is a true n-ary arithmetical relation f(x1, ..., xn) under the standard interpretation[19].
Further, since the conclusion in (a), and the premise in (b), are not effectively verifiable, the two meta-theses are essentially undecidable. Prima facie, we thus have two, apparently equally sound, but essentially different, systems of Arithmetic based on constructive, and intuitionistically unobjectionable[20], interpretations of standard PA; reasonably, they should have significantly different consequences.
1.7 Three theorems
We argue that this is, indeed, so, in the following:
Theorem 1: The principle of Quantum Uncertainty is inconsistent
with the Classical Halting Thesis.
Theorem 2: The principle of Quantum Uncertainty is logically
consistent with the Quantum Halting Thesis.
Theorem 3: The Quantum Halting Thesis implies that every
partial recursive function can be effectively defined as a total[21] function.
2. Defining a
deterministic universe
We start by introducing the critical concept of a Gödelian proposition, first constructed by Gödel in his seminal 1931 paper, as the basis for defining a mathematical function that, loosely speaking, corresponds to the classical quantum state of a particle.
Definition 1: The standard interpretation of a unary PA-formula is a quantum signature if, and only if, the formula is PA-unprovable but effectively true under the standard interpretation.
The
significance of a quantum signature [F(x)][22] is that, given any natural number k, we can always construct an individually[23] effective[24] method for determining that F(k) holds, but there may be no uniformly effective
method, even for Laplace’s intellect Li, such that, given any k, we can always determine that F(k) holds.
Treating
the terms “particle”, “property” and “value” as undefined primitives, we
intuitively assert the:
Expressibility
Thesis: Any physical property of any given particle that can
be measured by some effective method can be expressed by a finite string in a
suitably defined, recursively enumerable, language[25] L, which has a finite alphabet. Similarly, any value
of such a physical property that can be determined by some effective method can
also be expressed by a finite string in L.
It
immediately follows that:
Lemma 1:
Any property can be uniquely Gödel-numbered[26] by a natural number, and any value of a property can,
similarly, be uniquely Gödel-numbered by a natural number.
We note
that not every natural number need correspond to a property, and that not every
natural number need correspond to the value of a property. We also note that,
even if the number of all possible particles, properties and values in a
deterministic universe U are not denumerable, since we can only express
denumerable strings within L, in this essay we only consider the recursively
enumerable set of properties for which we have an effective method of
measurement.
We next
define:
Definition
2: The signature of a given particle Pi at any inter-action t is a number-theoretic relation p(i, t, x, y) that
holds if, and only if, for any given property represented by x, there is a unique natural number y, where the domain of x is assumed to be a recursively enumerable set[27] of the Gödel-numbers of all the properties that are
expressible in L[28], and the domain of y is assumed to be a recursively enumerable set of the
Gödel-numbers of all values that are expressible in L.
We can
now define:
Definition
3: The history Hi, t of any
given particle Pi[29] at any specified inter-action t[30] is the set of particle signatures {p(i, s, x, y): 0 =< s =< t-1}.
Definition
4: For any given particle Pi, we define a given property[31] k as
deterministic if, at any given inter-action t, there is an effective method[32] to determine a unique value mk, t such that p(i, t, k, mk, t) holds.
Definition
5: A given universe U is deterministic if, for any
given particle Pi, any given property k, and any given inter-action t, there is an effective method to determine a unique
value mk, t such that p(i, t, k, mk, t) holds.
We now
note that:
Lemma 2:
If the properties expressible in L are recursively enumerable, and we assume
Turing’s Thesis[33], then we can effectively determine whether p(i, t, k, m) holds or not
in a given deterministic universe U for any given natural numbers i, t, k, m.
Proof: By
definition, and the Turing Thesis, there is a classical Turing machine T such
that, given any natural numbers i, t, k, m as input,
T halts if, and only if, p(i, t, k, m) holds. If
k is not the Gödel-number of a property, then p(i, t, k, m) is
undefined, and we assume that, in such a case, T will loop on the input k.
We note that the definition of any classical Turing
machine T can be extended to allow for recording, in its unlimited memory[34], of every instantaneous tape description[35] during the operation of a classical Turing machine.
Now, T can always be meta-programmed to compare its
current instantaneous tape description with the finite set of previous
instantaneous tape descriptions during its current operation, and to
self-terminate if an instantaneous tape description repeats itself, i.e. at the
onset of a loop[36]. We can, thus, use this condition to effectively
define p(i, t, k, m) as holding vacuously if k is not the Gödel-number of a property.
