[lioeters]

There's a section in the linked Wikipedia article that describes Jorge Luis Borges tracing the origin of the concept back to Aristotle's Metaphysics.

That's the best thing I've read today. I've often heard the monkeys-on-typewriters imagery, but never related it to an idea of a "total library" which contains everything that can ever be written, probably even itself. I suppose the digits of Pi may be considered such a "library".

"Strictly speaking, one immortal monkey would suffice."

[srtjstjsj]

The "total" library (which is mathematically paradoxical) can't contain itself because it has higher cardinality than any of its contents.

[lioeters]

When I wrote that a total library would contain "probably even itself", I had a nagging feeling that may lead to a contradiction.

Here is the essay "Total Library" by Borges:

https://www.gwern.net/docs/borges/1939-borges-thetotallibrar... (PDF)

It mentions "abnormal transfinite numbers (whose parts are no smaller than the whole)". This seems to be a reference to Russell's Paradox:

https://en.wikipedia.org/wiki/Russell's_paradox

> Let us call a set "normal" if it is not a member of itself, and "abnormal" if it is a member of itself.

> Now we consider the set of all normal sets, R, and try to determine whether R is normal or abnormal. If R were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if R were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal.

> This leads to the conclusion that R is neither normal nor abnormal: Russell's paradox.

That proves your point, that it's a mathematical impossibility.

Borges calls it "the vast, contradictory Library", so clearly he was aware of this fact.

[gwern]

Borges's library can't contain itself because it is of a strictly finite size: the books are only 410 pages, or whatever, and so while astronomically large, the total number of sequences is finite. His point was that for any sub-book-sized claim, there would be another book contradicting it, if only through prefixing "it is not true that" or something.

His Book of Sand would, however, appear to be vulnerable to diagonalization.