Now, on to the actual mathematics!
"Understand thus..." says the first line of the treatise, giving the definitions of the earliest level; of course, without a formalized tetrational notation, he had to improvise. In modern notation, we can summarize: hn=n^^n (now we would call the same function "megafuga").
Now we have hhn=h(hn), hhhn=h(hhn), and so on; culminating in the general case mhn=hm(n).
[I use the superscript here in the modern sense; a different meaning occurs in the treatise itself - though that doesn't really create a conflict, since it fortunately doesn't include this case!Read more >
Specifically, it does not include cases like hxn, with only one h to the left of the superscript and one number to the right - which, for obvious reasons, inc…
In this series of blog posts, I will try to summarize what might well have been the world's first googological notation to reach beyond the tetration level (some years before the Steinhaus notation that was so famously expanded by Moser) - what I call the Kharms notation (after its author, Daniil Kharms). But first, a bit of background...
The classic (at least relative to your typical online history) Russian source on very large numbers is the Stas article from 2003. It is in the errata to that article that I first found out about Daniil Kharms' 1931 treatise "Lifting of a Number".
Knowing what I knew about (a rather famous absurdist poet), I was not expecting any particularly serious mathematics. What I found, however, was an article that,…Read more >