Gödel's Theorem and its Implications

Abstract

Gödel's theorem, to be specific Gödel's incompleteness theorem, was proved and published as "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" in 1931 by an Austrian mathematician, Kurt Gödel. The "Principia Mathematica" (PM) authored in three volumes by Russell and Whitehead is a huge work that intends to reduce mathematics to logic. The PM system, however, leaves some of the statements within the system as unprovable. Gödel establishes that there are propositions which are neither provable nor disprovable within the system, but are necessarily true (true without being proved so!). Even though Gödel's theorem specifically addresses the arithmetic system, it is extendable to all kinds of axiomatic systems- to physical universe as a whole as conceived by modern science. Gödel 's incompleteness theorem includes two theorems as given below: Assuming that the axiomatic system A is arithmetically consistent, it is possible to construct a sentence G which is a true but is not provable in A. In other words, given any consistent arithmetic system, there always exists an expression which is true, but not provable within that system. The consistency of an arithmetical system cannot be proved within that system. The implications of both these theorems are far reaching. According to the first theorem, there is always a sentence G (known as Gödel sentence) which is true but neither provable nor disprovable. This implies that the system always remains incomplete. What makes the Gödel sentence significant is that even though not provable within the system, by using its own tools, it can be intuitively known to be true since the denial of the Gödel sentence will produce a contradiction. This implies that human intuitive capacity is alien to mathematical/ logical systems and hence no replica of human, with all its capabilities, can be re-constructed artificially. Gödel 's findings are applicable not only to mathematical systems but also to all rational and scientific systems that are axiomatic in nature.

Headnote

Abstract

Gödel's theorem, to be specific Gödel's incompleteness theorem, was proved and published as "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" in 1931 by an Austrian mathematician, Kurt Gödel. The "Principia Mathematica" (PM) authored in three volumes by Russell and Whitehead is a huge work that intends to reduce mathematics to logic. The PM system, however, leaves some of the statements within the system as unprovable. Gödel establishes that there are propositions which are neither provable nor disprovable within the system, but are necessarily true (true without being proved so!). Even though Gödel's theorem specifically addresses the arithmetic system, it is extendable to all kinds of axiomatic systems- to physical universe as a whole as conceived by modern science.

Gödel 's incompleteness theorem includes two theorems as given below:

Theorem 1: Assuming that the axiomatic system A is arithmetically consistent, it is possible to construct a sentence G which is a true but is not provable in A. In other words, given any consistent arithmetic system, there always exists an expression which is true, but not provable within that system.

Theorem 2: The consistency of an arithmetical system cannot be proved within that system.

The implications of both these theorems are far reaching. According to the first theorem, there is always a sentence G (known as Gödel sentence) which is true but neither provable nor disprovable. This implies that the system always remains incomplete. What makes the Gödel sentence significant is that even though not provable within the system, by using its own tools, it can be intuitively known to be true since the denial of the Gödel sentence will produce a contradiction. This implies that human intuitive capacity is alien to mathematical/ logical systems and hence no replica of human, with all its capabilities, can be re-constructed artificially. Gödel 's findings are applicable not only to mathematical systems but also to all rational and scientific systems that are axiomatic in nature.

Key Words: Gödel's theorem, Axiomatic systems, Consistency, Completeness, Gödel sentence

Introduction

The analytic tum in Philosophy came into being in the beginning of 20th Century as a revolt against the idealistic metaphysics, especially of German traditions. One of the earliest forms of this movement was early realism and analysis, which abandoned the speculative metaphysics and adopted a realist view of the world. This stage of development, contributed mainly by GE Moore and Bertrand Russell, is closely related to the stage succeeded, which is aimed at the construction of formal language or the ideal language. The movement of ideal language Philosophy can be traced back to its origin in the German philosopher Gottlob Frege, who worked at the area overlapping philosophy and mathematics.

