GODEL’S INCOMPLETENESS THEOREM – Part 1/2

[vishnudaivata]

BRIDGE

Dedicated to: Dr. T. D. Singh

Center for BHAKTI Studies

and Sri Ranganatha Gaudiya Matha

Monthly e-news letter, Volume: Jan/2007

 

 

Contents

 

Message from Srila Gurudev

Page 1

 

Note from Editor

Page 2

 

GODEL’S INCOMPLETENESS THEOREM1 – Part 1/2

Sripad BV Dandi Swami

(Sri Ranganatha Gaudiya Matha, Bangalore, India)

Page 2

 

 

Thanks

Proof reading and editing by Sripad Janardan Prabhuji

 

 

Message from Srila Gurudev

 

All Glories to Sri Sri Guru & Gauranga!

 

Swami B.V. Narayan

Founder Acharya of Bhaktivedanta Trust International [bHAKTI] and International Gaudiya Vedanta Trust

 

Sri Kesavji Gaudiya Math - Mathura [uP] 281001 India - Ph.: + 91565.2502334

 

President of Sri Gaudiya Vedanta Samiti Trust and Vice President of Sri Gaudiya Vedanta Samiti

 

My dear devotees and friends,

 

My heartly blessings are for you. All glories to Sri Sri Guru and Gauranga, all glories to Sri Sri Radha Vinode Bihariji.

 

I became very happy that a news-letter is provided for all who are participating and assisting with the Center for BHAKTI Studies. I pray to our Guruvarga, to Sri Sri Gaura Nityananda Prabhu and Sri Sri Radha Krsna Yugal to bestow Their causeless mercy upon all of you that your desire to serve Hari, Guru and Vaisnavas will increase by participating in the programs that are offered by the Center for BHAKTI Studies.

 

 

 

Your ever well-wisher,

 

Swami B.V. Narayan

 

 

Message from the Editor

 

Hare Krishna and Dandvats Dear All:

 

All glories to Srila Gurudev and Srila Prabhupada.

 

Welcome to the third edition of BRIDGE a monthly e-news letter from Center of BHAKTI Studies, Harrisburg, and Sri Ranganatha Gaudiya Matha, Bangalore, India.

 

 

Begging your mercy & blessings,

Editor

 

 

GODEL’S INCOMPLETENESS THEOREM1 – Part 1/2

 

Are people just complicated robots… or do we have souls? On the one hand, being conscious certainly feels like something more than the mechanical working out of a computer program. But on the other hand, what could a soul possibly be? How could it act on matter? Might a machine have a soul?

 

Kurt Godel was unquestionably the greatest logician of the century. He may also have been one of our western greatest philosophers. A theorem of mathematical logic, Godel’s Incompleteness Theorem, is widely thought to have a bearing on this constellation of problems. In this and subsequent issues we will begin by investigating famous Godel’s theorem and its implications.

 

Background1: Before Godel came up with his theorem in the summer of 1930, the thinkers of the Industrial Revolution liked to regard the universe as a vast pre programmed machine. It was optimistically predicted that soon scientists would know all the rules, all the programs. Logical positivism was being founded and developed by a group of philosophers known as the “Vienna Circle”. The Vienna Circle was developing a program to unify all of science, using the language of symbolic logic. The inspiration came from Whitehead and Russell’s Principia Mathematica of 1910. In this monumental work (three mammoth volumes), it is shown how all of our familiar mathematical concepts and facts can be logically derived from certain very simple and primitive principles of reasoning. The logical positivists hoped to put other branches of science, including physics and psychology, on a similarly rigorous footing.

 

The principal achievement of Whitehead and Russell was to get a mechanically precise definition of what we mean when we say that some statement follows logically from other statements. With this definition in hand, the “formalist” mathematician David Hilbert pointed out that mathematics was now only a matter of choosing the right axioms and examining the logical consequences of these axioms. The positivists hoped to extend this approach to all the sciences, and even to all of human thought.

