Friday, October 12, 2012

"The Incompleteability or Inexhaustibility of Mathematics" -- Goedel

Full Title:  ‘The Gödelian Dialectic’, or, “The Incompletability of Mathematics”.

Dear Readers,

The following is an excerpt on the unconscious Marxian-dialectical essence of what the mathematical Platonist, Kurt Goedel, called "The Incompletability of Mathematics", or "The Inexhaustibility of Mathematics".

It is excerpted from F.E.D. Vignette #4 --,Part_I_of_II,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_10OCT2012_1.pdf

"Kurt Gödel, arguably the contributor of the greatest leaps forward in the science of formal logic since classical antiquity, described an ‘axiomatic dialectic’ of mathematics, albeit in “[‘early’-]Platonic”, ‘‘‘a-psychological’’’ and ‘‘‘a-historical’’’ terms, hence in a-psychohistorical terms as well, as follows --

It can be shown that any formal system whatsoever — whether it is based on the theory of types or not, if only it is free from contradiction — must necessarily be deficient in its methods of proof.

Or to be more exact: For any formal system you can construct a proposition — in fact a proposition of the arithmetic of integers — which is certainly true if the system is free from contradiction but cannot be proved in the given system [F.E.D.: the foregoing summarizes Gödel's "First Incompleteness Theorem"].

Now if the system under consideration (call it S) is based on the theory of types, it turns out that exactly the next higher type not contained in S is necessary to prove this arithmetic proposition, i.e., this proposition becomes a provable theorem if you add to the system the next higher type and the axioms concerning it.

[Kurt Gödel, “The Present Situation of the Foundations of Mathematics (*1933o)”, in S. Feferman, et. al., editors, Kurt Gödel: Collected Works (Volume III: Unpublished Essays and Lectures), ibid., page 46; bold, italics, underline, and color emphasis added by F.E.D.].

Again --

If we imagine that the system Z [F.E.D.: a formal, logical, propositional-/predicate-calculus system inclusive of Natural Numbers’ Arithmetic, not the full system of the positive and negative Integers, and zero ... standardly also denoted by Z] is successively enlarged by the introduction of variables for classes of numbers, classes of classes of numbers, and so forth, together with the corresponding comprehension axioms, we obtain a sequence...of formal systems that satisfy the assumptions mentioned above, and it turns out that the consistency ... of any of these systems is provable in all subsequent systems.

Also, the undecidable propositions constructed for the proof of Theorem 1 [F.E.D.: Gödel’s “First Incompleteness Theorem”] become decidable by the adjunction of higher types and the corresponding axioms; however, in the higher systems we can construct other undecidable propositions by the same procedure. ...

To be sure, all the propositions thus constructed are expressible in Z (hence are number-theoretic propositions); they are, however, not decidable in Z, but only in higher systems...

[Kurt Gödel; On Completeness and Consistency (1931a), J. van Heijenoort, editor, Frege and Gödel: Two Fundamental Texts in Mathematical Logic, Harvard University Press [Cambridge: 1970], page 108, bold, italic, underline, and color emphasis [and square-brackets-enclosed commentary] added by F.E.D.].

The cumulative «aufheben»-progression of ‘‘‘conservative extensions’’’, i.e., the advancing ‘ideo-cumulum’ of axiomatic systems which Gödel describes above was viewed ahistorically by him.

Gödel was an avowed ‘‘‘neo-Parmenidean’’’, and a professed “mathematical Platonist” [in the sense of the earlier rather than of the later Plato].

He didn't intend this meta-system — this cumulative micro-diachronic progression of [axioms-] systems — to serve as a temporal or psychohistorical model of the stages of human mathematical cognition, as reflective of the stages of the self-development of humanity's collective cognitive powers as a whole; of the knowledges to which each such epoch of those powers renders access, and of the “historically-specific” ideologies [or pseudo-knowledges] to which human thinking is susceptible within each such epoch.

We of F.E.D., however, do wish to explore its efficacy as such.

Note how, as Gödel narrates this axioms-systems-progression above, each successor system «aufheben»-contains its immediate predecessor system, and, indeed, all of its predecessor systems; how each higher logical type «aufheben»-contains all predecessor logical types.