Further development of the ideal language philosophy was mainly due to the contributions of Russell and the early Wittgenstein. Both of them were the advocators of logical atomism1, even though they slightly differ in their approaches. Logical atomism as a programme divides the world into logical simples called logical atoms. These logical atoms exist independent of each other, but can stand in relation with each other. The independent existence with the possibility of coming into relationship with each other makes it possible to express the world in language. In the representation of the world in language, the words correspond to the logical simples and propositions to the states of affairs (Wittgenstein, 1974, 2.0272; 4.026 ). The isomorphism11 of thought, language and reality is a necessary presupposition of logical atomism. The programme of ideal language philosophy is to show that the problems in philosophy, that are essentially caused by the incorrect use of language, can be resolved/ dissolved through the correct/ precise use of language.

Mathematics and Truth

Many philosophers see mathematical truths as necessary truths. They are a priori and analytic. Hume held the view that mathematics deals with the relations of ideas. Mathematics is expected to give true and certain knowledge. This is possible due to the reason that, mathematics studies the necessary interrelationship between ideas. This makes mathematics a coherent system possibly free of errors.

However, the relationship between mathematical truths and empirical truths becomes problematic. How can we establish a relationship between the two? How does the mathematical truth related to the empirical truth and how can the mathematical proof be related to empirical reality? Newton's title, The Mathematical Principles of Natural Philosophy and Galileo's dictum that "The book of nature is written in the language of mathematics"" give us hope to find such a relationship. Considering mathematics as the language of nature and considering the isomorphism of language and reality, we may assume that the nature shares the structure of mathematical reality. If so, the truth in mathematics ought to correspond with the empirical truth, of course, at the structural level. The isomorphism of thought and language on the one hand and the notion that mathematics is a form of language on the other, lead us to evoke the possible structural relationship between thought and mathematics. The structure of thought, then, must govern the possibility in mathematics. As we know, the study of correctness of the structure of thought, the study that differentiates between correct and incorrect reasoning is taken up by Logic. The fact that the structure of thought is central to mathematics leads to the possibility of mathematical principles being grounded in the principles of logic.

Reality matters in the sciences; it has no place in mathematics. When a mathematician deduces a new statement, that new statement is true provided that no errors were made in the logic that connected premise and conclusion. In mathematics, logic alone determines what is true from what is false (Tabak, 2011, p. 8)iv.

Logical Foundations of Mathematics

The relationship between logic and mathematics, however, was not established during the dawn of analytical movement in philosophy. The first attempt to explain the logical foundations of mathematics was made by Frege in his two volume book The Foundations of Arithmetic, of which the first one appeared in 1884. The effort did not lead to much success.

... focused on logic and the foundations of mathematics. Their [Concept-Script and The Foundations of Arithmetic] aims were (i) to set out a formalized language and proof procedure sufficient for mathematics, and (ii) to derive arithmetic from the axioms of, and definitions available in, this system-and thereby to provide a logical basis for all of mathematics. Although the degree to which Frege achieved (ii) is a matter of continuing debate, the degree to which he achieved (i) is not (Soames, 2010, p. 7).

The next attempt to base mathematics on the logical foundations is done by Alfred North Whitehead and Bertrand Russell through their celebrated, incomplete, three volumes work Principia Mathematica, published in 1910-13. "The formal system laid out in Principia Mathematica was sufficient for expressing all the truths of arithmetic; it was also, presumably, consistent" (Goldstein, 2005, p. 144). Even though the attempt was successful to a great extent, it has been identified that their system was incomplete due to its inability to accommodate self-referential statements that embodies liar paradox. The specific paradox in this context, known as Russell's paradox, goes like this:

Let S be the set of all sets that are not members of themselves. The question then is "Is S a member of itself?" If S is an element of S, then S is a member of itself and should not be in S. If S is not an element of S, then S is not a member of itself, and should be in S. this leads to no solution to the problem posed.

Russell approached this problem using his theory of types. According to this theory, each set is a set of a distinct types. Accordingly, the members of Type A sets can then only be the members of a set of Type B since sets of Type B, by definition, can contain only sets of Type A. thus, there exists no question of a set being a member of itself. The result, however, was counter intuitive and dissatisfactory.

