Most intelligent people consider that a set with an infinity of elements (e.g. the natural numbers) is the largest a set can be. But Cantor persuaded the maths establishment that infinity was tiny in comparison with Aleph 1.  Not only that.  Aleph 1 itself was uniimaginally tiny in comparison with Aleph 2.

Let’s pause at this point and consider the meaning of “greater than” and “tinier than”.  Both phrases are going to be put under intolerable pressure as we scan through the staircase of Alephs.

It is evident that Aleph 2 is unimaginably tinier than Aleph 3, and also that Aleph 3 is unimaginably tiny in comparison with Aleph 4.
So how do these unbelievably tiny magnitudes compare with Aleph 1000000?  Language fails us at this point because Aleph1000000 must make Aleph 4 super-unimaginably tiny.

But further strain falls onto these comparisons when we consider Aleph 1000000000.  Now Aleph 1000000 is far, far, far smaller than merely “unbelievably tiny” when compared with Aleph 1000000000.

And it doesn’t stop there. If G stands for the Google, Aleph G is pretty huge, but it shrinks into a mere dot compared with Aleph GG.
And if we let H stand for GG, Aleph H is considerably tinier than Aleph HH.

We find ourselves in a mindset which feels like the Sorcerer’s Apprentice. It is hardly surprising that Georg suffered from mental breakdown. But it is hardly what you would expect after his theory of the transfinite had been praised to the skies and described as a “Paradise” by David Hilbert.  It was praise from the most talented mathematician of the modern era. It should have put George on cloud nine… but evidently it didn’t.
When we try to get a mental grip on this see-saw of extreme comparisons, we are up against the unfortunate fact that the bumpiness goes on for ever… There is no point at which the Alephs come to an end. If Hilbert had got out of bed on the wrong side in 1900 he might just as well have decided to call Georg’s theory a “nightmare”.

We are operating in fantasyland… a verdict confirmed by the tendency of transfinite apologists when they admit that the elements of these mythical transfinite sets are “indefinable”.  They can’t be definable, because there is only an ordinary infinity of finite maths-symbolic permutations, and only a small subset of these are real-number generators.

To comment on the reasoning, email:  per4group@gmail.com    CHRISTOPHER ORMELL around August 1st 2026.