It was a variation on these sad apparitions that came to wake him one night late in January 1933. This was the dream that he termed in retrospect 'the herald of defeat'.
He dreamed of 2^100 (2 to the hundredth power, an enormous number) personified as two identical, freckled, beautiful dark-eyed girls, looking straight into his eyes. But now there wasn't just sadness in their look, as there had been in his previous visions of the Evens; there was anger, hatred even. After gazing at him for a long, long while (this in itself was sufficient cause to brand the dream a nightmare) one of the twins suddenly shook her head from side to side with jerky, abrupt movements. Then her mouth was contorted into a cruel smile, the cruelty being that of a rejected lover.
'You'll never get us,’ she hissed.
At this, Petras, drenched in sweat, jumped up from his bed. The words that 2^99 (that's one half of 2^I00) had spoken meant only one thing: He was not fated to prove the Conjecture. Of course, he was not a superstitious old woman who would give undue credence to omens. Yet the profound exhaustion of many fruitless years had now begun to take its toll. His nerves were not as strong as they used to be and the dream upset him inordinately.
Unable to go back to sleep, he went out to walk in the dark, foggy streets, to try to shake off its dreary feeling. As he walked in the first light among the ancient stone buildings, he suddenly heard fast foot-steps approaching behind him, and for a moment he was seized by panic and turned sharply round. A young man in athletic gear materialized out of the mist, running energetically, uttered a greeting and dis-appeared once again, his rhythmic breathing trailing off into complete silence.
Still upset by the nightmare, Petros wasn't sure whether this image had been real, or an overflow of his dream world. When, however, a few months later the very same young man came to his rooms at Trinity on a fateful mission, he instantly recognized him as the early-morning runner. After he was gone, he realized with hindsight that their first, dawn meeting had cryptically signalled the dark forewarning, coming as it did after the vision of 2^100, with its message of defeat.
The fatal meeting took place a few months after the first, early-morning encounter. In his diary Petros marks the exact date with a laconic comment – the first and last use of Christian reference I discovered in his diaries: '17 March 1933. Kurt Gödel's Theorem. May Mary, Mother of God, have mercy on me!'
It was late afternoon and he had been in his rooms all day, sitting forward in his armchair studying parallelograms of beans laid out on the floor before him, lost in thought, when there was a knock on the door.
'Professor Papachristos?'
A blond head appeared. Petros had a powerful visual memory and immediately recognized the young runner, who was full of excuses for disturbing him. 'Please forgive my barging in on you like this, Professor,’ he said, 'but I am desperate for your help.' Petros was quite surprised – he'd thought his presence at Cambridge had gone completely unremarked.
He wasn't famous, he wasn't even well known and, except at his almost nightly appearances at the university chess club, he hadn't exchanged two words with anyone other than Hardy and Littlewood during his stay.
'My help on what subject?'
'Oh, in deciphering a difficult German text – a mathematical text.' The young man apologized again for presuming to take up his time with such a lowly task. This particular article, however, was of such great importance to him that when he heard that a senior mathematician from Germany was at Trinity, he couldn't resist appealing to him for assistance in its precise translation.
There was something so childishly eager in his manner that Petros couldn't refuse him.
Td be glad to help you, if I can. What field is the article in?'
'Formal Logic, Professor. The Grundlagen, the Foundations of Mathematics.'
Petros felt a rush of relief that it wasn't in Number Theory – he'd feared for a moment the young caller might have wanted to pump him on his work on the Conjecrure, using help with the language merely as an excuse. As he was more or less finished with his day's work, he asked the young visitor to take a seat.
'What did you say your name was?'
'It's Alan Turing, Professor. I'm an undergraduate.'
Turing handed him the journal containing the article, opened at the right page.
'Ah, the Monatshefte für Mathematik und Physik,' said Petros, 'the Monthly Review for Mathematics and Physics, a highly esteemed publication. The title of the article is, I see, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme". In translation this would be… Let's see… "On the formally undecidable propositions of Principia Mathematica and similar Systems". The author is a Mr Kurt Gödel, from Vienna. Is he well known in this field?'
Turing looked at him surprised. 'You don't mean to say you haven't heard of this article, Professor?'
Petros smiled: 'My dear young man, mathematics too has been infected by the modern plague, overspecialization. I'm afraid I have no idea of what's being accomplished in Formal Logic, or any other field for that matter. Outside of Number Theory I am, alas, a complete innocent.'
'But Professor,' Turing protested, 'Gödel's Theorem is of interest to all mathematicians, and number theorists especially! Its first application is to the very basis of arithmetic, the Peano-Dedekind axiomatic system.'
To Turing's amazement, Petros also wasn't too clear about the Peano-Dedekind axiomatic system. Like most working mathematicians he considered Formal
Logic, the field whose main subject is mathematics itself, a preoccupation that was certainly over-fussy and quite possibly altogether urmecessary. Its tireless attempts at rigorous foundation and its endless examination of basic prindples he regarded, more or less, as a waste of time. The piece of popular wisdom, 'If it ain't broke, don't fix it,' could well define this attitude: a mathematician's job was to try to prove theorems, not perpetually ponder the status of their unspoken and unquestioned basis.
In spite of this, however, the passion with which his young visitor spoke had aroused Petros' curiosity. 'So, what did this young Mr Gödel prove, that is of such interest to number theorists?'
'He solved the Problem of Completeness,' Turing announced with stars in his eyes.
Petros smiled. The Problem of Completeness was nothing other than the quest for a formal demonstration of the fact that all true statements are ultimately provable.
'Oh, good,' Petros said politely. 'I have to tell you, however – no offence meant to Mr Gödel, of course – that to the active researcher, the completeness of mathematics has always been obvious. Still, it's nice to know that someone finally sat down and proved it.'
But Turing was vehemently shaking his head, his face flushed with excitement. "That's exactly the point, Professor Papachristos: Gödel did not prove it!'
Petros was puzzled. 'I don't understand, Mr Turing… You just said this young man solved the Problem of Completeness, didn't you?'
'Yes, Professor, but contrary to everybody's expectation – Hilbert's and Russell's included – he solved it in the negative! He proved that arithmetic and all mathematical theories are not complete!'
Petros was not familiar enough with the concepts of Formal Logic immediately to realize the full implications of these words. 'I beg your pardon?'
Turing knelt by his armchair, his finger stabbing excitedly at the arcane symbols filling Gödel's article. 'Here: this genius proved – conclusively proved! – that no matter what axioms you accept, a theory of numbers will of necessity contain unprovable propositions!'