Hence we can effectively determine whether p(i, t, k, m) holds or
not for any given natural numbers i, t, k, m.¶[37]
We thus
have:
Lemma 3:
If we assume the Church Thesis, then the number-theoretic relation p(i, t, x, y), describing
the signature of the particle Pi at inter-action t, is recursive[38] in a given deterministic
We note that, in view of Lemma 3, the particle signature p(i, t, x, y) is
essentially Laplace’s formula “Elf” for a given deterministic universe
U, since it essentially condenses “into a single formula the movement of the
greatest bodies of the universe and that of the lightest atom”, and “all the
forces that animate Nature and the mutual positions of the beings that comprise
it”.
We also
note that the particle signature p(i, t, x, y), at any
inter-action t in a given deterministic universe U, can be
viewed as a point in an infinite dimensional
Hilbert space, whose co-ordinates are (mt, 1, mt, 2, ...),
where mt,
k is the
value along the k axis that
satisfies p(i, t, k, mt, k). We can thus define:
Definition 6:
A Gödelian point gi(t) of the
particle Pi is a
point in an infinite dimensional Hilbert space, whose co-ordinates are (mt, 1, mt, 2, ...),
where mt,
k is the
value along the k axis that satisfies
p(i, t, k, mt, k).
Definition
7: The Gödelian world-line[39] G(i) of the
particle Pi is
the non-terminating sequence of particle signatures p(i, 0, x, y), p(i, 1, x, y) ... in a given
deterministic
Definition
8: A Gödelian world-line is a fundamental particle Pi in a given deterministic universe U if, and only if,
the non-terminating sequence of the Gödelian points, gi(t), t >=
0, is a well-defined mathematical entity.
2.1 Defining a quantum universe
We next
define:
Definition
9: The PA-formula [Qt(x, y)], which expresses[40] the recursive relation pi(t, x, y) in PA, is
a quantum signature if, and only if, the
formula [(E!y)Qt(x, y)][41] is PA-unprovable, but true under the standard
interpretation of PA.
We note
that, by definition, if the PA-formula [Qt(x, y)] expresses the recursive relation pt(x, y), then, for any given any natural numbers k, m:
(i) (E!y)Qt(k, y) holds, and
(ii) Qt(k, m) holds if, and only if, pt(k, m) holds.
We can
now define:
Definition
10: A Gödelian world-line is quantum if the
PA-representation of its signature at every inter-action is a quantum
signature.
Definition
11: A deterministic universe is pre-destined if it does
not contain any quantum Gödelian world-line.
Definition
12: A deterministic universe is partially pre-destined
if, and only if, it is not pre-destined.
Definition
13: A partially pre-destined universe is quantum if, and
only if, it contains only quantum Gödelian world-lines.
Lemma 4:
A quantum universe Q contains no fundamental particles.
3. The Classical Halting Thesis
in a quantum universe
We now
have that:
Lemma 5:
If we assume the Classical Halting Thesis (a) in a deterministic
universe, then, for any given particle P, there is a classical Turing
machine T such that, for any input k, T will halt and return the value m if, and only if, pt(k, m) holds.
Proof: T
will halt on input k
if, and only if, (E!y)Qt(k, y) holds. By our hypothesis, for any given k, (E!y)Qt(k, y) holds
uniquely for some value m. Hence,
for given input k, T must
run a sub-routine that will halt on, and can be programmed to return, m.¶
This
would clearly imply that:
Lemma 6:
If we assume the Classical Halting Thesis (a), then there is some
effective method that can completely determine the values of any finite number
of, arbitrarily selected, properties simultaneously for any given particle in a
deterministic universe.
We
define:
Definition
14: The principle of Quantum Uncertainty holds in a
universe if, and only if, there are properties of some particles whose values
cannot be completely determined simultaneously.
It
follows that:
Theorem 1:
The principle of Quantum Uncertainty is inconsistent with the Classical Halting
Thesis.
4. The Quantum Halting Thesis in a quantum universe
However,
in a quantum universe, the formula [(E!y)Qt(x, y)] is not
necessarily PA-provable for any given particle. It follows that:
Lemma 7:
If we assume the Quantum Halting Thesis (b) in a deterministic universe,
then, for any given particle P, there is no classical Turing machine T
such that, for any given input k, T will halt and return the value m if, and only if, pt(k, m) holds.
