Tarski, Wittgenstein and Undecidable Sentences in Affine Relation to Gödel’s. Thought of the Day 65.0


I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system.’ Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.” — Wittgenstein

Any language of such a set, say Peano Arithmetic PA (or Russell and Whitehead’s Principia Mathematica, or ZFC), expresses – in a finite, unambiguous, and communicable manner – relations between concepts that are external to the language PA (or to Principia, or to ZFC). Each such language is, thus, essentially two-valued, since a relation either holds or does not hold externally (relative to the language).

Further, a selected, finite, number of primitive formal assertions about a finite set of selected primitive relations of, say, PA are defined as axiomatically PA-provable; all other assertions about relations that can be effectively defined in terms of the primitive relations are termed as PA-provable if, and only if, there is a finite sequence of assertions of PA, each of which is either a primitive assertion, or which can effectively be determined in a finite number of steps as an immediate consequence of any two assertions preceding it in the sequence by a finite set of rules of consequence.

The philosophical dimensions of this emerges if we take M as the standard, arithmetical, interpretation of PA, where:

(a)  the set of non-negative integers is the domain,

(b)  the integer 0 is the interpretation of the symbol “0” of PA,

(c)  the successor operation (addition of 1) is the interpretation of the “ ‘ ” function,

(d)  ordinary addition and multiplication are the interpretations of “+” and “.“,

(e) the interpretation of the predicate letter “=” is the equality relation.

Now, post-Gödel, the standard interpretation of classical theory seems to be that:

(f) PA can, indeed, be interpreted in M;

(g) assertions in M are decidable by Tarski’s definitions of satisfiability and truth;

(h) Tarskian truth and satisfiability are, however, not effectively verifiable in M.

Tarski made clear his indebtedness to Gödel’s methods,

We owe the method used here to Gödel who employed it for other purposes in his recently published work Gödel. This exceedingly important and interesting article is not directly connected with the theme of our work it deals with strictly methodological problems the consistency and completeness of deductive systems, nevertheless we shall be able to use the methods and in part also the results of Gödel’s investigations for our purpose.

On the other hand Tarski strongly emphasized the fact that his results were obtained independently, even though Tarski’s theorem on the undefinability of truth implies the existence of undecidable sentences, and hence Gödel’s first incompleteness theorem. Shifting gears here, how far was the Wittgensteinian quote really close to Gödel’s? However, the question, implicit in Wittgenstein’s argument regarding the possibility of a semantic contradiction in Gödel’s reasoning, then arises: How can we assert that a PA-assertion (whether such an assertion is PA-provable or not) is true under interpretation in M, so long as such truth remains effectively unverifiable in M? Since the issue is not resolved unambiguously by Gödel in his paper (nor, apparently, by subsequent standard interpretations of his formal reasoning and conclusions), Wittgenstein’s quote can be taken to argue that, although we may validly draw various conclusions from Gödel’s formal reasoning and conclusions, the existence of a true or false assertion of M cannot be amongst them.

Is Depeche Mode an Alt-Right Band?

The members of Mode all emerged from fashy signalling New Romantic and avant grade electronic milieu. The band’s first album, mainly written by the synth pop guru and genius Vince Clarke of later Yazoo (Yaz in the U.S) and Erasure fame, launched the band with their first album Speak and Spell.  Politics was not so present on the first album, but was more reflected the band’s name a reference to Fast Fashion and New Romance – a pre-Bret-Easton-Ellis type notion that celebrated the decadent 80s love of surface, fast living, young love, good looks, and high times. But, as soon as Vince Clarke left the band and Martin Gore took over the songwriting slot, they began signalling political ideas of both the Left and Right.