... there are other problems, which afflict even this modified theory. Once individuals and functions (or sets) are divided into exclusive types, there has to be a separate, though isomorphic, arithmetic for each type, an idea that is highly counterintuitive. Furthermore the validity of standard arithmetic as applied to individuals requires the assumption that there is an infinity of such individuals (Baldwin, 2001, p. 36).

Formalisation of Mathematics

In 1920s, a new approach to mathematics, known as metamathematics or metalogic, became influential among mathematicians. David Hilbert (1862-1943) was the chief exponent of this approach. This programme of formalization of mathematics was intended to solve the foundational issue in mathematics raised by set theory. Hilbert took up the task through axiomatisation of mathematics.

The axiomatic system consists of accepted presumptions (axioms or postulates) and theorems that are deduced from those axioms. This deduction of theorem from axioms are to be done within an axiomatic system with rigour, by purely depending on the principles of logic. Since the elements of the system depends not on any empirical data but on logic alone, and theorems are deduced rigorously, the system is of general nature and can be used for any interpretations wherein all the axioms are true. Once the formalisation of mathematical system is complete, they belong to what Hilbert calls meta-mathematics (Nagel, 2001, p. 27).

There are two important characteristics of an axiomatic system, namely, consistency and completeness. Consistency remains the most important feature among these. A deductive system is said to be consistent if it contains no formula such that both the formula and its negation are provable as theorems within it (Copi I. M., 1996, p. 164). The importance of consistency arises from the fact that in an inconsistent system, a theorem and its negation can both be proved. With a theorem and its negation in the same system, the system can accommodate contradictions. That makes the system not reliable for any proof, since anything, say 'x' can be proved within that system (Copi I. M., 2002, pp. 375-77). The second important feature of an axiomatic system is completeness. A deductive system can be said to be complete, if all the desired formulas can be proved within the system (Copi I. M., 1996, p. 166). When we consider the fact that an inconsistent system can prove any theorem, whatever it may be, we can conclude that all inconsistent systems are complete. Hence the completeness of a system is worth only if it is consistent. Deductive completeness of a system is important and preferable from practical standpoint, since it can address all the problems within the system.

Hilbert's programme aimed at establishing the consistency of mathematics and its solvable ability.

If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction. This is the basic philosophical position that I consider requisite for mathematics and, in general, for all scientific thinking, understanding and communication^

Hilbert's programme was intended to decide the logical validity or invalidity (the proof for the truth or falsity) of a formula in first order predicate logic, by a mechanical, algorithmic procedure. Because the rules of the formal system are our own creation, we would be able to police them, to examine them as pure signs to see that they did not lead to inconsistency, to contradiction (Yourgrau, 2006, p. 56). He was optimistic and declared "We must know. We shall know" (Urquhart, 2006, p. 309). The axiomatic system, in principle, captures the essence of all deductive systems and are not only applicable to mathematical systems but also to all other formalisms. All scientific systems, especially the physical systems, are considered to be axiomatic and are expected to provide solutions to all physical problems.

Gödel's Incompleteness Theorems

Kurt Gödel, an Austrian mathematician, published a paper entitled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" in 1931, later known as Gödel's incompleteness theorem, challenged the conclusions of Hilbert. He established that there will at least be one formula in every axiomatic system, which is necessarily true and is neither provable nor disprovable within the system. Even though Gödel's theorem specifically addresses the arithmetic system, it is extendable to all kinds of axiomatic systems- even to physical universe as a whole as conceived by modern science.

Gödel's incompleteness theorem includes two theorems; the simplified semantic versions are given below:

First incompleteness theorem: Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.

Second incompleteness theorem: For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself (Raatikainen, 2015).

The first theorem is often called Gödel 's incompleteness theorem and the second one, Gödel 's consistency theorem.vi The implications of both these theorems are far reaching. According to the first theorem, there is always a sentence G (known as Gödel sentence) which is true but neither provable nor disprovable. This implies that that system always remains incomplete. Gödel and later on Turing held that such features are not system specific but is extendable to all formal systems. The second theorem implies that only an inconsistent formal system can prove its own consistency! (Yourgrau, 2006, p. 57).

