"It seems to be the dominant data configuration," Mace called. "The analogue of the tetrahedral motif on the physical level. It represents a theorem. See, the heart of the structure is the core statement, the petals corollaries, endlessly thrown off and lost..."

"What theorem?"

"Godel's Incompleteness. We think. We're guessing, extrapolating on hints of structure we've picked up elsewhere... But it's not really a theorem, here. It's merely a statement of the result. Like an axiom; a given."

"I don't understand."

Mace laughed, briefly and scornfully. Wriggling before the landscape of information he pointed again. Amid a meadow of data structures, Kapur picked out another sunflower, the characteristic Godel shape. Mace jabbed both arms against the vast data diorama, again and again. "There, and there! What do you see, Kapur?"

Godel, Kapur saw, repeated over and over; there was a fractal spiral of Godel sunflowers here, embedded in this chill web of data.

"There's more, of course," Mace said. "We've recognized a lot of physical understanding in here, particularly representations of cosmic events. See that starburst?" A firework of red and yellow, endlessly dynamic, scattered a hundredfold through the 'Flake. "That's Mach's principle: that the inertia of an object is induced by the net gravitational attraction of the rest of the Universe--"

"Tell me about Godel," Kapur said patiently.

On the low-quality laser link, Mace's voice was like a buzzing insect. "Godel was a genius. An Austrian; a Mozart of his subject. In the middle of the twentieth century he produced a theorem on undecidability.

"Godel studied mathematics in the abstract. Think about that, policeman: not just the mathematics you studied at high school; not the maths I studied in the Navy college -- but any sort of mathematics which it is possible to construct."

"You have my attention," Kapur said dryly. "Go on."

"Godel showed that within any mathematical scheme you can write down statements which it would be impossible to prove or disprove. They are undecidable, you see. And so mathematics can never be made complete. You could never deduce everything from a finite set of axioms; there would always be new statements to make... new facts to record, if you like."

Kapur shook his head. "I cannot imagine how it is possible even to begin to frame such a theorem, let alone to prove it."

"It isn't that difficult," Mace said lightly. "It's rather like the standard proof that the real numbers are uncountable; you make a list of all possible statements within your general mathematical scheme -- and from that list generate another statement which isn't in the list--"

"Never mind." Kapur let the terrifying implications sink in. How could there be a hole in mathematics -- in the most fundamentally abstract of human inventions? He felt as if the floor had fallen away from his Universe.

What kind of people had these Snowmen become, to hold such an awesome, nihilistic theorem at the heart of their philosophy?