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 incompleteness 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 meta-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 mathematical 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 mathematical 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 preliminary defini- question was established after the develop- tions of the consistency and truth of axiomatic ment and general recognition of the revolu- theories: tionary theories of theoretical physics of the x An axiomatic theory is consistent, if it is 20th century. Today, all educated people are in agreement with the facts of the field well aware that the applicability of classical under study. mechanics is limited to the field of phenome- x An axiomatic theory is a serious pretend- na of the macrocosm and is completely inap- er to the status of a true theory only inso- plicable neither to the world of atoms and el- far as it is successful in solving the main ementary particles, nor to the world of sub- problems of the field under study. light speeds. Formation of the algebra of logic (The expression “this theory is in agree- in the 19th century and the victorious advance ment with the facts of the field under of mathematical logic in the 20th century are study” means that in this theory no state- reasonably perceived as the emergence of log- ment is derived that is denied by any ical systems more powerful than the Aristote- known fact of the field under study.) lian syllogistic. And even the ideal of scientific 2.2. True theory, absolute truth, and sci- knowledge – geometry – had to leave the ped- entific progress. In connection with the pro- estal of the absolutely true knowledge when posed preliminary definitions, several ques- mathematicians recognized the consistency of tions arise. First, does an axiomatic theory non-Euclidean systems of geometry developed remain for all time in the status of a “candi- in the 30s of the 19th century by Nikolai Loba- date” for becoming a true theory, or from a chevski and János Bolyai, and especially when certain point in time and, according to a cer- Bernhard Riemann presented to the society of tain criterion, it deserves to be awarded the mathematicians his grandiose system of geom- honorary title of a true theory? In a sense, it etries in 1854 (Riemann, 2004). can be argued that in the eyes of the scientific From what has been said, it can be con- community, theory achieves the status of “au- cluded that there are no absolutely true gen- thentic truth” from the moment it is included eral statements, laws and principles in the in official university handbooks. This actual scope of scientific knowledge. And the fact state of affairs necessarily assumes both the that individual empirical judgments serve as 20 WISDOM 2(9), 2017 20 The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics

case if he can present at least one fact (a true decades of successful activity of this theory in the starting material of human theoretical x The universal definition of truth goes singular statement) that contradicts any axiom solving the important problems of the field knowledge leads to the conclusion that human back to Aristotle: “Truth is the adequacy or theorem of this theory. under research, and its “impeccable behav- cognition is not given either the knowledge of of the thought and reality.” (“Veritas est Until contradictory facts are found, a iour” – complete absence of contradictory an absolute truth or a universal criterion for adequatio ratio et rei”) well-developed axiomatic theory is rightly re- facts. absolutely true general statements. x One needs to resolve specific questions of garded as a serious contender for the status of The second difficulty is expressed by the 2.3. The essence of mathematical truth. the theory of truth, and only then take up the true theory of the field under study. We following question: “Are not the logic of Ar- In the modern formulation, this problem most the development of a general conception. emphasize that the theory achieves this high istotle, the geometry of Euclid and the me- often comes down to the question “How is x The central problem of the theory of truth status precisely due to the successful axiomat- chanics of Newton completely true and in this truth defined in mathematics?”. The complex- in mathematics is expressed by the ques- ic resolution of the problems of the field un- sense the ultimate and absolute truths of theo- ity of the issue can be seen already in the fact tion: “What does exactly mean the notion der study. Based on the presented analysis, we retical knowledge?” A negative answer to this that the author of the article “Truth” in Stan- reality in elucidating the truth of a math- can propose the following preliminary defini- question was established after the develop- ford Encyclopedia of Philosophy Michael ematical statement?” tions of the consistency and truth of axiomatic ment and general recognition of the revolu- Glanzberg was forced to turn to the help of x The staircase of abstractions suggests the theories: tionary theories of theoretical physics of the 119 primary