While Hilbert was attempting to save mathematics from its foundational issues it faces, Gödel showed that such an attempt essentially is in vain. Consistency and completeness remained as two necessary features of an axiomatic system in order for it to be perfect one. While Mathematics is a discipline of perfection, it cannot aim anything less. With the two theorems proposed by Gödel, the hope for a perfect mathematical system is lost for ever, since, either we should have an incomplete system that is consistent or a complete system that is inconsistent. Hermann Weyl described this as Gödel "debacle", the Gödel "catastrophe"™

The method of proof in Gödel's theorem, known as Gödel numbering, is highly complex and is beyond the scope of this paper. However, the general methodology can be described here. Gödel distinguishes between three levels of mathematical theories; intuitive mathematics, formal mathematics and metamathematics (Yourgrau, 2006, pp. 60-61). In an arithmetical system there are corresponding intuitive arithmetic (IA), formal arithmetic (FA) and meta-formal arithmetic (MFA). The intuitive arithmetic contains propositions with content and are either true or false. These are seen in arithmetic text books. When this arithmetic is formalised using symbols and syntaxes, the next higher level of arithmetic system, called formal arithmetic, is formed. The formalised arithmetic does not have any contents within itself, but has the structure that necessarily leads to the proof of the statements in intuitive arithmetic. They are neither true nor false, but are formally correct or incorrect. The meta-formalised arithmetic is statements about formalised arithmetic and are having statements of formalised arithmetic as its content. The statements in meta-formalised arithmetic again are either true or false.

As arithmetisation of geometry done by Descartes, Gödel, through the arithmetisation of mathematics, expressed the theorems of mathematics in numerical form. Using this method, he constructed a formula in FA that does have a simultaneous meaning in IA and MFA. The process of construction of such formulas is known as Gödel numbering. Through Gödel numbering, Gödel devised a formula G that is unprovable in FA but is intuitively true in both IA and MFA.

This Gödel number G says that "G is not provable." The very meaning of G attained through arithmetisation is "G is not provable", and hence it is meaningful in IA and MFA. But in FA, where the proof for G is to be established, if G is proved, it leads to the conclusion that G is not proved, since G means "G is not provable". It is also impossible to prove the negation of G since, if "G is not provable" is disproved, then that amounts to giving a proof for G, which means "G is not provable". This goes against the very proof. Hence, the formula that is intuitively true in IA and MFA is neither provable nor disprovable in FA. If, for any reason, a proof is established, then the system is forced to include a contradiction and will become inconsistent.

Using the same method, Gödel established that, it is not only that a system is incomplete but also that the consistency of any axiomatic system cannot be proved within that system. Hence, if a system is consistent, we will not be able to establish the consistency within the system. The non-establishment of inconsistency of a system within that system, however, doesn't allow us to infer its consistency. This is because, it may be the intellectual limitation from the part of the one who attempts to establish the inconsistency, that the inconsistency could not be established. Paradoxically enough, we will be able to prove the consistency within a system itself, only if it is inconsistent. Hence, as of the consistency of the system, we all are always in dark.

The Gödel sentence G is distinct from the liar paradox, since it speaks of provability rather than truth. Gödel himself was aware of this and he himself differentiates between proof and truth. According to him, it is the non-differentiation of proof from truth, that pushed his theory to controversies and also to the unacceptability among his contemporary mathematicians. Proof, while belongs to a system and established within that system syntactically, without any content, truth is semantic and has content and meaning that lies outside the syntax. Mathematical proof is not reduceable to mechanical (algorithmic) proof. That in a sense leads to the impossibility of deriving semantics from syntax, and brings in the role of intuition in mathematics.

A possible solution to the incompleteness theorem is to consider G0 (the first Gödel number) as a valid additional theorem in the system. This however will generate another Godel number Gi, and if we include Gi in our system, it generates G2 and so on infinitely (Penrose, 1999, pp. 142-43). Hence there is no way to get out of the incompleteness and it may be considered to be intrinsic to axiomatic systems.

