Bayliss frowned. Her augmented eyes were blank, reflecting the washed-out light as she considered. "An accident, then? But Marsden was no fool. Whatever he was up to with this clumsy implant in his scalp, I cannot believe he would be so careless as to let it kill him."

"What was he 'up to'?" Chen asked sourly. "Have you figured that out yet?"

Bayliss rubbed the bridge of her small, flat nose. "There is an immense amount of data here. Much of it not indexed. I've sent data-mining authorized-sentience algorithms into the main stores, to establish the structure."

"Your preliminary thoughts?" Hassan demanded.

"Metamathematics."

Hassan looked blank. "What?"

"And many experimental results on quantum nonlinearity, which--"

"Tell me about metamathematics," Hassan said.

The patches of woven metal over Bayliss's corneas glimmered; Chen wondered if there was any sentience in those augmentations. Probably. Such devices had been banned on Earth since the passing of the first sentience laws, but they could still be found easily enough on Mars. Bayliss said, "Marsden's data stores contain a fragmented catalogue of mathematical variants. All founded on the postulates of arithmetic, but differing in their resolution of undecidable hypotheses."

"Undecidability. You're talking about the incompleteness theorems," Chen said.

"Right. No logical system rich enough to contain the axioms of simple arithmetic can ever be made complete. It is always possible to construct statements which can be neither disproved nor proved by deduction from the axioms; instead the logical system must be enriched by incorporating the truth or falsehood of such statements as additional axioms..."

The Continuum Hypothesis was an example.

There were several orders of infinity. There were "more" real numbers, scattered like dust in the interval between zero and one, than there were integers. Was there an order of infinity between the reals and the integers? This was undecidable, within logically simpler systems like set theory; additional assumptions had to be made.

Hassan poked at the corpse with his booted toe. "So one can generate many versions of mathematics, by adding these true-false axioms."

"And then searching on, seeking out statements which are undecidable in the new system. Yes." Icons scrolled upwards over Bayliss's eyes. "Because of incompleteness, there is an infinite number of such mathematical variants, spreading like the branches of a tree..."

"Poetry," Hassan said; he sounded lazily amused.

"Some variants would be logically rich, with many elegant theorems flowing from a few axioms -- while others would be thin, over-specified, sterile. It seems that Marsden has been compiling an immense catalogue of increasingly complete logical systems."

Silence fell; again Chen was aware of the sour stink of the body at her feet. "Why? Why come here to do it? Why the implant? And how did he die?"

Hassan murmured, "Bayliss said the catalogue was fragmented. This -- metamathematical data -- was stored carelessly. Casually." He looked to Bayliss for confirmation; the little woman nodded grudgingly.

"So?" Chen asked.

"So, Susan, perhaps this metamathematical experiment was not Marsden's primary concern. It was a byproduct of his core research."