sources. The sub-section of this existence of a principle possibility for the x An axiomatic theory is consistent, if it is 20th century. Today, all educated people are topic – “axiomatic theory of truth” – includes reduction of any abstraction to the level of in agreement with the facts of the field well aware that the applicability of classical 15 different conceptions of truth. The state of elementary ones. Apparently, this is con- under study. mechanics is limited to the field of phenome- affairs is extremely complicated by the fact firmed by the reduction of all mathemat- x An axiomatic theory is a serious pretend- na of the macrocosm and is completely inap- that the problem of truth in its genesis and ics to the theory of sets. er to the status of a true theory only inso- plicable neither to the world of atoms and el- nature, without doubt, has an inextricable link x For the elementary abstractions, it is not far as it is successful in solving the main ementary particles, nor to the world of sub- with philosophy, where diversity and contra- difficult to find a correspondence with ob- problems of the field under study. light speeds. Formation of the algebra of logic dictory opinions are an unwritten norm. This jects of reality. (The expression “this theory is in agree- in the 19th century and the victorious advance article begins with the following paragraph: In formalized theories, a central role is ment with the facts of the field under of mathematical logic in the 20th century are “Truth is one of the central subjects in philos- played by models - theories with simpler ab- study” means that in this theory no state- reasonably perceived as the emergence of log- ophy. It is also one of the largest. Truth has stractions with strictly defined properties. With ment is derived that is denied by any ical systems more powerful than the Aristote- been a topic of discussion in its own right for the help of models, one can answer the ques- known fact of the field under study.) lian syllogistic. And even the ideal of scientific thousands of years. Moreover, a huge variety tion of the correspondence of abstract (mathe- 2.2. True theory, absolute truth, and sci- knowledge – geometry – had to leave the ped- of issues in philosophy relate to truth, either matical) statements to the real state of things. entific progress. In connection with the pro- estal of the absolutely true knowledge when by relying on theses about truth, or implying Thus, an unambiguous conclusion fol- posed preliminary definitions, several ques- mathematicians recognized the consistency of theses about truth. It would be impossible to lows from the principle of reducibility: the tions arise. First, does an axiomatic theory non-Euclidean systems of geometry developed survey all there is to say about truth in any truth of any abstract (mathematical) state- remain for all time in the status of a “candi- in the 30s of the 19th century by Nikolai Loba- coherent way” (Glanzberg, 2013). ment is established by its correspondence to date” for becoming a true theory, or from a chevski and János Bolyai, and especially when The situation described does not inspire some simpler abstract model, which ulti- certain point in time and, according to a cer- Bernhard Riemann presented to the society of one more attempt to solve the problem of truth mately means its reducibility to a certain ar- tain criterion, it deserves to be awarded the mathematicians his grandiose system of geom- as a whole, and of the mathematical truth as its ea of reality. honorary title of a true theory? In a sense, it etries in 1854 (Riemann, 2004). most difficult case, in particular. Yet let us That is, the whole peculiarity of mathe- can be argued that in the eyes of the scientific From what has been said, it can be con- make a new attempt by proceeding from the matical truth lies in the fact that in mathemat- community, theory achieves the status of “au- cluded that there are no absolutely true gen- following premises: ics the truth of assertions is verified usually thentic truth” from the moment it is included eral statements, laws and principles in the x The problem of truth is a problem of all with the help of simpler abstract models in official university handbooks. This actual scope of scientific knowledge. And the fact the sciences, and in this sense, a philo- which seem as if not having a direct relation state of affairs necessarily assumes both the that individual empirical judgments serve as sophical one. to reality. Just this seeming separateness from 20 21 21 WISDOM 2(9), 2017 Robert DJIDJIAN

