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:
“The next generation of women will enter a world in which they are perceived to have more opportunities for creating fulfilling lives than women have ever had before.”
—Elizabeth Debold (20th century)
“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)
“Good government cannot be found on the bargain-counter. We have seen samples of bargain-counter government in the past when low tax rates were secured by increasing the bonded debt for current expenses or refusing to keep our institutions up to the standard in repairs, extensions, equipment, and accommodations. I refuse, and the Republican Party refuses, to endorse that method of sham and shoddy economy.”
—Calvin Coolidge (1872–1933)