Implications of Gödel's Theorems

Since arithmetic is foundational to mathematics, the implications of Gödel's theorem are not limited to arithmetic alone but are extendable to all systems of knowledge that is axiomatic. This includes all science and all realms where reason is foundational. The whole universe, taken as a system governed by strict causal laws, is also axiomatic. The implications of Gödel's theorems are to be looked into from this position too.

Reliability of Mathematics and other Axiomatic Systems: What makes the Gödel sentence significant is that even though not provable within the system, by using its own tools, it can be intuitively known to be true since the denial of the Gödel sentence will produce a contradiction. This implies that human intuitive capacity is alien to mathematical/ logical systems and hence no replica of human intelligence, with all its capabilities, can be re-constructed artificially. This shows that mathematics, and thus all axiomatically built systems need intuition as a key tool not for establishing proof, but for establishing truth. A distinction between the notions of truth and proof is necessary here. While proof is axiomatic and constructed within a system, truth lies outside of the system. It is also implied that syntax cannot fully account for semantics. While syntax is closed and axiomatically governed, semantics is open and nonaxiomatic. Gödel's findings have implications on the reliability of Mathematics as a method of proof. Since the consistency of the system cannot be proved from within the system, mathematics remains unproven for its consistency and hence its reliability is questioned. It is more significant when we take into account that the inconsistency can produce a proof for anything, whatsoever.

.. .it seems to me that it is a clear consequence of the Gödel argument that the concept of mathematical truth cannot be encapsulated in any formalistic scheme. Mathematical truth is something that goes beyond mere formalism. This is perhaps clear even without Gödel's theorem. For how are we to decide what axioms or rules of procedure to adopt in any case when trying to set up a formal system (Penrose, 1999, p. 145)?

Halting Problem and the Limitations of Algorithmic Machines: Alan Turing, in 1936, invented the idea of Turing Machineviii and attempted to find the solution for the halting problem. Halting problem decides whether the computer, for a given algorithm and an input, will halt after making a decision or run for ever. Turing showed that halting problem is undecidable for Turing machines, and Gödel-type propositions would be caught in an infinite loop in the Turing machine (Driessen, 2005, p. 70). This clearly indicates the limitations of algorithmically governed processing that they cannot capture the features of human mind, even if they evolve to any extend in future. This shows the advantage of human mind in capturing mathematical truth with insights, over mechanically governed machines (Urquhart, 2006, p. 316). Yourgrau indicates that as a direct consequence of Gödel's theorem, a fool proof antivirus program becomes impossible in the field of current computer programming. Any program we make, being algorithmic, will have a 'hole' in it and can be exploited to harm the program (68).

Artificial Intelligence and Consciousness: If we subscribe to Ned Block's position of "mind as the software of the brain" (Bolck, 1995, p. 377), we may infer that mind functions algorithmically. If so, then what makes mind intuitive cannot be its algorithmic nature, but something different. This can be viewed in connection with Penrose's position regarding the infinite generation of new Gödel numbers, once we include a Gödel number as new theorem:

Does this ever end? In a sense, no; but it leads us into some difficult mathematical considerations... The critical issue, at each stage, is to see how to code the adjoining of an infinite family of Gödel propositions into providing a single additional axiom (or finite number of axioms). This requires that our infinite family can be systematized in some algorithmic way. To be sure that such a systematization correctly does what it is supposed to do, we shall need to employ insights from outside the system just as we did in order to see that Pk(k) was a true proposition in the first place. It is these insights that cannot be systematized and, indeed, must lie outside any algorithmic action (Penrose, 1999, p. 143)!

The seeing of the truth, Penrose holds that, lies outside the system of axioms, and he sees it as the very essence of consciousness (Penrose, 1999, pp. 540-41). This leads to the impossibility of algorithmic operations being conscious, for all algorithms works axiomatically.

