Dangerous fantasy

From Session 21 March 2026:

(Ark) Well, I would like to know what kind of physics I need to describe densities. What kind of math? How to get to densities?

A: Geometric algebra.

Q: (Ark) Geometric algebra. Finite dimensional or infinite dimensional? Can you help me?

A: Infinite.

And so I am from finite to infinite. Not that I was not there already long ago. While finite feels “safe”, the infinite attracts with its unpredictability. Our true free will, whenever we have one, resides there, in true infinity. One may try to argue that for us it is impossible to distinguish between something very vert large, and another something that is truly infinite, While on some level this may be so indeed, on another level there is an essential difference. Our AI is all finite. But there are no reasons to believe that our consciousness is finite. And to have a glimpse into what this infinite potential may lead to, we must not freak out. We must deal with the infinite and take all it can offer us.

Yet finite can help us here. So I asked finite Perplexity AI to tell me about George Cantor. Here is what it came with:

In the winter of 1884, Georg Cantor sat at his desk in Halle, surrounded by papers no one seemed to want. Outside, the town moved with ordinary purpose—carriages, shopkeepers, students—but inside his study, Cantor was wrestling with something that refused to stay finite.

He had written to Dedekind again.

“Je le vois, mais je ne le crois pas,” he confessed in one letter—“I see it, but I do not believe it.” What he saw was a hierarchy of infinities: not a vague philosophical abyss, but a structured, almost crystalline universe where some infinities were strictly larger than others. It was not speculation; it was proof. The real numbers were more numerous than the integers. Infinity, it turned out, was not a single monolith but a ladder without end.

And yet, nearly everyone recoiled.

Kronecker, once a respected senior figure in Berlin, dismissed Cantor’s work as a dangerous fantasy. “God made the integers,” he reportedly said, “all else is the work of man.” For Kronecker, mathematics was to remain grounded in the finite and constructive. Cantor’s infinities were not just incorrect—they were illegitimate.

But Cantor could not unsee what he had seen.

One evening, as the story goes—perhaps embroidered over time—he tried to explain his diagonal argument to a colleague over dinner. Taking a napkin, he listed decimal expansions, then carefully altered each diagonal digit. “You see,” he insisted, tapping the paper, “this number escapes the list. Always escapes. No matter how you try to capture them, the continuum exceeds enumeration.”

The colleague nodded politely, then changed the subject.

It was not hostility alone that isolated Cantor; it was incomprehension. His ideas did not merely extend mathematics—they altered its ontology. Infinity was no longer a direction or a limit, but an object that could be compared, ordered, even counted in its own strange way.

The cost was personal. Cantor’s letters from this period oscillate between exultation and despair. At times he believed himself chosen to reveal a deep truth about creation. At others, he feared he had lost his grip entirely. “My theory stands as firm as a rock,” he wrote—yet he sought constant reassurance, as if the clarity of his vision demanded witnesses who never came.

And still, he continued.

What sustained him was not agreement, but inevitability. The arguments did not depend on approval. Once seen, they unfolded with a kind of internal necessity. The infinite sets existed—not in the world of stones or stars, but in a realm of pure thought whose consistency was, to Cantor, indistinguishable from reality.

Years later, when Hilbert would declare, “No one shall expel us from the paradise that Cantor has created,” it would sound like triumph. But for Cantor himself, that paradise was first a wilderness—vast, silent, and almost entirely uninhabited.

He walked there largely alone.

More on this in my Wordpress blog.