Cantor's Dragon

An absolutely fabulous tale of a man outwitting the devil in a very creative way, reminiscent of the story, “The Devil and Simon Flagg” by Arthur Porges. George Cantor, who has been hospitalized with mental exhaustion from his acrimonious battles with Kronecker over the existence of a completed Infinity, is visited upon by a dragon who tells him that upon death, Cantor must choose to go to either heaven or hell. Hell is near, heaven infinitely far away. To reach heaven, one has to climb the infinitely long “Dragon’s Stair”. Each stair has a name carved on it. If the climber finds his name on a stair, he must wait there for all eternity, unable to climb any further. Ergo, Heaven has always been empty. Before going up the stairs, the climber must state his name. However, Cantor detects a loophole in this scheme and implements his Diagonal Proof judiciously to not only outwit the Dragon, but also put a reverse condition on the Dragon which ensures that all of future climbers would find their way to heaven.

Such a gem of a story! I was reminded also of Kafka’s story, “The Law”, where the dying man finds no recourse to reach the doors of justice. In the present story, the situation seems just as hopeless, but the ending mathematically brilliant.

Further Commentary: I see the story not as a supernatural occurrence but a hallucination - Cantor, after all, was in the sanitarium because of the enormous stress he had been and his arguments with Kronecker et al. Cantor was under intense attack about his ideas on Infinity of which he was the prophet. He was also an extremely devout Christian wedded to ideas of Heaven, Redemption, Judgment Day, etc. So very naturally, as he feels (in the story) that his end may be near (though in reality, he lived for long after that), he sees the dragon, a Satan-figure, come to test him. His mind, in his troubled state, would naturally contrive a hallucination where he comes out victorious, outwits Satan and charts a path to his God. It is then not a surprise that his mind formulates a problem set to it by the dragon which has a very convenient solution in the diagonal method. Since this is all in Cantor’s mind, there is no question on why this problem is contrived, why only Cantor gets this vision or how this is unfair to those before Cantor. This particular access to heaven is designed for Cantor (by his mind), just as in Kafka’s story, where the Gate of Law is personalized for each being.

Indeed, this exact reason applies to the fact that his mind very specifically creates the ambiguity on what a “name” means. Cantor has an explicit dialogue about this with the dragon:

“ “Braggart,” Cantor says. Then he switches direction: “Who decides my name?” The dragon understands the question immediately. “You do.” Cantor smiles. “Can I name myself while climbing the stair?” The dragon thrashes its tail in anger. It growls, and blows smoke, before it answers. “You must start to say your name before stepping upon the way.” “But I need not finish naming myself before starting on the way?” The dragon is silent an hour. Cantor listens to the fire fluttering in its lungs. He patiently writes out his proof as he waits for his answer. “No,” the dragon hisses, leaking flames that cast flickering shadows along the walls. “You need not finish naming yourself before starting on the way.” ”

Thus, the dragon explicitly acknowledges that a “name” is simply a label and there is nothing inherently required for it to be a finite length or said up-front. He also, very grudgingly, accepts that the person does not have to finish naming his name before starting. Therefore, the entire climb on the dragon’s staircase is designed to be a self-boot-strapping, unzipping process as Cantor walks on the stairs. Is it contrived? Yes! His mind is seeking salvation in a clever way and sets it up exactly so it can solve the puzzle.

The dragon also explicitly says that heaven is infinitely far away. So even if someone found a different scheme to evade the dragon’s condition, he would still need an anti-Zeno scheme to reach heaven in a finite time. For this, heaven-seekers can consult Rudy Rucker’s “White Light” :-). Finally, if there is an objection that he does not specify what to do if the nth step name is less than n-letters long, the dodge is easy - he simply adds a random letter (or any other way of selection. An axiom of Choice!).