Self-Knowledge and the Limitations of Human Knowledge: The incompleteness of proof within a system leads us to the impossibility of rational acquirement of all possible truths. Rationality, being taken as the paradigm of our understanding, implies that it is impossible to know everything about us rationally. This insufficiency of reason does not, however, leads to the impossibility of self-knowledge. The rational impossibility may be catered with the intuitional possibility. However, a proof for intuitive knowledge is impossible, for proof being always rational. Further, the human knowledge as a whole is limited due to both the problem of incompleteness and inconsistency. The incompleteness leads us to the possibility of generating new knowledge on accomplishing each and every element and the problem of inconsistency puts the validity of all such knowledge, including the rational one in question.

Are We Living in a Simulated Universe? The consistency theory, which holds that, the consistency of an axiomatic system cannot be proved within that system enable us to decide upon the hypothesis that we are living in a simulated universe (David, 2016). This hypothesis provides an interesting thought experiment, which asks us to question the very nature of our reality. But how might we deduce whether or not we're living in such a world? The consistency theorem gives a way out that there is no way to show that we are living a simulated/ illusionary universe. The universe as a whole may be taken as a system. According to Gödel, there is no way to establish the consistency of that system. Hence, we do not and cannot know whether universe is consistent or not. The only possibility is that to stablish it as inconsistency. But if that universe is inconsistent, how can we rely on the proof we derive from it? Hence, the truth about the reality (whether real or simulated) of the universe cannot be known from within.

The Problem of Freewill: The essence of derivation of Gödel's theorem may be taken like this:

1. G cannot be proved in the system S.

2. G can neither be disproved in S.

3. The negation of G in S makes the system inconsistent through the generation of contradiction.

4. Therefore, G must be true.

This method is applicable while exploring the question of freewill. If one asks the question "Is there freewill?", and if she gets the answer "No", then the answer ends up in contradiction. For in order to answer "No", the one who answers contradict herself as the answer presupposes the freedom to answer. Since the attempt to deny freewill ends up in contradiction, as shown by the methodology of Gödel's proof, it is intuitively assumed that freewill exists.

Impossibility of a Theory of Everything: In his article "Gödel and the End of the Universe", Stephen Hawking refers to Gödel to establish that there is no possibility for having a theory of everything as he held earlier. The impossibility of such a theory, he says, will save us from the stagnation of our knowledge.

Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Gödel's theorem ensured there would always be a job for mathematicians (Hawking, 2016).

Gödel's incompleteness theorems have ruptured all axiomatic systems at base. Since the systems are questioned at the fundamental level, and since almost all rational/ scientific systems are axiomatic, the implications of the theorems are unending. The implications are negative as far as the reliability of rational systems is considered. There are also the positive ones, since, the theorem endorses human mind's capacity to know intuitively and the possibility of freewill. However, there lies an uncertainty regarding Gödel's proof. The very proofs for incompleteness and consistency are constructed within an axiomatic system. The incompleteness theorem is true only if the axiomatic system Gödel based upon is consistent. However, as per the consistency theory, the consistency cannot be proved. Isn't that makes the proof susceptible!

Footnote

Notes:

i Logical atomism holds that the meaning of our sentences is rooted in a primitive relation between simple expressions and their simple worldly bearers known as the logical atoms. In a logically perfect language, atomic sentences describe configurations of these atoms, and complex sentences are combinations of the atomic sentences (Oliver, 2005, p. 603).

ii A term, generally used in Chemistry, to indicate crystals having the same crystalline structure.

iii Galileo quoted in Ehahard Behrends (88).

iv This position, however, is not established fully. The very paper is intended to explain Gödel's criticism of this view.

v van Heijenoort quoted in Alasdair Urquhart (308-09).

vi See Driessen (68)

vii Quoted in Palle Yourgrau (58).

viii The Turing machine is the end of any possible evolution with regard to software as well as hardware: with infinite speed and memory, using digital logic, neural networks, quantum logic or any other up to now known or unknown technology for data processing. It is the best what physical laws and the most advanced design can offer with in addition unlimited speed and unlimited memory. With this machine any finite sequence of processing steps could be carried out in an infinitesimally small amount of time (Driessen, 2005, p. 70).

References

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