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The Paradox of Gödel's Numbering and The Robert DJIDJIAN UDC 1/14:510.6 Robert DJIDJIAN THE PARADOX OF GÖDEL’S NUMBERING AND THE PHILOSOPHY OF MODERN METAMATHEMATICS Abstract The author of this article critically analyses the proof of Gödel’s famous theorem on the incom- pleteness of formalized arithmetic. It is shown that Gödel’s formalization of meta-mathematics pro- vides a proof of the incompleteness not of mathematical science but of the system of formalized me- ta-mathematics developed by Gödel himself. The arguments against the idea of the formalization of meta-mathematics are presented. The article suggests also an interpretation of the essence of math- ematical truth. It is noted that the refutation of Gödel’s proof does not suggest returning to Hilbert’s program of formalism since the formalization of an axiomatic theory can’t exclude the appearance of paradoxes within its framework. It is shown that the use of self-referential Gödel’s numbering in a formalized system leads to the emergence of a Liar type paradox – a self-contradictory formula that demonstrates the inconsistency of that same system. Keywords: formalized theories, metatheory, Gödel’s theorem, Gödel’s numbering, formalized metamathematics, inconsistency, paradoxes. Introduction completeness of formalized calculi (Gentzen, 1935). And the most independent minds, de- Gödel's theorem on the incompleteness of spite general adoration before Gödel's proof of formalized arithmetic is one of the most widely the incompleteness of formalized arithmetic, accepted and highly evaluated results of logi- tried to restore the shaken authority of mathe- cal and mathematical thought of the 20th cen- matical science by proving the completeness of tury (Gödel, 1931). It is this article, together formalized mathematical theories (Henkin, with the previously proven fundamentally im- 1950; Hasenjaeger, 1953). The wave of publica- portant theorem on the completeness of the tions on the significance of Gödel’s theorems first order predicate calculus that has deter- (Franzén, 2005), their philosophical interpreta- mined the philosophy and ideology of all sub- tion (Rucker, 1995; Wang, 1997), the current sequent studies on the foundations of mathe- demonstrative exposition (Smith, 2007), at- matics (Gödel, 1929). Encouraged by Gödel's tempts to apply them far beyond the reasonable results, many mathematicians began to investi- (Lucas, 1970; Feferman, 2011) continues to this gate the consistency of formalized systems day. (Smoryński, 1977; Smullyan, 1991). Other re- The first part of this article demonstrates searchers undertook an intensive study of the that Gödel’s proof of his incompleteness theo- 18 WISDOM 2(9), 2017 18 The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics UDC 1/14:510.6 rem is not valid, since in actuality it does not (Kleene, 1952). But in this kind enriched sys- Robert DJIDJIAN demonstrate the incompleteness of the Prin- tem one can build Gödel’s undecidable formu- cipia Mathematica and the related sufficiently la G which would demonstrate that this kind THE PARADOX OF GÖDEL’S NUMBERING rich systems. enriched system of formalized arithmetic is AND THE PHILOSOPHY OF MODERN METAMATHEMATICS The second part of this article discusses incomplete. It should be underlined here that the general methodological problems of for- this proof of incompleteness relates to the sys- Abstract malized theories and in its final section proves tem of formalized arithmetic enriched by Gö- that if it is allowed to use Gödel’s numbering del’s numbering, but not of the classical sys- The author of this article critically analyses the proof of Gödel’s famous theorem on the incom- in a system (including formalized axiomatic tem of formalized arithmetic of Principia pleteness of formalized arithmetic. It is shown that Gödel’s formalization of meta-mathematics pro- theories) then in this system one can build a Mathematica. vides a proof of the incompleteness not of mathematical science but of the system of formalized me- Liar type paradox demonstrating the incon- ta-mathematics developed by Gödel himself. The arguments against the idea of the formalization of sistency of this system. Part II meta-mathematics are presented. The article suggests also an interpretation of the essence of math- ematical truth. It is noted that the refutation of Gödel’s proof does not suggest returning to Hilbert’s Part I The first part of this article concludes my program of formalism since the formalization of an axiomatic theory can’t exclude the appearance refutation of Gödel’s proof of the theorem on of paradoxes within its framework. It is shown that the use of self-referential Gödel’s numbering in 1.1. Young Kurt Gödel in his famous the incompleteness of formalized arithmetic. a formalized system