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!'

'You mean, of course, the false propositions?'

'No, I mean true propositions – true yet impossible to prove!'

Petros jumped to his feet. 'This is not possible!'

'Oh yes it is, and the proof of it is right here, in these fifteen pages: "Truth is not always provable!"'

My uncle now felt a sudden dizziness overcome him. 'But… but this cannot be.'

He flipped hurriedly through the pages, striving to absorb in a single moment, if possible, the article's intricate argument, mumbling on, indifferent to the young man's presence.

'It is obscene… an abnormality… an aberration…'

Turing was smiling smugly. That's how all mathematicians react at first… But Russell and Whitehead have examined Gödel's proof and proclaimed it to be flawless. In fact, the term they used was "exquisite".'

Petros grimaced. '"Exquisite"? But what it proves – if it really proves it, which I refuse to believe – is the end of mathematics!’

For hours he pored over the brief but extremely dense text. He translated as Turing explained to him the underlying concepts of Formal Logic, with which he was unfamiliar. When they'd finished they took it again from the top, going over the proof step by step, Petros desperately seeking a faulty step in the deduction.

This was the beginning of the end.

It was past midnight when Turing left. Petros couldn't sleep. First thing the next morning he went to see Littlewood. To his great surprise, he already knew of Gödel's Incompleteness Theorem.

'How could you not have mentioned it even once?' Petros asked him. 'How could you know of the existence of something like that and be so calm about it?'

Littlewood didn't understand: 'What are you so upset about, old chap? Gödel is researching some very special cases; he's looking into paradoxes apparently inherent in all axiomatic systems. What does this have to do with us line-of-combat mathematicians?'

However, Petros was not so easily appeased. 'But, don't you see, Littlewood? From now on, we have to ask of every statement still unproved whether it can be a case of application of the Incompleteness Theorem… Every outstanding hypothesis or conjecture can be a priori undemonstrable! Hilbert's "in mathematics there is no ignorabimus" no longer applies; the very ground that we stood on has been pulled out from under our feet!'

Littlewood shrugged. 'I don't see the point of getting all worked up about the few unprovable truths, when there are billions of provable ones to tackle!'

'Yes, damn it, but how do we know which is which?'

Although Littlewood's calm reaction should have been comforting, a welcome note of optimism after the previous evening's disaster, it didn't provide Petros with a definite answer to the one and only, dizzying, terrifying question that had jumped into his mind the moment he'd heard of Gödel's result. The question was so horrible he hardly dared formulate it: what if the Incompleteness Theorem also applied to his problem? What if Goldbach's Conjecture was unprovable?

From Littlewood's rooms he went straight to Alan Turing, at his College, and asked him whether there had been any further progress in the matter of the Incompleteness Theorem, after Gödel's original paper. Turing didn't know. Apparently, there was only one person in the world who could answer his question.

Petros left a note to Hardy and Littlewood saying he had some urgent business in Munich and crossed the Channel that same evening. The next day he was in Vienna. He tracked his man down through an academic acquaintance. They spoke on the telephone and, since Petros didn't want to be seen at the university, they made an appointment to meet at the cafe of the Sacher Hotel.

Kurt Gödel arrived precisely on time, a thin young man of average height, with small myopic eyes behind thick glasses.

Petros didn't waste any time: "There is something I want to ask you, Herr Gödel, in strict confidentiality.'

Gödel, by nature uncomfortable at social intercourse, was now even more so. 'Is this a personal matter, Herr Professor?'

'It is professional, but as it refers to my personal research I would appreciate it – indeed, I would demand! – that it remain strictly between you and me. Please let me know, Herr Gödel: is there a procedure for determining whether your theorem applies to a given hypothesis?'

Gödel gave him the answer he'd feared. 'No.'

'So you cannot, in fact, a priori determine which statements are provable and which are not?'

'As far as I know, Professor, every unproved statement can in principle be unprovable.'

At this, Petros saw red. He felt the irresistible urge to grab the father of the Incompleteness Theorem by the scruff of the neck and bang bis head on the shining surface of the table. However, he restrained himself, leaned forward and clasped his arm tightly.

‘I’ve spent my whole life trying to prove Goldbach's Conjecture,’ he told him in a low, intense voice, 'and now you're telling me it may be unprovable?'

Gödel's already pale face was now totally drained of colour.

'In theory, yes -'

'Damn theory, man!' Petros' shout made the heads of the Sacher cafe's distinguished clientele turn in their direction. 'I need to be certain, don't you understand? I have a right to know whether I'm wasting my life!'

He was squeezing his arm so hard that Gödel grimaced in pain. Suddenly, Petros felt shame at the way he was carrying on. After all, the poor man wasn't personally responsible for the incompleteness of mathematics – all he had done was discover it! He released his arm, mumbling apologies.