reality creates a sense of paradox: on the one dox, connected with the notion of the set of all hand, the reliability of mathematical truths sets, which was revealed in the first volume of seems unshakeable and absolute, on the other Frege’s work. hand, in view of the abstract nature of math- The discovery of similar paradoxes of the ematical statements, they are perceived as set theory and cardinal numbers, the Burali- completely detached from reality, and the Forti paradox and the Cantor paradoxes, also question of their truth appears hanging in the belong to the same period. In the first decades air. of the twentieth century, they were supple- 2.4. Formalism and formalization. Ac- mented by a popular analogue of Russell's par- cording to Stefan Kleene's authoritative work adox (the “paradox of barber”), while Richard “Introduction to Metamathematics”, the for- revealed a paradox in the theory of real num- malization of a theory means a formal repre- bers. It was easy to notice that all these various sentation of the corresponding axiomatic theo- paradoxes are very similar to the self- ry (Kleene, 1952, chapter III). This is achieved referential ancient Liar paradox (Kleene, 1952, in two steps. First, a complete symbolization of pp. 36-40). the axiom system of this theory is made. For This specific feature of almost all known this purpose, the initial symbols of the designa- logical-mathematical paradoxes suggests that tion of all concepts appearing in the system of the source of paradoxes is hidden rather in the axioms should be fixed; with the help of this self-referentiality of the corresponding con- symbolism each axiom is presented as the cer- cepts than in the axioms and logical means of tain initial formula. The second step of formal- inference. But the outstanding mathematician ization is the fixation of the rules of inference, of the twentieth century, David Hilbert, con- with the help of which only new formulas are ceived the idea of “completely eradicating” the obtained from the axiom system of the formal- paradoxes from the mathematical kingdom by ized theory. That is, thanks to formalization, building mathematical theories in the form of the original axiomatic theory is transformed extremely strict formalized calculi. Thanks to into a symbolic calculus. The philosophical Hilbert's authority and the fact that many para- strategy of constructing formalized calculi, doxes were perceived as belonging to the very originating from the works of Gottlob Frege foundation of mathematical knowledge, the and explicitly formulated by David Hilbert, is direction of formalism in studies of the founda- to achieve the greatest rigor in the formulation tions of mathematics began to gain increasing and development of axiomatic theories, which, recognition. This was facilitated by Kurt Gö- in theory, should eliminate the emergence of del's proof of the completeness of the first- contradictions and paradoxes in the developed order predicate calculus, the basis of the foun- formalized calculi. However, already on the dations of research on the foundations of eve of the publication of the second volume of mathematics (Gödel, 1929; Gödel, 1930; van his pioneering work “Grundgesetze der Math- Heijenoort, 1967). Naturally, the researchers ematik”, Frege was extremely disturbed by the had to cover the “gold rush” to be the first in letter of the young Bertrand Russell, in which proving the completeness of formalized arith- the German scholar was informed of a para- metic. But Gödel himself put the seemingly

22 WISDOM 2(9), 2017 22 The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics

reality creates a sense of paradox: on the one dox, connected with the notion of the set of all insurmountable barrier in front of the whole ture on foundations of mathematics, there is hand, the reliability of mathematical truths sets, which was revealed in the first volume of strategy of formalism by proving in 1931 his no objection to the evaluation of metatheory seems unshakeable and absolute, on the other Frege’s work. famous theorem on the incompleteness of for- as proof theory – an idea originating from the hand, in view of the abstract nature of math- The discovery of similar paradoxes of the malized arithmetic. work of D. Hilbert. Indeed, the only concrete ematical statements, they are perceived as set theory and cardinal numbers, the Burali- 2.5. Is the idea of formalization of meta- predicate used in constructing Gödel's un- completely detached from reality, and the Forti paradox and the Cantor paradoxes, also theories correct? Before we consider the me- solvable formula G is the predicate “to be a question of their truth appears hanging in the belong to the same period. In the first decades thodological aspects of Gödel's theorem, we proof”. air. of the twentieth century, they were supple- need to carefully consider the concept of met- However, Kleene, and apparently under 2.4. Formalism and formalization. Ac- mented by a popular analogue of Russell's par- amathematics and its natural generalization - his influence, many authors of articles on for- cording to Stefan Kleene's authoritative work adox (the “paradox of barber”), while Richard the concept of metatheory. A formalized axi- malization and metatheory believe that formal- “Introduction to Metamathematics”, the for- revealed a paradox in the theory of real num- omatic system is called object-theory. Since ized metatheory and metamathematics are part malization of a theory means a formal repre- bers. It was easy to notice that all these various the whole school of formalism arose to elimi- of mathematics. Klini insists that using sentation of the corresponding axiomatic theo- paradoxes are very