Sunday, July 13, 2008

The Upxkcd Number

Following on from the previous post, it seemed fitting to launch this blog's own stupidly big number.

Introducing the Up Function, Up(n): a Conway arrow chain (a very elegant, one could even say simple, way of producing very large numbers) containing n copies of any number n:

Up(1) = 1
Up(2) = 2→2
Up(3) = 3→3→3
Up(4) = 4→4→4→4

They look innocent enough, but the values of Conway chains grow real fast: Up(4) is already far bigger than Graham's Number.

What's more, there's nothing to stop us using the function more than once, thus:

Up(2)(4) = Up(Up(4))

The brilliant xkcd webcomic created their own xkcd number by sticking Graham's number in the wonderfully explosive Ackermann function:

xkcd = A(g64,g64)

Up likes the xkcd number, but – remarkably – even this is much smaller than Up(4). Indeed, in Conway notation, xkcd is barely distinguishable from g64. Therefore, naturally, we're proposing to make use of our nice new function to turbo-charge it up a bit.

Cue the Upxkcd Number, Ux:

Ux = Up(xkcd)(xkcd)

I think that's enough to be getting on with!

[back to blog]


...however, if you do want more, bigger, and better, and you want it now, here are the best pages I've found...

Scott Aaronson's excellent article on the game of "who can name the bigger number"
Robert Munafo's encyclopaedic website on large numbers. It starts on the ground, and it goes all the way to the top.

"I have this vision of hoards of shadowy numbers lurking out there in the dark, beyond the small sphere of light cast by the candle of reason. They are whsipering to each other; plotting who knows what. Perhaps they don't like us very much for capturing their smaller brethren with our minds. Or perhaps they just live uniquely numberish lifestyles, out there beyond our ken." – Douglas Reay
A sequence of increasingly nested omega infinities: 'countable' infinity (which is how many numbers you'd count if you counted for ever – this is the first member of the first class of infinities), the infinitieth class of infinity after that one, the infinitieth-infinitieth class of infinity, and so on. And at the limit of *this* infinite sequence, we find the first of the 'fixed points of the omega function' - infinite ordinal numbers N sufficiently large that the Nth infinity is equal to N itself


Tom said...

Big? Totally fuxkcd!

Louis Epstein said...

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