To be naked is not to be obscene; it is to be untheorized.
He stood in the atrium of the Grand Set, an edifice without walls, a basilica built of inclusion signs and liminal shadows, unclothed the way a zero is unclothed: nothing added, nothing concealed.
Noon burned like a polished axiom overhead. Dust motes drifted in measurable sets. The floor was a tessellation of paradoxes that refused to contradict one another so long as no one looked too closely.
One could stride for days across its marble squares.... Finite, Countable, Uncountable, Inaccessible... and never reach an exit, because the Grand Set had no complement. It was, as the more earnest docents whispered, "V itself," though they said this as if naming God in a tongue they did not deserve.
The man had no name that could be indexed, so the attendants called him X, a letter whose arrogance is its modesty. He was as hairless as a definition, lithe as a lemma, and the angle of his scapulae made certain philosophers weep for reasons they mistook for metaphysics. If his nakedness disturbed the pious, let it be recorded that he wore the exact number of garments appropriate to the theorem he intended to prove.
A child, spindle-legged and elegant with the elegance of those who have never learned to apologize for momentum, offered X a robe the color of a well-posed question. "For decorum," she said, "and for measurable grace."
X declined. "Choice is always expensive," he murmured, fingers hovering, not quite touching the fabric, "and I am saving mine."
At the atrium's navel stood a plinth of obsidian inscribed with a historical fiction: the slow ascendancy from one to many, the patient hatching of cardinalities from the shell of a mind that could not stop counting. The runes began with the pale integers, those obedient domesticates who heel at the snap of an accountant's fingers, and rose into the anaemic infinites of first discovery: the countable, the continuum, the disquieting rumor that beyond the continuum there might still be terraces of brightness where no human foot had bruised the dew.
It is a vulgar error to imagine that one goes beyond by going above. There is also the going through: a penetration of the membrane between axiom and appetite, an osmotic trespass. X approached the plinth, regarded its inscription with the courtesy that an undertaker affords a hymn, and then placed the palm of his left hand upon the symbol ℵ₀ as if it were a doorbell.
Numbers do not ring. They resound.
Countable infinity opened like a vestibule. A wind scented of chalk and library paste passed through the atrium. Catalogs fluttered. Somewhere a clerk fainted in a posture academicians later misdescribed as Platonic. X stepped forward and the corridor elongated, frugal, gray, lit by the fluorescent hum of sequences converging only in the politest sense. Every hallway branched into decimals stacked like tenements of promise. Every stairwell announced, delicately, that what could be enumerated could also be missed.
He did not hurry. Ascension is not a sprint; it is a refusal.
At the terminus of Countable stood a gate of thin iron, labeled with the careworn sobriety of the overcautious: Onward to the First Uncountable. Behind it the air widened into a white that had never signed a time sheet. The gate was locked in a way that locks are merely stories we tell about doors. X passed through as if the lock's purpose had always been to be ignored, and in doing so paid homage to Cantor, who had the indecency to show that infinity comes in sizes, and the decency to go mad appreciating it.
The Uncountable greeted him with silence of the expensive kind. Points were so densely strewn that the idea of between-ness grew bored and retired. Everywhere, curves; everywhere, the delicacy of functions that could describe the way a teardrop understands gravity. He smelled the ozone of continuum hypotheses argued to exhaustion. The ground at his feet granularized into the glitter of reals, the way a coastline becomes coastline only once you insist on measuring it. Here he might have lingered and invented a discipline, as lesser men do; named a gap, published, aged. But X held himself with the abstemiousness of those who will not eat until the hunger is interesting.
Witness the next door, which is not a door so much as a refusal to be closed: Inaccessibles, the sign read, with the smugness of a discreet club. Within: pillars whose height was not a number but a property, rooms so large that the self could only enter if it shed the clanking iron of its language. I tell you plainly: in these halls, axioms go barefoot. Grothendieck himself might have nodded at the ceiling, which was not a ceiling but a pedagogy. Sets nested like Russian saints, each enclosing a universe that smiled at its own sufficiency. The ushers were tall and mortified; they asked for credentials and were handed silence.