similar to the self- nate the possibility of a contradictions and par- Godel’s enumeration, “metamathematics be- ry (Kleene, 1952, chapter III). This is achieved referential ancient Liar paradox (Kleene, 1952, adoxes in mathematical sciences, it is natural comes a branch of number theory” (Klini, in two steps. First, a complete symbolization of pp. 36-40). that the primary task of the theory of formal- 1952, p. 205). The essence of the argument in the axiom system of this theory is made. For This specific feature of almost all known ized calculi was to establish their consistency. favor of this position is this: mathematized this purpose, the initial symbols of the designa- logical-mathematical paradoxes suggests that As the research of formalized theories has un- metatheory is almost no different from other tion of all concepts appearing in the system of the source of paradoxes is hidden rather in the folded, an equally important possibility of es- sections of mathematics and can quite be con- axioms should be fixed; with the help of this self-referentiality of the corresponding con- tablishing their completeness has emerged. sidered as a new section of mathematics. With symbolism each axiom is presented as the cer- cepts than in the axioms and logical means of These two principal qualities of formalized this argument one could agree, if not for one tain initial formula. The second step of formal- inference. But the outstanding mathematician systems are studied by metatheory, which, nat- important circumstance. In metamathematics ization is the fixation of the rules of inference, of the twentieth century, David Hilbert, con- urally, should be outside the studied formal- in general, and in formalized metamathemat- with the help of which only new formulas are ceived the idea of “completely eradicating” the ized system. Otherwise, the formalized theory ics, in particular, no mathematical axiom or obtained from the axiom system of the formal- paradoxes from the mathematical kingdom by would study itself, which is illogical. Moreo- theorem is used, and no mathematical state- ized theory. That is, thanks to formalization, building mathematical theories in the form of ver, if the metatheory is formalized, it will lose ment is derived from them. The whole content the original axiomatic theory is transformed extremely strict formalized calculi. Thanks to the sense of distinguishing between metatheo- of metamathematical discussions, even after into a symbolic calculus. The philosophical Hilbert's authority and the fact that many para- ry and object-theory. the formalization of metamathematics, is a strategy of constructing formalized calculi, doxes were perceived as belonging to the very But it was precisely the path of formaliz- question of consistency, completeness, solva- originating from the works of Gottlob Frege foundation of mathematical knowledge, the ing metamathematics that Gödel and his fol- bility and similar characteristics directly relat- and explicitly formulated by David Hilbert, is direction of formalism in studies of the founda- lowers chose. True, in his famous article Gö- ed and even expressible with the predicate “to to achieve the greatest rigor in the formulation tions of mathematics began to gain increasing del does not use the terms metatheory and ob- be a proof”. That is, after formalization, meta- and development of axiomatic theories, which, recognition. This was facilitated by Kurt Gö- ject-theory, but by constructing his unsolvable theory remains a tool for studying the demon- in theory, should eliminate the emergence of del's proof of the completeness of the first- formula G with the help of Gödel’s number- strative characteristics of formalized systems, contradictions and paradoxes in the developed order predicate calculus, the basis of the foun- ing and recursive functions, he actually real- including formalized mathematics. But how formalized calculi. However, already on the dations of research on the foundations of izes the formalization of metamathematics. else could it be? After all, formalization chan- eve of the publication of the second volume of mathematics (Gödel, 1929; Gödel, 1930; van The overwhelming part of the excellent work ges only the language of presentation, but not his pioneering work “Grundgesetze der Math- Heijenoort, 1967). Naturally, the researchers of S. K. Kleene “Introduction to metamathe- the subject of research. Would the presentation ematik”, Frege was extremely disturbed by the had to cover the “gold rush” to be the first in matics” is devoted precisely to the strict and of the Homer Iliad in the language of formal- letter of the young Bertrand Russell, in which proving the completeness of formalized arith- complete realization of Gödel’s line of math- ized mathematics enrich the mathematical sci- the German scholar was informed of a para- metic. But Gödel himself put the seemingly ematization of metamathematics. In the litera- ences? And after the rhymed presentation of

22 23 23 WISDOM 2(9), 2017 Robert DJIDJIAN

