My suggestion
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It would be so mind boggling that I struggle to think of useful examples that illustrate how insane it is (like odds of every person on Earth choosing one particle out of the universe at random and everyone happens to pick the same particle), that don't apply just as well to something like 9↑↑↑9, or even 9!!!! for that matter.Regarding this as a stepping stone towards Graham's number, how about utterly ruin that by allowing to replace 9 with G - Graham's number itself?
My intuition said so, but proving it does not sound intuitive. The constraint "5 symbols" should probably allow any digit, constant, operator or function even if several signs are used to express the function. After all, the factorial could have been written as a function !(x), but by convention we use postfix notation. Conceptually, operators and functions are equal. This means that 𝚻(G!) really is only three symbols, so we can have 𝚻(𝚻(𝚻(𝚻(G)))), which we could write as G𝚻𝚻𝚻𝚻 in postfix notation. Digit also becomes superfluous since we include constants. We still have to keep a somewhat loose requirement that the function or constant must have some kind of use in reasoning about something. Otherwise there is nothing stopping us from defining endless new functions (consider TREEω using the reasoning in the video, and so on).Yup! And 𝚻(G!) with G standing for Graham's number (g_64) is by far the better option than g(𝚻!), with 𝚻 standing for TREE(3).
I think I made a pretty lame definition of ↟ in saying that that a↟b is simply TREE(TREE(...TREE(a)...) with depth b. How about something like this?:G↟G Oh God...
Would Graham's number even exceed the number of all subatomic particles in the multiverse if we consider our universe to be average and string theory landscape's prediction of 10^500 universes to be correct, Wats?It would be so mind boggling that I struggle to think of useful examples that illustrate how insane it is (like odds of every person on Earth choosing one particle out of the universe at random and everyone happens to pick the same particle), that don't apply just as well to something like 9↑↑↑9, or even 9!!!! for that matter.Regarding this as a stepping stone towards Graham's number, how about utterly ruin that by allowing to replace 9 with G - Graham's number itself?
I was thinking as well of other methods of generating stupidly large numbers, but the constraint of using just 5 symbols makes it interesting. TREE(3) came to mind for instance, but you would have to discount the parentheses. I don't know if TREE(3) is bigger than G↑(G!)G, though I would not be surprised if it is. (If it's not then just feed it a slightly bigger number, since TREE(n) grows faster than the tetration process behind Graham's number.) Rather, what I find totally surprising is that it manages to be so huge (especially given that TREE(2) is 3 and TREE(1) is 1), and yet it isn't infinite.
You have no idea how small that number of subatomic particles is compared to Graman's number. Not even the number of permutation's of Planck volumes over the number of Planck times in the universe since the Big Bang will even begin to describe Graham's number. There simply isn't a direct way to relate Graham's number with anything physical. To give you an idea: The number of subatomic particles in the universe is much smaller than even the number of digits in Graham's number. Much smaller than the number of digits of the number of digits. And so on for a number of times greater than the number of subatomic digits. Only now are we approaching the order (or shall I say order-order or order-order-order) of magnitude of Graham's number. And we were just getting started with Graham's number using it as seed for the most rapidly growing function imagined with a simple description, and then applying recursion on that.Would Graham's number even exceed the number of all subatomic particles in the multiverse if we consider our universe to be average and string theory landscape's prediction of 10^500 universes to be correct, Wats?
The argument can be made that if the universe/multiverse is mathematical than rather than the Big Bang, there must be a Big Bounce because if the universe/multiverse is mathematical than all numbers must be able to be described within the entirety of its existence. So would say Max Tegmark. We could apply a recursive function to the multiverse also, as you can envision universes existing inside black holes inside universes, and so on (and on and on lol)You have no idea how small that number of subatomic particles is compared to Graman's number. Not even the number of permutation's of Planck volumes over the number of Planck times in the universe since the Big Bang will even begin to describe Graham's number. There simply isn't a direct way to relate Graham's number with anything physical. To give you an idea: The number of subatomic particles in the universe is much smaller than even the number of digits in Graham's number. Much smaller than the number of digits of the number of digits. And so on for a number of times greater than the number of subatomic digits. Only now are we approaching the order (or shall I say order-order or order-order-order) of magnitude of Graham's number. And we were just getting started with Graham's number using it as seed for the most rapidly growing function imagined with a simple description, and then applying recursion on that.Would Graham's number even exceed the number of all subatomic particles in the multiverse if we consider our universe to be average and string theory landscape's prediction of 10^500 universes to be correct, Wats?
That doesn't make sense to me. I don't see why a "mathematical" universe couldn't be a subset of mathematics. If "mathematical" universe means that the universe can be fully described by mathematics, why couldn't mathematics also describe abstract, non-physical things?The argument can be made that if the universe/multiverse is mathematical than rather than the Big Bang, there must be a Big Bounce because if the universe/multiverse is mathematical than all numbers must be able to be described within the entirety of its existence. So would say Max Tegmark. We could apply a recursive function to the multiverse also, as you can envision universes existing inside black holes inside universes, and so on (and on and on lol)