 

In 1930 Kurt Godel proved that there can never be a UTM (Universal Truth Machine). There can’t even be an MTM (Mathematical Truth machine). There is no complete set of axioms for mathematical truth. Any system of knowledge about the world is, and must remain, fundamentally incomplete, eternally subject to revision. There can never be a final, best system of mathematics. Every axiom-system for mathematics will eventually run into certain simple problems that it cannot solve at all. This is Godel’s Incompleteness Theorem. If Godel’s Theorem tells us anything, it is this: Man will never know the final secret of the universe. Of course, anyone can say that science does not have all the answers. What makes Godel’s achievement so remarkable is that he could rigorously prove this, stating his proof in the utterly precise language of symbolic logic. His basic procedure is as follows:

Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.

1. Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.

2. Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."

3. Now Gödel laughs his high laugh and asks UTM whether G is true or not.

4. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.

5. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").

6. "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

Think about it - it grows on you.

With his great mathematical and logical genius, Gödel was able to find a way (for any given P (UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics.

Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth. A bit more precisely, the Incompleteness Theorem shows that human beings can never formulate a correct and complete description of the set of natural numbers, {0,1,2,3,…}. But if mathematics cannot ever fully understand something as simple as number theory, then it is certainly too much to expect that science will ever expose any ultimate secret of the universe.

But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.

For Godel the mathematical entities exist independently of the activities of mathematicians, in much the same way that the stars would be there even if there were no astronomers to look at them. For Godel, mathematics, even the mathematics of the infinite, was an essentially empirical science. Godel’s feeling that there is a nonmaterial component to human consciousness. The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute.

The concept that emerges out of Godel’s theorem is: No finite system can generate arbitrarily complex patterns, No finite system can understand everything, and No finite system can define Truth5.

Kurt Godel's achievement in modern logic is singular and monumental: indeed it is more than a monument, it is a landmark, which will remain visible far in space and time... The subject of logic has certainly completely changed its nature and possibilities with Godel's achievement.

Overcoming Godel’s incomplete and inconsistency and It’s Extension:

Before we do that consider Berry’s2 paradox forerunner to Godel’s theorem.

Sets are defined by the unique properties of their elements. One may not mention sets and elements simultaneously, but one notion has no meaning without other. The widely used Peano's notation

A = {x: x has property P}

incorporates all the pertinent attributes: a set A, a property P, elements x. But, of course, one does not always use the formal notations. For example, it's quite acceptable to talk of the set of all students at the East Brunswick High or the set of fingers I use to type this sentence. The space being limited, some sets are described on this page and some are not. Let's call russell the set of all sets described on this page. Just driving the point in: russell's elements are sets described on this page. Note that this page is where you met russell. For it's where it was defined after all. So russell has an interesting property of being its own element: russell russell.

With the example of russell it's apparent that some sets contain themselves as elements while others do not. Let RUSSELL stand for the set of all sets that are not their own elements. What may be said about RUSSELL? Which is it?

Assuming RUSSELL RUSSELL leads to a contradiction for, by definition, RUSSELL does not contain itself. Assuming RUSSELL RUSSELL implies that RUSSELL satisfies the definition and, hence, RUSSELL RUSSELL. Impossibility.

That RUSSELL is such a set that neither RUSSELL RUSSELL nor RUSSELL RUSSELL has been discovered by Bertrand Russell (1872-1970) in 1901.

Krsna = A= {x: x has property P such that it has all possibilities –belonging to Krsna)

Illustration: Story of 1. The pastime of stealing of calf and friends:

Story of Brahma Stealing the Cowherd Boys

“After killing Aghasura, Krishna, along with his associates the cowherd boys, went for a picnic within the forest. The calves, being allured by green grasses, gradually went far away, and therefore Krishna’s associates became a little agitated and wanted to bring back the calves. Krishna, however encouraged the boys by saying. ‘You take your tiffin without being agitated. I shall go find the calves.’ Thus the Lord departed. Then, just to examine the potency of Krishna, Lord Brahma took away all the calves and cowherd boys and kept them in a secluded place.

“When Krishna was unable to find the calves and boys, He could understand that this was a trick performed by Brahma. Then the Supreme Personality of Godhead, the cause of all causes, in order to please Lord Brahma, as well as His own associates and their mothers, expanded Himself to become the calves and boys, exactly as they were before. In this way, He discovered another pastime. A special feature of this pastime was that the mothers of the cowherd boys thus became more attached to their respective sons, and the cows became more attached to their calves. After nearly a year, Baladeva observed that all the cowherd boys and calves were expansions of Krishna. Thus He inquired from Krishna and was informed of what had happened.”