This,
now, clearly implies that:
Lemma 8:
There is no effective method that can always determine the values of a finite
number of, arbitrarily selected, properties of any given particle P
simultaneously, even though, by our premise, given any property k, there is always some effective method that will
completely determine its value mk at inter-action t.
We thus
conclude that:
Theorem 2:
The principle of Quantum Uncertainty is consistent with the Quantum Halting
Thesis[42].
5. Some consequences
We note that t in Definition
2 is a natural number that is assumed to be the t’th inter-action of the particle P
with some other particle in the universe, necessarily with reference to some
initial signature p0(k, mk,
0)[43]. In other words, we also assume that, for any given
particle P and any given property k, the signature p0(k, mk,
0), p1(k, mk,
1), ..., pt-1(k, mk, t-1), is
assumed known to
5.1 The Gödel β-function
We note
further that, for any given property k and inter-action t = tn, we
can define a recursive Gödel β-function[44] qP, k(b, c, i), such
that qP, k(b, c, i) = mk, i for all 0 =< i =< n[45], where mk, i is the value of the property k at the inter-action ti, and b, c are natural
numbers that can be constructively determined by the sequence mk, 0, mk , 1, ...,
mk, n.
5.2 Properties with random values in a deterministic universe
If we
assume that the inter-actions t0, t1, ..., tn refer to the series of inter-actions that occur
during a particles lifetime[46], then we may reasonably define:
Definition
15: A property k is predictable if the sequence of values represented by the
sequence mk,
0, mk , 1, ...,
mk, n, as the number n of inter-actions increases indefinitely, is a Cauchy sequence;
otherwise k is random.
Prima
facie, it thus follows that:
Lemma 9:
Both predictable and random properties are consistent with a deterministic
universe.
5.3 Probability of an inter-action yielding a given value
We
note, further, that, for any given sequence mk, 0, mk , 1, ...,
mk, n, there are denumerable pairs of natural numbers (b, c) that
define Gödel β-functions relative to the sequence; however,
although the functions will all yield identical values of mk, i for 0 =< i =< n, each of them will yield different sequences mk, n+1, mk, n+2, ....
The challenge
of any theory, thus, would be to determine, firstly, which of these functions
are such that all the terms of the non-terminating sequence are Gödel-numbers
of values of the property k; secondly,
the probability that the result mk, n+1 of the
inter-action tn+1, for any pair (b, c), will represent
any given value of the property k.
We thus
have that:
Lemma 10:
A sufficient condition for a deterministic universe to be pre-destined is that
every property of every particle P have a characteristic Gödel β-function
that determines the value of the property at any inter-action tn for all n >= 0.
Lemma 11:
A necessary condition for a deterministic universe to be pre-destined is that
the initial signature p0(k0, m0), p0(k1, m1), ..., p0(kt-1, mt-1), ...
of every particle P is assumed known.
If we
now define:
Definition
16: A universe is predictable if there is always an
effective method such that, for any given particle P, any given
deterministic property k, and any
given inter-action t, we can
determine a unique value mk
such that pt(k, mk)
holds.
We then
have that:
Lemma 12:
A sufficient[47] condition for a deterministic universe to be
unpredictable is that the initial signature
p0(k0, m0), p0(k1, m1), ..., p0(kt-1, mt-1), ... of
every particle P is assumed unknowable.
5.4 Superposition
We also
note that, if qP, k(b, c, i) = mk, i for all 0 =< i =< n, where mk, i
is the value of the property k at the inter-action ti, and q'P, k(b', c', i) = m'k, i for all 0 =< i =< n', where m'k, i is the value of the property k at the inter-action t'i, we can always combine the two sequences of
inter-actions mk,
0, mk, 1, ..., mk, n, and m'k, 0, m'k, 1, ..., m'k, n', appropriately[48] to yield a sequence m"k, 0, m"k, 1, ..., m"k, n'', where n'' = n + n'.
We thus
have a Gödel β-function q''P, k(b'', c'', i) = m''k, i for all 0 =< i =< n'', where m''i is the value of the property k at the inter-action t''i, which may be considered as the superposition of the
Gödel β-function q'P, k(b', c', i) on qP,
k(b, c, i).