This Left and Right synthesis was both progressive and forward-looking for the era, and really added to the band’s power level, intellectual weight, longevity, and the ability of their work to sound as relevant today as ever. A Broken Frame, their second LP, featured a Neo-Realist folk type cover, reminiscent of both Nazi art and the Communist “Realism” that was favoured by the Stalin and subsequently China and North Korea. The follow up Construction Time Again was an open rebellion to Jacques Derrida’s openly nihilistic and destructive deconstructionism that was all the rage in the 80s intellectual scene. It also featured a fascistic cover of an Aryan man smashing down a hammer. From that image alone the Alt-Right could have been born. Again, the Left and Right symbolism were being mixed together. The album Music for the Masses featured a kind of overarching, fashy motif of a loudspeaker in the wilderness on the cover and an anthem and theme song on the record, Pimpf, given visual expression with the help of the wonderful Anton Corbijn. This was quite openly the most fascist reference in their whole oeuvre. Pimpf was named after a Nazi Youth Movement, and at this time Martin Gore began making his most fashy statements in the media about politics. Gore, the rumour goes, was getting into fascist aesthetics, fashion, and ideas from the mid to late 80s until the early 90s, until he discovered his real father was of mixed race, or something along those lines. Then he went silent on the issue. But he still continued to signal these ideas in his art, albeit in a slightly more diffused and subterranean way. But he was also signalling some left-wing Socialist ideas. With him, it seems, there’s always been a kind of dialectic at play.

Exceprted from and denials here, here against the claim by Richard Spencer.


Prometheus and Hinduism. Note Quote.

Prometheus: Yes, I caused mortals to cease foreseeing their doom.
Chorus: Of what sort was the cure that you found for this affliction?
Prometheus: I caused blind hopes to dwell within their breasts.
Chorus: A great benefit was this you gave to mortals.
Prometheus: In addition, I gave them fire.
Chorus: What! Do creatures of a day now have flame-eyed fire?
Prometheus: Yes, and from it they shall learn many arts.
Aeschylus, Prometheus Bound


After the coming into being of the world, the stories tell us, everything had found its place. But one creature, capable of lofty thought, was still missing. When Kronos ruled Olympus, the deathless gods decided to fashion a golden race of mortal men. Knowing of the divine seed that slumbered in the Earth, only recently separated from the heavenly aether, Prometheus mixed clay and running water. He shaped this into likenesses of the all-controlling gods, including also qualities taken from all the animals. This is how the first humans came into existence. In the course of ages, after Zeus had banished the ancient gods to Tartarus, humans populated the Earth, but lumbered witless as if in a dream. They did not know how to see, hear, or understand, or how to create things with their hands. Prometheus’s empathy led him to steal for them the forbidden divine fire hidden by Zeus, which allowed him to teach them skills and sciences that used all their potentials.

Angered by the theft, Zeus plotted revenge on both humanity and its benefactor. He had Hephaestus, smith of the gods, create a beautiful virgin, Pandora, who was furnished with disastrous gifts by Athena, Aphrodite, and other gods. When a box she was carrying was opened on Earth, all evils and diseases escaped from it and spread among mankind. One single good thing, Hope, had not escaped when she clapped the lid closed. Meanwhile Prometheus was dragged to earth’s remotest wilderness and bound to a rock over a terrifying abyss with chains that could not be undone. Every day an eagle came and ate from his liver, which would regenerate each night. He endured this torment for centuries until the hero Hercules set him free.

This myth calls to mind stories of divine fire brought to mankind in many other traditions.

The allegory of the fire of Prometheus is another version of the rebellion of the proud Lucifer [“light-bringer”], who was hurled down to the bottomless pit, or simply unto our Earth, to live as man. The Hindu Lucifer, the Mahasura, is also said to have become envious of the Creator’s resplendent light, and, at the head of inferior Asuras (not gods, but spirits), to have rebelled against Brahma; for which Siva hurled him down to Patala. But, as philosophy goes hand in hand with allegorical fiction in Hindu myths, the devil is made to repent, and is afforded the oppor­tunity to progress.

Also in Hinduism are the Manasaputras or “sons of mind,” who brought mankind the fire of thought. In the Nordic Edda the name of the god Loki – a blood brother of Odin – comes from the old word liuhan, “to illuminate.”

