Roots : Some disquieting features of Gödel’s reasoning and conclusions
This is a reproduction of the PREFACE from the author’s earlier collection of notes and formally unpublished papers, compiled in DE-MYSTIFYING GÖDEL’S THEOREM VI.
Although much of the material is irrelevant in the light of later work, and contains gross errors and inaccuracies (as patiently pointed out by Professor Martin Davis in correspondence with the author), this preface does effectively outline the various disquieting features of Gödel's reasoning and conclusion which the author was unable to retrace and accept with conviction.
Since these features resisted reconciliation despite sustained efforts over the years to develop an increasing understanding of Gödel's intentions, reasoning and conclusions, they may be of interest to some in suggesting new avenues of investigation.
“ This work is a collection of notes and unpublished papers covering thirty years of private, isolated study of texts on the logical and philosophical foundations of mathematics.
A result crucial not only to logic, but of significance to the evolution of scientific thought in any discipline, is developed in Chapter_2, where we faithfully mirror Gödel's original argument to show that the primitive recursive formula used in his proof of Theorem VI :
Q (x, y ) ºM ~ [ x B k [ Sb ( y : (19, z(y)) ) ] ]
cannot be decided intuitively in a 2-valued logic, as is wrongly assumed by Gödel whilst affirming the existence of UNDECIDABLE sentences in formal systems.
( This is also the thesis underlying Chapters_3, 5 and 8.)
In Chapters_4 and 6 we focus on another inconsistency - one that follows from Gödel's postulated equivalence between his provable formulas and his PROVABLE FORMULAS.
In Chapter_1 we trace the root of such inconsistency to the fact that the formal definition of provable formulas, used by Gödel whilst defining his formal system, makes these indistinguishable from the identically defined TRUE formulas of classical systems.
When subsequently equating the former, and hence the latter, to his arithmetical definition of constructively PROVABLE FORMULAS, Gödel is then inadvertently, contrary to his own intention and thesis, postulating the very equivalence in his system that he seeks to avoid.
As is possibly true in general of non-academic work, this collection suffers from both an inherent lack of awareness and authority concerning the scope and current status of textual and non-textual work, and the focus and specialised format expected by practicing logicians.
Paradoxically, this should make most chapters in this collection accessible to a wider audience of serious multi-disciplinary scholars, mathematicians, philosophers and scientists.
For all of them, it contains significant new mathematical and philosophical results, emerging out of the attempts, at both elementary and technical levels, to critically review and address various anomalies and invalid assumptions underlying Gödel’s reasoning on the purported existence of ‘formally undecidable’ sentences in classically consistent formal systems.
At various levels of understanding and interpretation, the implicit acceptance of Gödel’s conclusions on the existence of such ‘formally undecidable’ sentences is reflected in (and so, reasonably, must influence) the mind-set behind the philosophical thought underlying current writings in not only the purely scientific disciplines, but also studies of the more wide-ranging cross-cultural subjects such as modernism and deconstruction.
Hence the focus of these notes is almost entirely on Theorem VI of Gödel’s ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems I’ [Gödel_2].
Underlying this work is the assessment, and premise, that if all Gödel achieves in his paper is to introduce a new concept of PROVABLE FORMULAS for which he can construct a sentence p such that ‘p is not a PROVABLE FORMULA’ and ‘~ p is not a PROVABLE FORMULA’ are both classically TRUE, the result would have little extended significance.
To give the concept significance in existent mathematical systems, he needs to define his PROVABLE FORMULAS in terms of existing concepts, relate them to classically TRUE formulas, and then derive consequences extending existing results significantly.
We show, in Chapters_1 and 9, how Gödel attempts this by, firstly, representing the classical formula P arithmetically by a unique numeral p in a formal system S.
We see how he then represents and defines arithmetically in S various meta-concepts, such as the concept of ‘p is a PROVABLE FORMULA’ which is TRUE (under classical interpretation) if and only if it is the last FORMULA of a finite sequence of FORMULAS, each of which is an AXIOM or an IMMEDIATE CONSEQUENCE of two of the FORMULAS preceding it in the sequence.
He calls any such sequence a formal PROOF of the FORMULA p.
