“Luminous” and “Dark Integers” are two short stories by Greg Egan about a different mathematical universe and its interaction with ours. The best way to read them now seems to be this book:

• Greg Egan, *Dark Integers and Other Stories*.

(1) how large a Church numeral one could get with a lambda-expression of limited size

(2) how small an expression was sufficient t express particular integers.

And I was interested in integers that were the simpler than the others in their neighbourhood.

It didn’t take long before I realized that passing an integer as a parameter to another gave me exponentiation, which was thus a lot simpler than using addition of multiplication.

Nor did it take long to realize that actually comparing the large integers represented by these lambda expressions quickly became infeasible.

I assigned a cost of 1 for each application and each lambda.

I did tinker with a computer program that would generate

At some point I figured that I’d really have to impose some information-theoretic cost to the use of variables, perhaps a logarithm depending on how many contextually bound variables were available to choose from… this got inconveniently complicated and the project was laid to rest. ]]>

This URL appears to have had “…” inserted into it. Anyway, it doesn’t work (although it perhaps used to). I may remember reading the story — is it the one where our computational world was being invaded by a different set of integers; the invasion becoming evident starting with the *huge* integers used in cryptography?

Thanks, Hendrik! Fixed!

]]>Your link to Hilbert Levitz’s paper, Transfinite ordinals and their notations: for the uninitiated, is broken. If I chop off several parts at the start of the URL it works. So I suspect the software generating this web page is prefixing something to the URL that it shouldn’t.

]]>J. E. Littlewood acknowledged the difficulty in the chapter Large Numbers of his book “Littlewood’s Miscellany.” Here’s his comment (plus a bit of the rest of the chapter, to give you a taste)-

“We can now say: scrap the existing definitions, as scaffolding, and define *this* to be , and carry on as before. We can scrap again, and so on: here I decide to stop. Once we stop we may take , or (what we take does not matter provided only ).

The reader will agree that the numbers mentioned are large: it is not possible to say how large; all that can be said about them is that they are defined as they are defined. If it were desired to compare terms in two rival systems a considerable technique would have to be developed.”

]]>That’s the story “Dark Integers”, anyway.

]]>The new URL is now broken, too. Is this the correct new one? http://will.tip.dhappy.org/blog/Compression%20Trees/…/book/by/Greg%20Egan/Dark%20Integers/Greg%20Egan%20-%20Dark%20Integers.html

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