the Euclid Principles, would it lose its mathe- tion is simply impossible, for Gödel's formula matical content? contains only one concrete predicate “to be a At first glance, it may seem that there is proof.” This unequivocally indicates that Gö- not much difference whether to say “Formal- del's formula belongs to the formalized theory ized metamathematics is a part of mathemat- of proof, and not to mathematics. ics” or to say “Formalized metamathematics is If the supporters of Gödel’s proof of in- fully stated mathematically”. In fact, the dif- completeness paid sufficient attention to this ference is fundamental. In the first case, the circumstance and adequately interpreted it, construction of Gödel’s undecidable formula then from the fact that Gödel’s formula be- within the framework of formalized meta- longs to the formalized theory of proof they mathematics speaks not only of the incom- should conclude that just Gödel’s formalized pleteness of formalized metamathematics, but theory of proof (formalized meta-mathematics) also of the mismatch of the mathematics itself, is incomplete, and not the system of formal- of which it is supposed to be a part. That is, ized arithmetic. That is, Gödel's theorem assuming that the formalized metamathematics proves the incompleteness of the formalized is part of mathematics, the undecidability of theory of proof, but not of mathematics. Gödel's formula can be represented as evi- 2.6. Refuting Gödel’sproof does not mean dence of the incompleteness of mathematics the necessity of vitalizing David Hilbert’s pro- itself. gram of formalization. Bad concepts remain In the light of modern meta-theoretical bad concepts also after their formalization. For approach, it is of principle importance to dis- example, the concept of material implication tinguish mathematically stated theory of proof of propositional logic is not adequate to the (meta-mathematics) from mathematics itself, concept of logical inference and remains non- even if the former is presented mathematically. adequate also after the formalization in the Taking this differentiation for granted, the con- frame of propositional calculus. It is rigorous- struction of Gödel's undecidable formula with- ly proved that the propositional calculus is a in the framework of formalized meta-mathe- consistent and complete formalized system. matics will only demonstrate the incomplete- Yet the paradox of material implication is al- ness of formalized meta-mathematics, but not so well known. of the incompleteness of mathematics itself. Another example of a dubious instrument We again came to the sacramental ques- of argumentation is presented by self-reflexive tion “Does Gödel’s formula belong to mathe- expressions. In the context of this paper the matics or to the theory of proof?” As was most problematic instance of self-reflexivity is proved above, the mathematization of any the- Gödel’s numbering. This self-reflexive instru- ory does not in any way change its content. ment was used in formalizing the system of This conclusion is even more convincing in the metamathematics in his famous article on the case of Gödel’s formula. As Kleene explains, incompleteness of formalized mathematics. All when interpreting the Gödel formula, it means mathematicians knew about the affinity of self- a statement that “establishes its unprovability” reflexive expressions with the Liar paradox; (Klini, 1952, pp. 205; 207). Another interpreta- nevertheless, they in the overwhelming majori-

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the Euclid Principles, would it lose its mathe- tion is simply impossible, for Gödel's formula ty consider it permissible to use this very any mathematical content. How could an out- matical content? contains only one concrete predicate “to be a doubtful means. standing mathematician not notice this? At first glance, it may seem that there is proof.” This unequivocally indicates that Gö- The reason for such a protectionist atti- This could be facilitated by the following not much difference whether to say “Formal- del's formula belongs to the formalized theory tude to the use of a very doubtful means is that circumstances. Apparently, the main thing ized metamathematics is a part of mathemat- of proof, and not to mathematics. without the use of self-reflective expressions, it was that Gödel’s theorem, elementary in its ics” or to say “Formalized metamathematics is If the supporters of Gödel’s proof of in- becomes very difficult to construct certain im- formulation, was proved by a grandiose and fully stated mathematically”. In fact, the dif- completeness paid sufficient attention to this portant sections of modern mathematical sci- super complicated mathematical construction. ference is fundamental. In the first case, the circumstance and adequately interpreted it, ence. After mentioning that the intuitive argu- It is quite natural that both its author and his construction of Gödel’s undecidable formula then from the fact that Gödel’s formula be- ment of Gödel’s proof “skirts so close and yet followers should have the impression that this within the framework of formalized meta- longs to the formalized theory of proof they misses a paradox” S. Kleene himself couldn’t is just the proof of a mathematical