We also see how he postulates the class of all such PROVABLE FORMULAS as being equivalent, in the meta-theory M, to the impredicatively defined minimal class Flg(f) of FORMULAS that contains all of the AXIOMS of S and which is closed with respect to the relation of IMMEDIATE CONSEQUENCE.
Thus, according to Gödel, ‘p is a PROVABLE FORMULA’ is TRUE (interpreted intuitively in M), if and only if ‘pÎFlg(f)’ is TRUE (interpreted similarly).
In Chapter_1 we critically review this postulated equivalence :
‘pÎFlg(f) ºM there is a finite sequence of FORMULAS of S, ending with p, each of which is either an AXIOM or an IMMEDIATE CONSEQUENCE of two of the preceding FORMULAS’.
We show there that Flg(f) is, by Gödel’s explicit definitions, in fact implicitly equivalent to the class of classically defined TRUE formulas of S.
Now Intuitionism already restricts itself to, essentially, considering as reliably TRUE only those FORMULAS of Flg(f) that meet Gödel’s finitary criteria for PROVABLE FORMULAS. It does not accept as being similarly TRUE those FORMULAS which can be shown to be in Flg(f) only by non-constructive reasoning such as, for instance, involves the Law of the Excluded Middle.
Thus Gödel goes beyond both classical and Intuitionist doctrines when he equates constructively defined PROVABLE FORMULAS to those in Flg(f).
Since the mathematical and logical antinomies are deeply rooted in classical systems, it follows that, by Gödel’s postulation of classically TRUE formulas as PROVABLE, these antinomies should also be formalisable within S, and hence reflected there as formal logical contradictions.
That this is indeed so is variously shown in Chapters_2, 3, 4, 5, 6 and 8.
In Chapter_2, further, we use Gödel’s original proof of his Theorem VI to specifically trace the root of the antinomies to his premise on the intuitive decidability of recursive functions, stated as a ‘fact’ in his Theorem V, on which his Theorem VI is critically dependent.
We show there that Gödel’s Theorem VI in fact establishes the existence of well-formed, but ill-defined, CONTRARY (intuitively inconsistent) primitive recursive sentences that arise out of the language of any Arithmetical system, and which do not obey a two-valued logic.
It follows that a consistent system of Arithmetic can only be defined impredicatively as ‘consisting’ of those well-formed FORMULAS of a suitably rich language that are not CONTRARY (ill-defined FORMULAS such that P =M ~P).
From this Gödel’s second thesis, that formal CONSISTENCY is not provable within these languages, follows trivially.
We further reason heuristically in Chapter_8 that, as a consequence of the existence of CONTRARY sentences in languages rich enough to formalise Arithmetic, the Principle of Comprehension, an axiom of current formal and naive Set Theories found necessary for the creation of sets needed to develop the Theory significantly, cannot hold if the system is to be formally CONSISTENT.
Consequently, we would have formulas in such systems that cannot be associated with a set of precisely their TRUTH values, thus effectively undermining the basis for both a Platonist interpretation of mathematical and logical systems that views sets as existing as really mathematically as the real world does physically, and the current extensional definition in various mathematical disciplines of a ‘function’ as a unique mapping (set) from the domain of its variable into the range of the variable concerned.
As a further consequence of the above (not treated here but developed in a separate collection of notes and papers under preparation), we construct discontinuous Cauchy sequences in the Real number space for which the Theorem on the Completeness of Metric Spaces does not hold, as its proof breaks down in such systems, being essentially based on the premise that ‘equivalent’ Cauchy sequences necessarily have a common ‘limit’, so there are no discontinuous Cauchy sequences in the space concerned.
In work fragmented over thirty years, it is not only presumptuous but impossible to fairly and honestly acknowledge the general debt owed to individuals, whether family, friends or associates, without whose continuing active support, nurturing, inter-actions and inspiration such effort cannot sustain.
A specific debt that can be more easily acknowledged than repaid is to my life-long friend Dr. Chetan H. Mehta, who not only has been, constantly, the devil’s effective advocate, but has both encouraged and urged, as a last resort, publishing of these investigations as they are, despite obvious limitations.
B.S.Anand Mumbai : June 1997”
(Updated : 10/11/01 1:57:50 AM by email@example.com)