Can Gödel's theory of this cumulative, ‘evolute’, «aufheben» progression of axioms-systems, which we term The Gödelian Dialectic, or The Gödelian [Idea-Systems' Ideo-]Metadynamic, provide at least an idealized [i.e., a distorted] image of actual history, of the actual psychohistorical struggle, process, and progress of mathematical aspects of the self-development of a humanity's universal labor [Marx]; of its collective cognitive capabilities, hence of its knowledges and ideologies?

We say “yes!”.

[To clarify our terminology here: we mean, by an ‘ideo-dynamic, a pattern that characterizes the ‘intra-systemic’ process of, e.g., deductively proving ever-more theorems within a single axioms-system, in the context of a diachronically presented ‘meta-system[atic]’ progression of axioms-systems.

We mean, on the contrary, by the neologistical term ideo-meta-dynamic, the pattern characterizing the ‘inter-systemic’, non-deductive-logic-al process of moving from inside each given predecessor axioms-system, to outside it, to inside its successor axioms-system].

Each of Gödel's "undecidable" propositions of arithmetic that plague each ‘‘‘epoch’’’ of this formal axiomatic expansion is a proposition asserting, of itself -- ‘‘‘self-reflexively’’’ -- that it is a formula which cannot be deductively demonstrated from the axioms of its axioms-system of arithmetic.

That is, each ‘‘‘Gödel formula’’’ states that I am not a theorem of my axioms-system -- not a provable statement of the axioms-system of arithmetic inside which it immanently arises, as a well-formed formula within that system.

If such a proposition is true -- if the proposition actually cannot be deductively demonstrated from the axioms of the axioms-system in which it arises -- then things are very bad from an immanent, formal-logical point-of-view; then the system in which that proposition arises immanently is logically incomplete -- i.e., is incapable of deductively demonstrating all of the true propositions which arise as well-formed formulas within the rules of that system, because its Gödel proposition exhibits that there is at least this one true proposition within it that it cannot deductively prove.

If such a proposition is false -- if the proposition actually can be deductively demonstrated from the axioms of the axioms-system in which it arises immanently -- then matters are even worse from an immanent formal-logical point-of-view; then the axioms-system of arithmetic in which this proposition arises immanently is even contradictory; ‘‘‘self-inconsistent’’’, i.e., is capable of deductively demonstrating at least one false proposition, as well as the [true] negation of that proposition, and thus of deductively “demonstrating” two [or more] mutually formally contradictory propositions, because its Gödel proposition exhibits that there is at least this one false proposition within it that it can, erroneously, deductively “prove”.

That is, the axioms-system can derive, as a theorem, a proposition that says of itself “I am not a theorem of this axioms-system.”, and which, thus proven as a theorem, is thereby shown to be a false proposition.

That is the “meta-mathematical” content of a ‘‘‘Gödel Formula’’’, its meaning at the level of the “meta-language” which talks about axioms-systems, provability from axioms, etc.

But the real secret of the meaning of such ‘‘‘Gödel Formulae’’’, seldom mentioned in the standard accounts of Gödel’s “First Incompleteness Theorem”, is their “deformalized” content, their “mathematical”, i.e. arithmetical and algebraic content; their meaning in the “object language”, which talks about numbers, algebraic [“diophantine”] equations, and their solvability, etc.

At that deeper, more concrete level, a ‘‘‘Gödel Formula’’’ is a proposition asserting the unsolvability of a specific diophantine” [F.E.D.: referring to the circa 250 C.E. proto-ideographic-algebraic work by Diophantus of Alexandria, the «Arithmetica»] algebraic equation.

I.e., each “Gödel formula”, or “Gödel sentence”, which, at the “meta-mathematical” level, asserts the either-self-incompleteness-or-self-inconsistency of its axioms-system, “deformalizes” to one which asserts the unsolvability -- within that system of arithmetic, and given the limitations of the kinds of numbers that are at that axioms-system’s disposal -- of a specific, algebraic, “diophantine equation” --

... The Gödel sentence φ... asserts its own undeducibility from the postulates....