statement. mathematics speaks not only of the incom- should conclude that just Gödel’s formalized miss the feeling that proving a fundamental They could not have imagined that the grandi- pleteness of formalized metamathematics, but theory of proof (formalized meta-mathematics) statement of incompleteness of formalized ose and super-complex mathematical construc- also of the mismatch of the mathematics itself, is incomplete, and not the system of formal- mathematic with the help of self-referential tion of Gödel’s proof is only a rigorous repre- of which it is supposed to be a part. That is, ized arithmetic. That is, Gödel's theorem formula is rather undesirable taking into ac- sentation of the self-reflective assertion that “G assuming that the formalized metamathematics proves the incompleteness of the formalized count the strictness of mathematical sciences. claims to be unprovable”, which had only an is part of mathematics, the undecidability of theory of proof, but not of mathematics. Apparently trying to justify the use of self- indirect link with mathematics through the Gödel's formula can be represented as evi- 2.6. Refuting Gödel’sproof does not mean referential numbering in Gödel’s proof, S. predicate “to be a proof” – the basic category dence of the incompleteness of mathematics the necessity of vitalizing David Hilbert’s pro- Kleene writes: “Its proof, while exceedingly of formal logic since the days of the Aristoteli- itself. gram of formalization. Bad concepts remain long and tedious in these terms, is not open to an Analytics. In the light of modern meta-theoretical bad concepts also after their formalization. For any objection which would not equally involve The bias in Gödel's assessment of his approach, it is of principle importance to dis- example, the concept of material implication parts of traditional mathematics which have formula as of a mathematical statement could tinguish mathematically stated theory of proof of propositional logic is not adequate to the been held most secure” (Kleene, 1952, p. 206). be facilitated also by the fact that the young (meta-mathematics) from mathematics itself, concept of logical inference and remains non- Suppose the position of mathematicians genius of mathematical science had the follow- even if the former is presented mathematically. adequate also after the formalization in the in relation to self-reflexive expressions can be ing psychological trait. From the school years, Taking this differentiation for granted, the con- frame of propositional calculus. It is rigorous- understood. But how to go on with the phi- his each opinion the young genius considered struction of Gödel's undecidable formula with- ly proved that the propositional calculus is a losophers and methodologists of science who the last truth, not subject to revision. His in the framework of formalized meta-mathe- consistent and complete formalized system. have learned from mathematicians that there brother Rudolf Gödel mentioned that young matics will only demonstrate the incomplete- Yet the paradox of material implication is al- are used highly questionable self-reflexive Kurt had “a very individual and fixed opinion ness of formalized meta-mathematics, but not so well known. expressions in the grandiose building of mod- about everything and could hardly be con- of the incompleteness of mathematics itself. Another example of a dubious instrument ern science? vinced otherwise. Unfortunately he believed We again came to the sacramental ques- of argumentation is presented by self-reflexive 2.7. Could the best logician of the twen- all his life that he was always right not only in tion “Does Gödel’s formula belong to mathe- expressions. In the context of this paper the tieth century make a simple logical mistake? mathematics” (O'Connor & Robertson, 2003). matics or to the theory of proof?” As was most problematic instance of self-reflexivity is In the context of this article we have in mind Unconventional progress in the study of proved above, the mathematization of any the- Gödel’s numbering. This self-reflexive instru- the question, “Could Gödel miss that his for- school subjects could not but strengthen this ory does not in any way change its content. ment was used in formalizing the system of mula, with its meaningful interpretation, does trait of the character of the young Gödel. The This conclusion is even more convincing in the metamathematics in his famous article on the not express a mathematical statement?” After most difficult tests of school education have case of Gödel’s formula. As Kleene explains, incompleteness of formalized mathematics. All all, the interpretation of Gödel's formula G always been associated with the assimilation of when interpreting the Gödel formula, it means mathematicians knew about the affinity of self- means only one thing: “G asserts that it is un- Latin grammar, and young Kurt managed to a statement that “establishes its unprovability” reflexive expressions with the Liar paradox; provable”. It does not take much mental effort cope unerringly with this difficult matter. Ru- (Klini, 1952, pp. 205; 207). Another interpreta- nevertheless, they in the overwhelming majori- to notice that this statement does not contain dolf Gödel had recalled later: “Even in High