It
follows that:
Lemma 13:
The probability[49] that q''P, k(b'', c'', n''+1) = m for a given m is always equal to, or greater than, the probability that qP,
k(b, c, n+1) = m.
Proof: The
sequence m0, m1, ..., mn is a proper sub-sequence of m''0, m''1, ..., m''n''. Hence, if q''P, k(b'', c'', n''+1) = m, then there is always some pair (b, c) such that
qP, k(b, c, n''+1) = m. The converse is not true.
Hence the set of Gödel β-functions for
which q''P, k(b'', c'', i) = m''i for all 0 =< i =< n'', where m''i is the value of the property k at the inter-action t''i, is a proper sub-set of the set of Gödel β-functions
for which qP, k(b, c, i) = mi for all 0 =< i =< n, where mi
is the value of the property k at the inter-action ti.¶
Appendix 1: Defining every partial recursive function as total
We note that, prima facie, the above consequences depend critically on the following argument, which effectively defines every partial recursive function as total.
Theorem 3: The Quantum Halting Thesis implies that every partial recursive[50] number-theoretic function f(x1, ..., xn) is constructively definable as a total function.
Proof: We assume that f is obtained from the recursive function g by means of the unrestricted μ-operator, so that f(x1, ..., xn) = μy(g(x1, ..., xn, y) = 0).
Given any sequence k1, ..., kn of natural numbers, we can then define:
(i) a classical Turing machine T1 that will halt if, and only if, (g(k1, ..., kn, k) = 0) holds for some natural number k;
(ii) a classical Turing machine T2 that will halt if, and only if, ~(g(k1, ..., kn, k) = 0) holds for some natural number k;
(iii) a classical Turing machine T3[51] that will halt if, and only if, [H(k1, ..., kn, y)] is PA-provable, where [H(x1, ..., xn, y)] is the PA-formula that represents (g(x1, ..., xn, y) = 0) in PA.
If we now assume the Quantum Halting Thesis, it follows that:
(a) if T1 halts, without looping, on all natural numbers, then [H(k1, ..., kn, y)] is PA-provable;
(b) similarly, if T2 halts, without looping, on all natural numbers, then [H(k1, ..., kn, y)] is PA-provable.
Hence, if (g(k1, ..., kn, y) = 0) holds for all natural numbers y, then:
(c) either [H(k1, ..., kn, y)] is PA-provable, and T3 will halt;
(d) or, if [H(k1, ..., kn, y)] is not PA-provable, then T2 will loop, and self-terminate, for some natural number k.
If we now run T1, T2 and T3 simultaneously, then we are assured that at least one of them will halt, or self-terminate, for some natural number k. We can thus use the condition indicated by the halting, or self-terminating, state to effectively define the partial recursive function f(x1, ..., xn) appropriately as a total partial recursive function.¶
Appendix 2: Collapse of the wave function
The above arguments also admit a mathematical description of the concept of the collapse of a wave function at a measurement as follows:
(i) Gödel has proved that there is an
arithmetic formula [G(x)] such that, for any given k, [G(x)] is provable.
(ii) Hence, for any given k, there is always some effective method for evaluating the arithmetic expression G(k).
(iii) Gödel has also proved that [(Ax)G(x)] is not provable.
(iv) Thesis: There is no uniform effective method (algorithm/Turing machine) that can evaluate the arithmetic expression G(x) for any given x.
(v) Thus, G(n) is individually computable, but not uniformly computable.
(vi) Theorem (provable by induction): For any given k, we can always find some effective method (algorithm/Turing machine) T(k) that can compute G(n) for all n<k, i.e. T(k) terminates for all n<k, but it “loops” on input k. (Note: All methods that evaluate G(n) for all n<k cannot be non-terminating on input k; this would imply that G(k) is undefined, which contradicts (ii).)
(vii) Quantum interpretation: The process of finding T(k+1) can be corresponded, firstly, to the act of finding a suitable method of measuring the value G(k) precisely, and, secondly, to the collapse of the wave function at k as a result of the measurement; we then have the new “state” T(k'), which can evaluate the value of G(n) for all n<k', where k<k', but not beyond!
(viii) If, now, we have some law that determines the state T(k') from the state T(k) and the interaction at k, we have a deterministic interaction that is, nevertheless, absolutely unpredictable, where we may then define free will as absolute unpredictability. (We note that, if k' > k+1, we have a language that admits inter-actions that can leave the state T(k') unchanged.)
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