Deformalizing φ... we see that under the standard interpretation it expresses a fact of the form [F.E.D.: for every n-ary list of number-components of x such that each number-component is a member of the set of ‘diophantine numbers’, e.g., of integers] ...ƒx ~= gx... , where ƒ and g are n-ary polynomials [F.E.D.: I.e., are n-ary polynomial functions, whereas ƒx and gx denote their function-values]....

An equation ƒx = gx, where ƒ and g are two such polynomials, is called diophantine ....

By a solution of the equation we mean an n-tuple α of natural numbers such that ƒα =... .

So φ... asserts the unsolvability of the...equation ƒx = gx, and the proof of [F.E.D.: Gödel’s “First Incompleteness Theorem”] produces... a particular diophantine equation that is really unsolvable, but whose unsolvability cannot be deduced from the postulates...

[Moshé Machover, Set Theory, Logic, and their Limitations, Cambridge University Press [Cambridge: 1996], pages 268-269,emphasis and square-brackets-enclosed commentary added by F.E.D.].

Each of the [‘“locally”’] unsolvable algebraic equations that we will evoke, in the core section of this essay, in our narration of the Dyadic Seldon Function ‘meta-model’ of the dialectic of the Standard Arithmetics, will be seen to formulate, and to ‘explicitize’, a paradox for that concept of [counting] number which is implicit in the “[syn]thesis” arithmetical axioms-system term
X#, of the ‘aporial antithesis-sum’, X# + x#.

Per the modern definition, a "diophantine equation" is an equation whose parameters [e.g., coefficients] and whose solutions are restricted to the “integers”, or, sometimes, to “rational” numbers [in the case of Diophantus himself, positive rational numbers only, excluding zero, etc.].

Each “Gödel sentence”-encoded equation truly is unsolvable within the given axioms-system.

However, the proposition that it is so, cannot be deductively proven within that axioms-system.

But that proposition can be so proven within the next axioms-system, its immediate successor — the latter being created through the «aufheben» self-internalization of the ‘‘‘vanguard’’’, ‘meristemal’, highest [in “logical type”] set idea-objects of the universe of discourse of the predecessor axioms-system, a ‘self-internalization’ which produces sets of the next higher “logical type”.

That proposition can also be so proven within all subsequent successor-systems, created by yet-further such «aufheben» self-internalizations. “Logical type” works like this.

If we say that the “universal set”, or “universal class”, containing all of the “logical individuals”, e.g., all of the individual numbers, that constitute a given “universe of discourse [of arithmetic]”, e.g., {0, 1}, for the “universe of discourse” of the arithmetic of Boolean algebra, is of “logical type” one, then the set of all of its subsets,

{ {0}, {1}, {0, 1}, { _ } },

with ‘{ _ }’ denoting the “empty set”, is of “logical type” two, and the set of subsets of that set of subsets --

{ {{0}}, {{1}}, {{0, 1}}, {{ _ }}, { {0}, {1}, {0, 1}, { _ } }, {{0}, {1}}, {{0}, {0, 1}}, {{0}, { _ }}, ... }

-- is of “logical type” three.

That is, we are, in effect, counting the ‘“depth”’ of the braces of a set, including counting the main, outer, braces, to assess the logical type of that set, i.e., to measure the ‘‘‘depth’’’ of ‘sets-as-elements content of the set in question.

If the “logical individuals”, or ‘arithmetical idea-objects’, “existing” per the comprehension axioms of a given axioms-system, are limited to “Natural” Numbers, classes of “Natural” Numbers,..., all the way up to classes of classes of... of “Natural” Numbers, e.g., to ‘class-objects’ up to a given “logical type”, then the next system will cumulatively expand those ‘‘‘existential’’’ limits by one step, to include also classes of classes of classes... of “Natural” Numbers, i.e., ‘class-objects’ of next-higher “logical type”.

Starting from the “universal class”, each second and higher class-inclusion of previous ‘class-objects’ can model [including via adjunction of those object’s corresponding comprehension axioms, defining the ‘computative behavior’ of these new entities] -- e.g., via the new, higher logical type-level of those special kinds of sets called “ordered pairs”, that arise, for the first time, in the second step of universal class self-inclusion -- a new kind of arithmetical ‘idea-object’; indeed,a new, higher kind of number.

