Systematically Creating Ever Faster Increasing Sequences
Given a strictly increasing integer sequence/function (n≥1) we can produce a faster growing sequence (where the superscript n denotes the nth functional power). This can be repeated any number of times by letting, each sequence growing much faster than the one before it. Then we could define, which grows much faster than any for finite k (here ω is the first infinite ordinal number, representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.
For example, starting with f0(n) = n + 1:
- f1(n) = f0n(n) = n + n = 2n
- f2(n) = f1n(n) = 2nn > (2 ↑) n for n ≥ 2 (using Knuth up-arrow notation)
- f3(n) = f2n(n) > (2 ↑)n n ≥ 2 ↑2 n for n ≥ 2.
- fk+1(n) > 2 ↑k n for n ≥ 2, k < ω.
- fω(n) = fn(n) > 2 ↑n - 1 n > 2 ↑n − 2 (n + 3) − 3 = A(n, n) for n ≥ 2, where A is the Ackermann function (of which fω is a unary version).
- fω+1(64) > fω64(6) > Graham's number (= g64 in the sequence defined by g0 = 4, gk+1 = 3 ↑gk 3).
- This follows by noting fω(n) > 2 ↑n - 1 n > 3 ↑n - 2 3 + 2, and hence fω(gk + 2) > gk+1 + 2.
- fω(n) > 2 ↑n - 1 n = (2 → n → n-1) = (2 → n → n-1 → 1) (using Conway chained arrow notation)
- fω+1(n) = fωn(n) > (2 → n → n-1 → 2) (because if gk(n) = X → n → k then X → n → k+1 = gkn(1))
- fω+k(n) > (2 → n → n-1 → k+1) > (n → n → k)
- fω2(n) = fω+n(n) > (n → n → n) = (n → n → n→ 1)
- fω2+k(n) > (n → n → n → k)
- fω3(n) > (n → n → n → n)
- fωk(n) > (n → n → ... → n → n) (Chain of k+1 n's)
- fω2(n) = fωn(n) > (n → n → ... → n → n) (Chain of n+1 n's)
Read more about this topic: Large Numbers
Famous quotes containing the words creating, faster and/or increasing:
“There are characters which are continually creating collisions and nodes for themselves in dramas which nobody is prepared to act with them. Their susceptibilities will clash against objects that remain innocently quiet.”
—George Eliot [Mary Ann (or Marian)
“Wooing, wedding, and repenting, is as a Scotch jig, a measure, and a cinquepace; the first suit is hot and hasty, like a Scotch jig, and full as fantastical; the wedding, mannerly-modest, as a measure, full of state and ancientry; and then comes repentance and, with his bad legs, falls into the cinquepace faster and faster, till he sink into his grave.”
—William Shakespeare (1564–1616)
“A point has been reached where the peoples of the Americas must take cognizance of growing ill-will, of marked trends toward aggression, of increasing armaments, of shortening tempers—a situation which has in it many of the elements that lead to the tragedy of general war.... Peace is threatened by those who seek selfish power.”
—Franklin D. Roosevelt (1882–1945)