What then is the inner significance behind these particular stories? Long ago the early human race had undergone a certain amount of evolution but “thinkers” had not yet been born: nature had succeeded in developing a suitable body but the soul-giving principle, the fire of self-conscious thought, had not yet been awakened. Adam and Eve, to a certain extent, existed in paradise without self-awareness. Like Lucifer, Prometheus is an allegorical representation of the incarnation of our higher self, the awakening of the active, self-reflective capacity for thought. This subject is therefore of the highest significance for human evolution.

It is owing to this rebellion of intellectual life against the morbid inactivity of pure spirit, that we are what we are — self-conscious, thinking men, with the capabilities and attributes of Gods in us, for good as much as for evil. Hence the rebels are our saviours. . . . It is only by the attractive force of the contrasts that the two opposites — Spirit and Matter — can be cemented on Earth, and, smelted in the fire of self-conscious experience and suffering, find themselves wedded in Eternity.

By the gods allying themselves with us for this period of development, it became possible for us to attain knowledge and wisdom. But why was Prometheus harshly punished? Other legends suggest a motive; for instance Genesis reports:

And the Lord God said, “The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever.” So the Lord God banished him from the Garden of Eden to work the ground from which he had been taken. After he drove the man out, he placed on the east side of the Garden of Eden cherubim and a flaming sword flashing back and forth to guard the way to the tree of life. – 3:22-4

As with Pandora’s box, all evil in this world arose from the joining of the spiritual with the human world. Many commentators see in this a question of guilt, but this misses the crux of the matter: Adam and Eve are driven out of paradise because, with the power of thought, there is no longer any paradise for them. The human being, equipped with the divine capacity for self-reflective thought, can use this newly-won strength to create or destroy, to accomplish wonderful things or great crimes. One day we will ascend again and establish a human race worthy of the gods, but there is still a long way to go in overcoming “I am I” (egoic awareness) to reach “I am” (universal consciousness).

This evolutionary process is also clearly reflected in symbolism. Spirit, represented by a vertical line, is linked with the material world, represented by a horizontal line. Together these give rise to the cross, the son or third logos. If this logos becomes active, as with the awakening capacity for thought through the incarnation of the higher self, then this cross begins to turn. The turning of the cross produces the swastika, a symbol found in many religions. Quite a number of terracotta discs were found under the ruins of ancient Troy that contained this symbol in two forms:  Svastika1 and Svastika2. Again, Pramantha, the Vedic divine carpenter, unites himself with Arani, nature or Maya (illusion). They produce the divine boy Agni, god of fire. In the Bible too, Joseph is a carpenter, a master builder, and Mary is very reminiscent of Maya. Their child is mankind, with the fire of self-aware thought bestowed by the Holy Spirit. The son of the creator nailed to the cross – is he not a symbol of this process that speaks the same clear language as the legend of Prometheus, spirit chained to the cross of matter? The Crucified Titan is the personified symbol of the collective Logos, the ‘Host’, and of the ‘Lords of Wisdom’ or the heavenly man, who incarnated in Humanity.

We are also told that the suffering of Prometheus – the “one who foresees” – will end. He who has sacrificed himself for mankind is redeemed out of pity by the demigod Hercules, a son of Zeus, despite the fact that he may not take off the indestructible chains (karma). Still, the vulture – our base nature – will no longer come to eat the Titan’s liver. Human mental development, accelerated by the incarnation of the higher self, became unbalanced, with physical and moral development unable to keep pace. Once we regain our inner equilibrium, we will recognize our true destiny and nature, release the god chained within us, and as mankind conquer the darkness of ignorance.



Every country has its own methods of preserving the knowledge and tradition of the Mysteries. The degrees are variously reckoned, sometimes four, five, seven, or even ten; but whatever the divisions, during the days of their purity they all honored the one divine purpose of consummating the spiritual marriage of the higher self with the awakened human soul, from which union springs the seer, the adept, the master of life. Through the ravages of time and priestcraft, and the tangle of intrigue and ignorance in which exoteric rites are enmeshed, one perceives the venerable tradition.