Thereby, this qualitative expansion of each predecessor axioms-system, in the formation of its successor axioms-system, together with the adjunction of the additional, comprehension axioms to the previous, predecessor axioms, corresponds to a qualitative expansion of the ‘idea-ontology’ -- of the ‘arithmetical ontology’, i.e., of the number-ontology -- of that predecessor axioms-system, thereby transforming it into its successor axioms-system.

Hypothesis: Specifically, the diophantine equation that was unsolvable as such within the predecessor axioms-system itself becomes solvable, albeit in a non-diophantine sense, i.e., by a non-diophantine type of number -- a number beyond the ‘“Natural”’ Numbers -- within the next [as well as within all subsequent] successor axioms-systems in this cumulative sequence of axioms-systems, precisely by means of these next higher kinds / logical types of “ordered pair” sets, and by means of the new kinds of numbers which they ground, which will not be ‘diophantine numbers’, e.g., not ‘“Natural”’ Numbers.

The result is a progression of qualitatively-distinct, ‘ideo-ontologically’ distinct, ‘number-space distinct’, axioms-systems of arithmetic.

Each successor arithmetical axioms-system “contains” the kinds of numbers, and at least some of the axioms, of all of its predecessor arithmetical axioms-systems.

It also contains a new kind of number, “absorbing” and “converting” into itself -- into its kind -- all of the previous-systems’ kinds of number that it “contains”, and, with this new kind of number, solving a kind of algebraic equation that was unsolvable in its immediate predecessor arithmetical axioms-system, and proving a kind of proposition that was “undecidable” -- neither provable nor dis-provable -- in its immediate predecessor arithmetical axioms-system.

But this successor arithmetical axioms-system also has its own, new kind of undecidable theorem, and its own, new kind of unsolvable algebraic equation, a ‘self-incompleteness’ which leads on to its own successor axioms-system of arithmetic.

We term such a progression a Gödelian Dialectic.

What we present herein, in the core section of this essay, is, precisely, The Gödelian Dialectic of the Standard Arithmetics, as encoded in a ‘dialectical meta-model’ of its systematically-ordered method-of-presentation, expressed via the ‘Dyadic Seldon Function Dialectical Equation’ introduced in the third sub-section of this background section.

Given the [potentially-infinite character of this dialectical, «aufheben» systems-progression of axioms-systems of arithmetic, driven by the provability-incompleteness and the equational-unsolvability that characterizes every possible axioms-system of arithmetic in this progression of axioms-systems of arithmetic, Gödel himself describes this progression as manifesting the incompletability or inexhaustibility of mathematics,which, from our point of view, represents also the potential  interminability of [The Gödelian ]Dialectic--

The metamathematical results I have in mind are all centered around, or, one may even say, are only different aspects of, one basic fact, which might be called the incompletability or inexhaustibility of mathematics. ...

The phenomenon of the inexhaustibility of mathematics, however, [is] always present in some form, no matter what standpoint is taken.

So I might as well explain it for the simplest and most natural standpoint, which takes mathematics as it is, without curtailing it by any criticism.

From this standpoint all of mathematics is reducible to abstract set theory. ...

So the problem at stake is that of axiomatizing set theory.

Now, if one attacks this problem, the result is quite different from what one would have expected.

Instead of ending up with a finite series of axioms, as in geometry, one is faced with [F.E.D.: a potentially] infinite series of axioms, which can be extended further and further, without any end being visible and, apparently, without any possibility of comprising all of these axioms in a finite rule producing them.

This comes about through the circumstance that, if one wants to avoid the paradoxes of set theory ... the concept of set must be axiomatized in a stepwise manner.

If, for example, we begin with the integers, that is, the finite sets of a special kind, we have at first the integers and the axioms referring to them (axioms of the first level), then the sets of sets of integers with their axioms (axioms of the second level), and so on for any finite iteration of the operation “set of”. ...”

[Kurt Gödel, “Some Basic Theorems on the Foundations of Mathematics and Their Implications (*1951)”, in S. Feferman, et. al., editors, Kurt Gödel: Collected Works (Volume III: Unpublished Essays and Lectures), ibid., pages 305-306].

¿Is this Gödelian ‘Dialectic of Arithmetics’ -- this “inexhaustibility” and “incompletability” of arithmetics -- still alive and in evidence today?