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School my brother was somewhat more one- cept of truth in a system of formalized arithme- sided than me and to the astonishment of his tic constructed using the predicate calculus of teachers and fellow pupils had mastered uni- the first order is not definable. However, the versity mathematics by his final Gymnasium concept of truth of first-order formalized years. ... Mathematics and languages ranked arithmetic can be defined in formalized arith- well above literature and history. At the time it metic constructed by means of the predicate was rumoured that in the whole of his time at calculus of the second order, etc. Actually, we High School not only was his work in Latin deal with a close analog of B. Russell’s hierar- always given the top marks but that he had chy of types (Bolander, 2017). made not a single grammatical error.” (O’Con- Here we digress from the essence of Tar- nor & Robertson, 2003). ski's theorem and confine ourselves to pointing 2.8. The paradox of Gödel’s numbering. out Tarski's those ideas that we need to Every reader familiar with the Liar paradox “L demonstrate the fundamental assertion that the claims that L is false”, on the first reading of use of Gödel’s numbering makes it possible to the meaningful interpretation of Gödel’s for- build a paradoxical (self-contradictory) formu- mula “G asserts that G is unprovable,” notices la in a system of formalized arithmetic. First, the close relationship of Gödel’s formula with in the very definition of his system, Tarski di- the Liar paradox. Therefore, many researchers rectly indicates that one of its main elements is have a natural question, if can Gödel's self- Gödel's numbering. Secondly, in his system of referential formula also lead to a paradox? formalized arithmetic, he uses the predicate “to Mathematicians reassure doubters that alt- be true”. Third, his proof of the theorem is hough the relationship to the paradox is close, based on the use of reductio ad absurdum, but Gödel's formula does not raise the paradox, which makes it absolutely unessential to resort since here it is a matter of unprovability, and to the complex tool of recursive functions. not falsity. Tarski’s theorem on the undefinability of It would seem that the question of the truth directly concerns only a formalized theo- paradox of Gödel's formula is in principle re- ry. But Gödel’s numbering can be used outside moved by Tarski's theorem. Alfred Tarski in formalized theories too. This means that using 1936 proved that the arithmetical truth is inde- the predicate “to be true” instead of the predi- finable in the system of formalized arithmetic cate “to be a proof”, we can use Gödel’s num- (Tarski, 1936; Tarski, 1983). It would seem bering to build in any system (including the that mathematicians should have rejoiced after formalized mathematics) the formula GT, proving a theorem on the indefinability of which is a complete analogue of Gödel's for- truth. Indeed, if the truth is indefinable in the mula G. For realizing this idea, it is sufficient formalized theory, then it means that within to replace the predicate “to be a proof” by the the framework of this system there cannot arise predicate “to be true” in all steps of construct- either contradictions, or paradoxes. Tarski ing Gödel's undecidable formula G. In the re- showed also that the concept of truth in this sult of these procedures we get the formula GT formalized theory can be determined by means which means “The proposition GT states that it of a richer formalized theory. Thus, the con- is false.” And then it is not difficult to come to

26 WISDOM 2(9), 2017 26 The Paradox of Gödel’s Numbering and the Philosophy of Modern Metamathematics

School my brother was somewhat more one- cept of truth in a system of formalized arithme- the paradox: “The proposition GT is both true damental means of set theory – the axiom of sided than me and to the astonishment of his tic constructed using the predicate calculus of and false.” choice and the continuum hypothesis. The teachers and fellow pupils had mastered uni- the first order is not definable. However, the Indeed, let us turn to the help of the meth- above demonstration of the inconsistency versity mathematics by his final Gymnasium concept of truth of first-order formalized od reductio ad absurdum. emerging from the use of Gödel’s numbering years. ... Mathematics and languages ranked arithmetic can be defined in formalized arith- First, let us suppose that the proposition in no way can reduce the merits of the great well above literature and history. At the time it metic constructed by means of the predicate GT is true. Then, according to its content, the logician and mathematician in the history of was rumoured that in the whole of his time at calculus of the second order, etc. Actually, we proposition GT must be false. the twentieth-century science. High School not only was his work in Latin deal with a close analog of B. Russell’s hierar- Now