In Asia Minor, Theon of Smyrna writes of five degrees in the initiatory cycle:

(1) “the preliminary purification,” because taking part in the Mysteries “must not be indiscriminately given to all who desire it”;

(2) “the tradition of sacred things” which constitutes the “initiation proper”;

(3) the “epoptic revelation,” where the candidate may experience direct intuition of truth;

(4) “the binding of the head and placement of the crown” — a clear reference to the mystical authority received with the crown of initiation to pass on the sacred tradition to others; and, finally,

(5) “friendship and interior communion” with divinity — this was considered the highest and most solemn mystery of all, the complete assimilation of the enlightened mind with the divine self — (Theon of Smyrna, Mathematics Useful for Understanding Plato + Isis Unveiled).

In Persia during the time of Mithraism, when the sun god was honored above earthly things, seven were the degrees, the candidate receiving a name relevant to each stage of interior growth. Using the Graeco-Latin names that have come down to us, the first-degree neophyte was called Corax, “raven” — the dark bird, one in whom the light of wisdom had not yet awakened in great measure. It signified likewise a servant: one who gives of his heart totally before receiving admission into the second degree which was termed Cryphius, “occult”: one accepted as a disciple of esoteric lore; the third was Miles, “soldier,” one who had received sufficient training and purification to become a worker for good. The fourth — Leo, “lion,” emblem of solar power — has reference to the fourth initiation in which the candidate begins the conscious solarizing of the nature through instruction and specialized training. The fifth degree was known as Perses, “Persian,” signifying to the Persians of the time one who was becoming spiritually human — manasaputrized, that is, mind-born. The sixth, Heliodromus, “messenger or runner of Helios (the sun)” is a reference to Mercury or Budha, as messenger between the sun in the cosmos and the sun in man: the bloom of buddhi. The final and seventh was called Pater, “father,” the state of a Full Initiate — (Esoteric Tradition 2:864).

The Hindus likewise had various names for their disciples as they passed from one degree to another. For instance, in one school the candidates received the names of the ten avataras of Vishnu. The first degree neophyte was termed Matsya, “fish”: one yet low in the scale of spiritual mastery. The second was Kurma, “tortoise”: one step higher in evolutionary development. The third degree was called Varaha, “boar,” a further advance in individualization, while the fourth was termed Nara-simha, “man-lion.” This fourth stage marks the turning point between the preliminary degrees of the Lesser Mysteries and the advanced degrees of the Greater Mysteries. This title of man-lion points to the choice demanded of the aspirant between dominance of animal soul qualities and the supremacy henceforth of the truly human attributes. Success in the fourth degree insured the entrance into the fifth called Vamana, “dwarf,” in which the candidate assumed the robes of occult humanhood, though such humanhood was as yet infantile compared to full mastery. Parasu-Rama, “Rama with an axe,” name of the sixth-degree neophyte, suggests one capable of hewing his way with equanimity through the worlds of both spirit and matter. In the seventh degree the disciple becomes fully humanized, receiving the name of Rama, hero of the Ramayana, an important epic of Hindustan. The last three degrees, the eighth, ninth, and tenth, are called respectively: Krishna, the avatara whose death ushered in the Kali yuga some 5,000 years ago; Buddha, whose renunciation of nirvana brought light and peace to a sorrowing world; and the final and tenth, Kalkin or Kalki, the “white-horse” avatara who is yet to come. As noted in the Vishnu Purana, he is destined to appear at the end of the Kali or Iron Age, seated on a white horse, with a drawn sword blazing like a comet, for the destruction of the wicked, the renovation of creation, and the restoration of purity. In ancient symbology the horse also symbolized the sun, hence the tenth avatara will come riding the steed of solar glory to usher in the New Age clothed with the sun of spiritual illumination.

While seven were the degrees usually enumerated in the Mysteries, hints have been given of three higher degrees than the seventh. But so esoteric would these be that only the most spiritualized of humanity could comprehend and hence undertake these divine initiations. Rare indeed are those who become avatara-like; rarer still, “as rare as are the flowers of the Udumbara-tree” are the Buddhas. As for the tenth and last — such has been left unmarred by description.