“When one full year had passed, Brahma returned and saw that Krishna was still engaged as usual with His friends and the calves and cows. Then Krishna exhibited all the calves and cowherd boys as four-armed forms of Narayana. Brahma could then understand Krishna’s potency, and he was astonished by the pastimes of Krishna. His worshipable Lord. Krishna, however, bestowed His causeless mercy upon Brahma and released him from illusion. Thus Brahma began to offer prayers to glorify the Supreme Personality of Godhead” (Srimad Bhagavatam 10.13, chapter summary).

Lord Brahma himself was astonished. How is that same cowheard boys are sleeping and at the same time playing with Krsna. This is similar to a statement, which is simultaneously true and also false, and much more.

Similarly many paradoxes may appear to conditioned souls. But as much as they appear as riddles impossible or beyond logic and comphrension of human mind yet they are insignificant to the one connected with Supreme Lord

Like infinity+ infinity=infinity-infinity

In the words Of Srila Sridhar Maharaj: Not only big and small, but even good and bad, even anti-parties, everything is harmonized in God. Good-bad, friend-enemy, everything is harmonized and accommodated there. And they lose their poison; all become good. He is all accommodating, all harmonizing, all-adjusting principle, both directly and indirectly. Thesis and antithesis find their highest synthesis in the Krsishna conception of divinity.

Let Krsna krsna: Krsna includes all, includes himself. Krsna –krsna =krsna, and krsna+krsna=krsna

Then krsna krsna and krsna krsna { In the above story, Sri Krsna, and His friends and calves certainly (many) Krsna’s from Krsna ‘s point, however, for Lord Brahma it appeared like ordinary boys and calves until Lord Sri Krsna revealed to Lord Brahma that they were all Sri Vishnu’s expansions of Sri Krsna.} Therefore no contradiction.

The nature of Lord Sri Krsna as described in Sri Brahma Samhita is follows:

I worship Govinda, the primeval Lord, whose transcendental form is full of bliss, truth, substantiality and is thus full of the most dazzling splendor. Each of the limbs of that transcendental figure possesses in Himself, the full-fledged functions of all the organs, and eternally sees, maintains and manifests the infinite universes, both spiritual and mundane. SBS 32

I worship Govinda, the primeval Lord, who is inaccessible to the Vedas, but obtainable by pure unalloyed devotion of the soul, who is without a second, who is not subject to decay, is without a beginning, whose form is endless, who is the beginning, and the eternal purus?a; yet He is a person possessing the beauty of blooming youth. SBS 33

I worship Govinda, the primeval Lord, who is Syamasundara, Kr?s?n?a Himself with inconceivable innumerable attributes, whom the pure devotees see in their heart of hearts with the eye of devotion tinged with the salve of love. SBS 38

He is an undifferentiated entity as there is no distinction between potency and the possessor thereof. In His work of creation of millions of worlds, His potency remains inseparable. All the universes exist in Him and He is present in His fullness in every one of the atoms that are scattered throughout the universe, at one and the same time. Such is the primeval Lord whom I adore. SBS 35

I worship Govinda, the primeval Lord, whose effulgence is the source of the nondifferentiated Brahman mentioned in the Upanis?ads, being differentiated from the infinity of glories of the mundane universe appears as the indivisible, infinite, limitless, truth. SBS 40

I worship Govinda, the primeval Lord, who is the absolute substantive principle being the ultimate entity in the form of the support of all existence whose external potency embodies the threefold mundane qualities, viz., sattva, rajas, and tamas and diffuses the Vedic knowledge regarding the mundane world. SBS 41

The light of one candle being communicated to other candles, although it burns separately in them, is the same in its quality. I adore the primeval Lord Govinda who exhibits Himself equally in the same mobile manner in His various manifestations. SBS 46

I adore the primeval Lord Govinda from whom the separated subjective portion Brahma receives his power for the regulation of the mundane world, just as the sun manifests some portion of his own light in all the effulgent gems that bear the names of suryakanta, etc. SBS 49

The three worlds are composed of the nine elements, viz., fire, earth, ether, water, air, direction, time, soul and mind. I adore the primeval Lord Govinda from whom they originate, in whom they exist and into whom they enter at the time of the universal cataclysm. SBS 51