suppose the converse that the propo- always given the top marks but that he had chy of types (Bolander, 2017). sition GT is false. This will mean “double ne- REFERENCES made not a single grammatical error.” (O’Con- Here we digress from the essence of Tar- gation” – the falsity of asserting the falsity of nor & Robertson, 2003). ski's theorem and confine ourselves to pointing the proposition GT. And then, according to the Bolander, T. (2017, August 31). Self-Refe- 2.8. The paradox of Gödel’s numbering. out Tarski's those ideas that we need to law of double negation, we have to conclude rence. Retrieved November 20, 2017, Every reader familiar with the Liar paradox “L demonstrate the fundamental assertion that the that the proposition GT must be true. from Stanford Encyclopedia of Philo- claims that L is false”, on the first reading of use of Gödel’s numbering makes it possible to We come to a paradox, a demonstration of sophy: the meaningful interpretation of Gödel’s for- build a paradoxical (self-contradictory) formu- the self-contradiction of the proposition GT. https://plato.stanford.edu/entries/self- mula “G asserts that G is unprovable,” notices la in a system of formalized arithmetic. First, The paradox of Gödel's numbering demon- reference/. the close relationship of Gödel’s formula with in the very definition of his system, Tarski di- strates the inconsistency of any system (includ- Feferman, S. (2011). Gödel’s Incompleteness the Liar paradox. Therefore, many researchers rectly indicates that one of its main elements is ing formalized mathematics) which allows us- Theorems, Free Will and Mathemati- have a natural question, if can Gödel's self- Gödel's numbering. Secondly, in his system of ing Gödel's numbering. cal Thought. (R. Swinburne, Ed.) referential formula also lead to a paradox? formalized arithmetic, he uses the predicate “to Free Will and Modern Science, 102– Mathematicians reassure doubters that alt- be true”. Third, his proof of the theorem is Conclusions 122. hough the relationship to the paradox is close, based on the use of reductio ad absurdum, Franzén, T. (2005). Gödel's Theorem: An In- but Gödel's formula does not raise the paradox, which makes it absolutely unessential to resort Gödel’s proof of his incompleteness the- complete Guide to its Use and Abuse. since here it is a matter of unprovability, and to the complex tool of recursive functions. orem is not valid, since in actuality it does not Wellesley: A. K. Peters. not falsity. Tarski’s theorem on the undefinability of demonstrate the incompleteness of the Prin- Gentzen, G. (1935). Untersuchungen über das It would seem that the question of the truth directly concerns only a formalized theo- cipia Mathematica and the related sufficiently logische Schließen, I & II. Mathe- paradox of Gödel's formula is in principle re- ry. But Gödel’s numbering can be used outside rich systems. matischeZeitschrift, 39, 176–210, moved by Tarski's theorem. Alfred Tarski in formalized theories too. This means that using If it is allowed to use Gödel’s numbering 405–431. 1936 proved that the arithmetical truth is inde- the predicate “to be true” instead of the predi- in a system (including formalized axiomatic Glanzberg, M. (2013, January 22).Truth. Ret- finable in the system of formalized arithmetic cate “to be a proof”, we can use Gödel’s num- theories) then in this system one can easily rieved November 22, 2017, from (Tarski, 1936; Tarski, 1983). It would seem bering to build in any system (including the build a Liar type paradox demonstrating the Stanford Encyclopedia of Philosophy: that mathematicians should have rejoiced after formalized mathematics) the formula GT, inconsistency of the system. https://plato.stanford.edu/archives/wi proving a theorem on the indefinability of which is a complete analogue of Gödel's for- The mathematical genius of Kurt Gödel is n2016/entries/truth/. truth. Indeed, if the truth is indefinable in the mula G. For realizing this idea, it is sufficient unquestionable. Grandiose proofs of the theo- Gödel, K. (1929). Über die Vollständigkeit des formalized theory, then it means that within to replace the predicate “to be a proof” by the rem on the completeness of the first order Logikkalküls (Doctoral dissertation). the framework of this system there cannot arise predicate “to be true” in all steps of construct- predicate calculus and theorems on incom- University оf Vienna. either contradictions, or paradoxes. Tarski ing Gödel's undecidable formula G. In the re- pleteness of formalized arithmetic, he received Gödel, K. (1930). Die Vollständigkeit der Ax- showed also that the concept of truth in this sult of these procedures we get the formula GT at the age of up to 25 years. A few years later iome des logischen Funktionenkal- formalized theory can be determined by means which means “The proposition GT states that it he received no less significant results, proving küls. Monatshefte für Mathematik, of a richer formalized theory. Thus, the con- is false.” And then it is not difficult to come to the consistency of systems using the most fun-

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