¿Are new, ‘ideo-ontologically expanded’ -- kinds-of-numbers-expanded -- arithmetics still being discovered?

¿I.e, do we today need to search for a Gödelian “next” arithmetic, with its new kinds of numbers: an arithmetic that has eluded us so far?

¿Do we need that next, new arithmetic, and its new kind of numbers, in order to solve equations that have remained “unsolvable” ever since their discovery, so far, and right up until to today?

¿Do we need it to enable calculations that we cannot perform even with today’s most advanced mathematics, hence to enable predictions that we cannot, even with today’s most advanced ‘ideo-technology’, discern, hence to enable new ‘physio-technologies’, new technologies that are crucial to the very survival of the Terran human «species», e.g., to mitigate external hazards -- ‘exolithic bombardments’, solar coronal mass ejections; solar and other-stellar mega-flares; magnetars; gamma ray bursts/hypernovae -- and to mitigate internal hazards, e.g., of a New/Final Ice Age, and of a New/Final Dark Age, due to dialectical, internal, immanent [self-]contradictions of the descendant phase of our present «species» of global [proto-]human civilization?

¿I.e., do we need this new math. to grow the social forces of production to the next level?

Consider the greatest scandal of modern science, which, given its vast magnitude, is mainly mentioned as such only in whispers: the “unsolvability” of “most” nonlinear integrodifferential equations.

This problem has fettered the advance of modern science since the inception of modern science, since the very discovery of such equations, over 300 years ago, and it encompasses the “unsolvability” of the most important of such equations, those that constitute Terran humanity’s most advanced formulations of its “laws” of nature to date.

These include the “many-body” Newtonian ‘‘‘gravitic’’’ equations, the Einsteinian General-Relativistic ‘‘‘gravitic’’’ equations, the Navier-Stokes equations of ‘“rheid”’ dynamics [the dynamics of rheological”, or flowing, matter, in the form of electrically nearly-neutralized liquids and gases, thus of weather, etc.], the Maxwell-Boltzmann-Vlasov equation [for electrically non-neutral, plasma ‘‘‘rheids’’’], etc.

This central failure of modern science is not some merely esoteric, ethereal, rarefied, merely conceptual failure, of concern only to specialists. It is a failure also in deeply practical terms.

Were we to discover how to analytically solve the Navier-Stokes equations, we would probably, given the disproportionalities of cause/effect inherent in nonlinearity, be enabled to stop hurricanes in their tracks, by applying a presently-‘harnessable’ amount of energy to their “Achilles-heals”; and to nip tornados in the bud.

Were we to learn to analytically solve the Vlasov equation, we could design a global grid of zero pollution fusion power reactors, harnessing radioactivity-free advanced fuel regimes, emitting only electrons.

The irruption of human capability to analytically solve “most” nonlinear integrodifferential equations would represent an enormous leap forward in “universal labor” [Marx], that could quickly translate into an enormous upsurge in the level of development of the human-social forces of [human society’s self-expanding self-re-] production, if the core -- the evil, craven, technodepreciation-terrified, and technologically-educated-middle-working-class-terrified ruling faction -- of the global ruling plutocracy could be prevailed upon, by popular insistence, to desist from their strategy of global enfetterment and reversal of those forces [negative growth].

Sans that irruption, and that persuasion, Terran humanity is headed downwards, into a new, global, and, this time, likely final Dark Age.

We have mentioned that the «aporia» of the “unsolvable equations” that motivate our [meta-]models’ dialectic movement from arithmetic to higher arithmetic will seem “muted” -- because we already know that they were eventually solved, and because we already know how they were solved, and because we already know the new kinds of numbers which made those solutions possible.

Many scientists and mathematicians are wont to say, today -- and without proof! -- that most nonlinear differential equations must remain ‘‘‘forever unsolvable’’’.

The «aporia» of the “unsolvability” of most nonlinear equations is drastically alivetoday, but the perception of that ‘drasticity’ is muted by despair, by the ideology of eternal unsolvability.