leads to the emergence of a Liar type paradox – a self-contradictory formula that 1931 paper presented an instrument of logical Dear readers, in the case you would like to ob- demonstrates the inconsistency of that same system. proofs – Gödel’s numbering – and constructed ject to my refutation, I ask you to apply only to with its help an undecidable formula G (Gödel, the argumentation of the above presented first Keywords: formalized theories, metatheory, Gödel’s theorem, Gödel’s numbering, formalized 1931). Presuming that this formula belongs to part of this article. metamathematics, inconsistency, paradoxes. the system of formalized arithmetic he con- In the second part of this article, I present cluded that the system of formalized arithmetic the general methodological considerations is incomplete (Gödel’s theorem). which help to demonstrate eventually that if it Introduction completeness of formalized calculi (Gentzen, The principle point here is that in actuality is allowed to use Gödel’s numbering in a sys- 1935). And the most independent minds, de- Gödel’s undecidable formula G does not be- tem (including formalized axiomatic theories) Gödel's theorem on the incompleteness of spite general adoration before Gödel's proof of long to the system of formalized arithmetic then one can build a Liar type paradox formalized arithmetic is one of the most widely the incompleteness of formalized arithmetic, presented in Principia Mathematica (Rusell & demonstrating the inconsistency of that sys- accepted and highly evaluated results of logi- tried to restore the shaken authority of mathe- Whitehead, 1910) because to construct Gödel’s tem. formula G one needs Gödel’s numbering 2.1. The consistency of axiomatic theo- cal and mathematical thought of the 20th cen- matical science by proving the completeness of which is absent in Principia Mathematica. ries. An axiomatic theory is considered as a tury (Gödel, 1931). It is this article, together formalized mathematical theories (Henkin, So we have to conclude that Gödel’s matter of serious discussion only after it is able with the previously proven fundamentally im- 1950; Hasenjaeger, 1953). The wave of publica- to show solutions to the main problems of the portant theorem on the completeness of the tions on the significance of Gödel’s theorems proof of his incompleteness theorem is not valid, since it actually does not demonstrate given field of scientific knowledge. We call first order predicate calculus that has deter- (Franzén, 2005), their philosophical interpreta- the incompleteness of the Principia Mathemat- such a theory well-developed axiomatic theory. mined the philosophy and ideology of all sub- tion (Rucker, 1995; Wang, 1997), the current ica and the related sufficiently rich systems. Just at this stage, the axiomatic method demon- sequent studies on the foundations of mathe- demonstrative exposition (Smith, 2007), at- 1.2. On the other hand, if a mathematician strates its fundamental advantage. When a given matics (Gödel, 1929). Encouraged by Gödel's tempts to apply them far beyond the reasonable would like to use Gödel’s numbering in the axiomatic theory achieves such a weighty result results, many mathematicians began to investi- (Lucas, 1970; Feferman, 2011) continues to this system of formalized arithmetic he should do it – succeeds to solve the main problems of the gate the consistency of formalized systems day. by explicitly including Gödel’s numbering in field under investigation – one can express his (Smoryński, 1977; Smullyan, 1991). Other re- The first part of this article demonstrates the scope of means of that formalized system suspicion in regard of this theory only in the searchers undertook an intensive study of the that Gödel’s proof of his incompleteness theo- 18 19 19 WISDOM 2(9), 2017 Robert DJIDJIAN case if he can present at least one fact (a true decades of successful activity of this theory in singular statement) that contradicts any axiom solving the important problems of the field or theorem of this theory. under research, and its “impeccable behav- Until contradictory facts are found, a iour” – complete absence of contradictory well-developed axiomatic theory is rightly re- facts. garded as a serious contender for the status of The second difficulty is expressed by the the true theory of the field under study. We following question: “Are not the logic of Ar- emphasize that the theory achieves this high istotle, the geometry of Euclid and the me- status precisely due to the successful axiomat- chanics of Newton completely true and in this ic resolution of the problems of the field un- sense the ultimate and absolute truths of theo- der study. Based on the presented analysis, we retical knowledge?” A negative answer to this can propose the following
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