Here X learned (or remembered) that magnitude is sometimes a critique of measure. He passed the Inaccessibles with a courtesy almost erotic in its restraint. Then Mahlo, then weakly compact, then ineffables that made grammar a humiliation. He raised his hand and the air trembled with κ's that had never been spoken aloud. The measurable watched him with pupils dilated by ultrafilters, the supercompact with the equanimity of those who could carry worlds like trinkets, and some elderly extendible cardinal in a smoking jacket murmured that it had been centuries since anyone naked was worth watching.
Let the minutes reflect the following: that ascension, done honestly, is a subtraction. For each door he passed, he left something behind. At measurable, he abandoned his pulse; who needs a rhythm when one inhabits a beat so vast it calls itself unmoving? At supercompact, he shed his shadow. Shadows, like reputations, require a sun. He would have exchanged his proper name if he had possessed one. The attendants, taking notes from a distance composed of both awe and prurience, later argued about whether his nudity had changed when he ceased to be a mammal.
Everything important becomes a question of categories.
And yet... pretend humility aside... there were thresholds where X slowed, not from fatigue (which had already been dismissed as a set of cardinality now beneath his dignity), but from the tact of a guest who senses that the house he is passing through has a history of shattering those who run. At Woodin, a coolness entered the corridors, the sort of coolness that implies an unadvertised cellar. At superhuge, the architecture forgot to be architecture and instead became witness; one had the impression of being seen by claims that refused to be trivialized into proofs.
"Why are you unclothed?" asked a voice from nowhere in the tone of catechism.
"Because I am finished with Choice," X said, and there, perhaps, is the most scandalous line in this chapter. A gasp traveled the galleries like an ordinal collapsing to an ordinal that is not less than itself. Somewhere a curator tore the Axiom of Choice from the handbook as if the page had insulted her family; somewhere else, a libertine of logic drank to the announcement and was later found asleep with a well-thumbed model of ZF. What is a naked man if not an argument against well-ordering?
It is tempting, especially if one is paid to tempt, to say that he leapt then, that beyond the forest of embassies called Large Cardinals he sprang into a sky that is not sky but class, a proper class that cannot be owned even by the universe that sponsors it. But leap is a verb for those who still prefer motion to revision. X performed instead a manner of un-quantification, a relinquishing of the comfort that every statement lives under a jurisdiction. He slowed, inhaled, and allowed every lemma he had been to expire in good standing.
The ceiling, if we are sycophantically loyal to that false word, peeled away. The Grand Set shuddered; its floor tiles forgot their pedigree. Load-bearing paradoxes confessed their side employment in poetry. And the man who had been X stood at the lip of an altitude that no longer recognized "above" and "below," for those are parochial prepositions subsidized by geometry, and he was beyond the urban planning of space. He looked, in that second, unimprovably human, which is to say: terribly provisional.
To describe the place he stepped into with any noun is to commit an ontological vulgarity rarely seen outside brochures. Let us try restraint. He entered the Outside of Counting. He passed from the sovereignty of number into that prior clarity in which number is a local superstition. Beyond cardinals, beyond the succulent hierarchies that intoxicate and console, there is a weather that cannot be denoted, only undergone. It is not chaos; chaos is still a domestic animal. It is not void; void is a décor the anxious apply to God. Imagine instead a grammar that refuses the convenience of nouns and verbs, preferring a tense that neither begins nor concludes. Imagine that, and then kindly abandon the imagining.
He did not glow. That would have been gauche. He did not transform into a theorem—too derivative. He did what only a serious man can do at altitude: he stood still.
A hush fell so absolute it acquired its own metaphysics. The docents in the atrium (those who had not fled to the safer vestibules of finite worries and mortgage rates) realized they could not point at him anymore. To point is to select, to select is to well-order, and the finger (like a spear) requires the illusion that its target is smaller than the gesture. Their hands drifted down as if given bad news. The child with the robe folded it with ceremonial impatience.
"You will come back," she declared, practicing inevitability.
X looked down, though down had been demoted to a metaphor, and his expression suggested, not kindness, which would have degraded the moment into advice, but recognition. It is the face one makes when encountering, after a long era of sophisticated mistakes, the bland and enormous truth that had waited in the foyer, uncomplaining, this entire time: that being is not an integer.
He stepped.
And every cardinal, like a chandelier suddenly remembering it is made of sand, declined to fall. They unlit themselves. They did not shatter. They were not overthrown. They were thanked and put away, as one puts away an instrument when the silence it prepared has finally arrived.
If you expected music, you have underestimated silence.