The sun who is the king of all the planets, full of infinite effulgence, the image of the good soul, is as the eye of this world. I adore the primeval Lord Govinda in pursuance of whose order the sun performs his journey mounting the wheel of time. SBS 52

I adore the primeval Lord Govinda, by whose conferred power are maintained the manifested potencies, that are found to exist, of all virtues, all vices, the Vedas, the penances and all jivas, from Brahma to the meanest insect. SBS 53

"Listen, O Vidhi, I am the seed, i.e., the fundamental principle, of this world of animate and inanimate objects. I am pradhana (the substance of matter), I am prakr?ti (material cause) and I am purus?a (efficient cause). This fiery energy that belongs specially to the Brahman, that inheres in you, has also been conferred by Me. It is by bearing this fiery energy that you regulate this phenomenal world of animate and inanimate objects." SBS 62

He who has the power to do anything, to undo anything, or to change anything into anything else is Isvara. Although He is the doer, He is nonetheless the non doer. Paramatma sandarbha 93

The potency that makes the inconceivable conceivable and the impossible possible. BR 1.1 pt

Everything animate or inanimate that is within the universe is controlled and owned by the Lord. Isopanishad 1

The Absolute Truth is indeed one without a second Chandogya Upanishad 6.2.1

By Me, in My unmanifested form, this entire universe is pervaded. All beings are in Me, but I am not in them. SBG 9.4

Russell’s Antinomy:

Let be the set of all sets, which are not members of themselves. Then is neither a member of itself nor not a member of itself. Symbolically, let . Then iff .

Bertrand Russell discovered this paradox and sent it in a letter to G. Frege just as Frege was completing Grundlagen der Arithmetik. This invalidated much of the rigor of the work, and Frege was forced to add a note at the end stating, "A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press."

Barber Paradox

A man of Seville is shaved by the Barber of Seville if the man does not shave himself. Does the barber shave himself? This pseudo paradox was proposed by Bertrand Russell.

If the answer is yes, then barber is cutting the hair of someone who does cut their own hair. If the answer is no, then barber is not cutting the hair of someone who does not cut their own hair. Both of these are contradictions and that is the paradox - the answer is neither yes nor no.

The one way to resolve (similar to Sri Krsna’s pastime where Lord Brahma steals boys and calves): Two identical barbers are there. One barber from Seville shaves everybody that they do not shave themselves. Second barber from Spiritual World looking just like one from Seville shaves the first barber and so the first barber does not violate the logic. The second barber then leaves without needing a shave.(as he is eternally youth capable of assuming any shape as source of everything).

An example in the material world that appears to fulfill is the nature of light. Is light’s nature is wave? Or Particle? Earlier it was thought to be particles (corpuscular theory), later waves (Huygens wave theory) and again particle theory (quanta by Einstein). Now they have experiment to prove that light behaves as simultaneously particle and wave.

It also like in computer, one can open multiple copies of word or internet explorer windows. Although multiple instances exist on the desktop, of the same file – one may work with one of them independently without affecting the other. Of course in this sense we are only making inert duplicates in a limited sense. What is extraordinary is that the Supreme Lord can accomplish this with consciousness.

…Contd Feb 2007….GODEL’S INCOMPLETENESS THEOREM Part 2/2

 

 

? Nectar of Instruction by Srimati Syamarani didi

? Sanskrit classes in English (& Sanskrit) by Vishnu Daivata Maharaj

? Workshops on Ayurveda by Dr. Janardan (assisted by BV Dandi Maharaj)

? Sril Gurudev’s live lectures Feb 25 – March 3, Gaura – Mandala Parikrama, Navadvip, WB, India

 

 

 

 

1 The infinity and the mind by Rudy Rucker, 2005 edition, Chapter- Robots and Souls

2Berry’s paradox Russell's ParadoxPoincaré disliked Peano's work on a formal language for mathematics, then called "logistic." He wrote of Russell's paradox, with evident satisfaction, "Logistic has finally proved that it is not sterile. At last it has given birth - to a contradiction." from R.Hersh, What is Mathematics, Really? Oxford University Press, 1997

3 Galileo Galilei, Two New Sciences New York Macmillan, 1914 p.26

4 George Cantor, Gesammelte Abbandlulgen, p. 204

5. Infinity and the mind chapter unnameable Rudy Rucker

6. Jaiva Dharma Chapter 13 Dasa mula tattva