Still, new “closed-form”, “exact”, “analytical” solutions to vast classes of ‘‘‘minimally nonlinear’’’ partial-differential “evolution equations”, including to the Nonlinear [cubic] Schrödinger equation, keep being discovered -- e.g., those called “nonlinear wave”, or “solitary wave” solutions; the “soliton” solutions, for waves that act like “particles”.

Many such “exact solutions” don’t even require new “transcendental functions”.

Old “transcendental functions”, e.g., the hyperbolic-tangent & hyperbolic-secant functions, provide their “exact”, “analytical”, “closed-from” solutions.

Are nonlinear differential equations possibly a kind of Gödelian “diophantine equation”, such as could be the subject of a “Gödel Formula” for the de facto axioms-set of today’s most advanced system of mathematics?

Yuri Matiyasevich, on the way to his solution of Hilbert’s Tenth Problem, his proof that arbitrary diophantine equations are “semidecidable” as to their solvability [in terms of ‘diophantine numbers’] -- including the proof of the ‘diophantinicity’ of the prime numbers, and the derivation of diophantine algebraic [finite] polynomial equations with only integer coefficients, whose positive solution sets, the sets of all positive values that they yield for integer values of their variables, is exactly the set of all prime numbers -- found that, indeed, the problem of the existence of the solutions of a Diophantine equation in natural numbers can be reduced [sic] to the problem of the existence of a solution of a system of polynomial differential equations of first order.

[Yuri Matiyasevich, Hilbert’s Tenth Problem, MIT Press [Cambridge: 1994], pages xix-xx, 46, 54-56, 85, 176, emphasis added].

But let us consider an “unsolvable” diophantine equation that is, apparently, a little less “esoteric”, and “much simpler”, than a nonlinear differential equation -- that is a little “closer to home”:

=   z/0,

for any z in Z  [including for z = 0 in Z:   x = 0/0].

That kind of diophantine equation is unsolvable among the N, the W, the Z, the Q, the R, the C, the H -- is still, to this very day, unsolvable in any of the “Standard Arithmetics”.

This «aporia», the quandary of division by zero, is still very much alive in our own times, and, again, its intensity is muted only by despair, only by the hopelessness that most of us are taught to feel about ever “fixing” this locus where our Standard Arithmetic breaks down, where it fails to work; by the official, consensus denial that there can exist any axioms-system of arithmetic in which zero division could make sense; in which this kind of diophantine equation could be usefully solved.

Actually, our nonlinear differential equation example is not so remote from our “more homey” zero division example after all: Zero division is the proximate cause of the problem of singularity, of the “meaningless” or “undefined” values that plague, especially and essentially, the nonlinear differential equations, which are so inherently prone to singularities, helping to thwart their solvability/“integrability” under their present-day arithmetical undergirding.

If you want to experience the “bite”, the intensity, the fierceness, the searing mental mood, of a real dialectical «aporia» -- still alive and still burning -- unmuted, then simply let yourself let go of your despair about ever finding an arithmetic in whose context, e.g.,

r/0   =   x,

for r in R is rendered intelligible.

Simply imagine believing that a solution is “out there”, ready to be found, to that “diophantine equation”.

Simply imagine that finding that arithmetic will solve the singularity problem, and, even more, help to unlock the door to the secrets that will habilitate to a full breach of the ‘‘‘Nonlinearity Barrier’’’ -- including its aspect of the ‘patterned-ness’ of the never-repeating -- a breakthrough to beyond the world-historical «aporia» that presently blockades Terran humanity from access to the higher theories, and to the higher technologies, that reside beyond that Barrier, and upon whose acquisition the matriculation of this humanity, from its looming ‘Meta-Darwinian Planetary Selection Test’, and from its “prehistory”, in Marx’s sense, so vitally depends.

The “solution” that

r/0   =   oo   =   "infinity"   =    x,

for r in R is known to incur devastating problems in physical models -- indeed, it leads to ‘infinity residuals’, to ‘infinite error’, to ‘infinitely wrong answers’.

There is almost equal despair, in the consensus view of today’s physics and applied mathematics communities, that the singularities of the extant nonlinear partial differential equation models of the “laws” of nature can ever be “solved”.

¿Can the Gödelian Dialectic, pursued to a sufficiently advanced stage, overcome this ‘‘‘incompleteness’’’ too?"



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