Bill Dubuque on the Goodstein Theorem
Following is a post to sci.math by Bill Dubuque, 11th December 1995. It is still on Google Groups but the formatting was lost when they changed the font and removed extra whitespace.
Bill Dubuque to sci.math Dec 11, 1995, 3:00:00 AM From: "Kenneth A. Regas" <[email protected]> Date: 10 Dec 1995 21:17:14 GMT ... What I'm getting at here is that G [Godel's undecidable sentence] is not the kind of sentence that one would come up against and be obliged to evaluate in the world of physical phenomena. It seems to be purely a product of "formal systems." It is my layman's understanding that Goedel's proof relies upon constructing a proposition like G. In that case, does Goedel's theorem have anything to say about "old mathematics" the kind that scientists and engineers are taught, wherein which "statements" such as G don't come up? Yes, recent discoveries yield a number of so-called 'natural independence' results that provide much more natural examples of independence than does Godel's contrived example based upon the liar paradox. See the bibliography below. In particular, your question is essentially the title of Gina Kolata's article "Does Godel's theorem matter to mathematics?", referenced below. As an example of such results, I'll sketch a simple example due to Goodstein of a concrete number theoretic theorem whose proof is independent of formal number theory (following [Sim]). Let b be a positive integer >= 2. Any nonnegative integer n can be written uniquely in base b representation: n n 1 k n = c b + ... + c b 1 k where k >= 0, 0 < c < b, n > ... > n >= 0, for i = 1, ..., k. i 1 k For example the base 2 representation of 266 is 8 3 266 = 2 + 2 + 2 We may extend this by writing each of the exponents n , ..., n 1 k in base b notation, then doing the same for each of the exponents in the resulting representations, ... until the process stops. This yields the so-called 'hereditary base b representation of n' For example the hereditary base 2 representation of 266 is 2 + 1 2 2 + 1 266 = 2 + 2 + 2 Let B[b](n) be the nonnegative integer which results if we take the hereditary base b representation of n and then syntactically replace each b by b+1, i.e. B[b] is a base change operator that 'Bumps the Base' from b up to b+1. For example bumping the base from 2 to 3 in the above equation yields 3 + 1 3 3 + 1 B[2](266) = 3 + 3 + 3 Consider a sequence of integers obtained by repeatedly applying the operation: bump the base then subtract one from the result. For example, iteratively applying this operation to 266 yields 2 + 1 2 2 + 1 266[0]:= 2 + 2 + 2 = 266 3 + 1 3 3 + 1 266[1] = 3 + 3 + 3 - 1 = B[2](266[0]) - 1 3 + 1 3 3 + 1 = 3 + 3 + 2 4 + 1 4 4 + 1 266[2] = 4 + 4 + 1 = B[3](266[1]) - 1 5 + 1 5 5 + 1 266[3] = 5 + 5 = B[4](266[2]) - 1 6 + 1 . 6 6 + 1 . 266[4] = 6 + 6 - 1 . 7 using 6 - 1 = 10000000 - 1 = 5555555 in base 6 6 + 1 6 6 5 = 6 + 5 6 + 5 6 + ... + 5 6 + 5 7 + 1 7 7 5 266[5] = 7 + 5 7 + 5 7 + ... + 5 7 + 4 . . . 266[k+1] = ... = B[k+2](266[k]) - 1 In general, if we start this procedure at the integer n then we obtain what is known as the Goodstein sequence starting at n. More precisely, for each nonegative integer n we recursively define a sequence of nonnegative integers n[0], n[1], ..., n[k], ... by n[0] := n [ B[k+2](n[k]) - 1 if n[k] > 0 n[k+1] := | [ 0 if n[k] = 0 for all k >= 0. If we examine the above Goodstein sequence for 266 numerically we find that the sequence initially increases quite rapidly: 2^2^(2+1)+2^(2+1)+2 ~= 2^2^3 ~= 3 10^2 3^3^(3+1)+3^(3+1)+2 ~= 3^3^4 ~= 4 10^38 4^4^(4+1)+4^(4+1)+1 ~= 4^4^5 ~= 3 10^616 5^5^(5+1)+5^(5+1) ~= 5^5^6 ~= 3 10^10921 6^6^(6+1)+5*6^6+5*6^5+..+5*6+5 ~= 6^6^7 ~= 4 10^217832 7^7^(7+1)+5*7^7+5*7^5+..+5*7+4 ~= 7^7^8 ~= 1 10^4871822 8^8^(8+1)+5*8^8+5*8^5+..+5*8+3 ~= 8^8^9 ~= 2 10^121210686 9^9^(9+1)+5*9^9+5*9^5+..+5*9+2 ~= 9^9^10 ~= 5 10^3327237896 10^10^(10+1)+5*10^10+5*10^5+..+5*10+1 ~= 10^10^11 ~= 1 10^100000000000 Nevertheless, despite appearances, it can be proved that this sequence converges to 0. In other words, 266[k] = 0 for all sufficiently large k. This surprising result is due to Goodstein (1944) who actually proved the same result for *all* Goodstein sequences: GOODSTEIN'S THEOREM. For all n there exists k such that n[k]=0. In other words every Goodstein sequence converges to 0. The secret underlying Goodstein's theorem is that the hereditary expression of n in base b mimicks an ordinal notation for ordinals less than eps[0]. For such ordinals, the base bumping operation leaves the ordinal fixed whereas the subtraction of one decreases the ordinal. But these ordinals are well-ordered and this allows us to conclude that a Goodstein sequence eventually converges to zero. Goodstein actually proved his theorem for a general increasing base-bumping function f:N->N (vs. f(b)=b+1 above) and he showed that convergence of all such f-Goodstein sequences is equivalent to transfinite induction below the ordinal eps[0]. One of the primary measures of strength for a system of logic is the size of the largest ordinal for which transfinite induction holds. It is a classical result of Gentzen that the consistency of PA (Peano arithmetic, or formal number theory) can be proved by transfinite induction on ordinals below eps[0]. But we know from Godel's second incompleteness theorem that the consistency of PA cannot be proved in PA. It follows that neither can Goodstein's theorem be proved in PA. Thus we have an example of a very simple concrete number theoretical statement in PA whose proof is nonetheless independent of PA. Another way to see Goodstein's theorem cannot be proved in PA is to note that the sequence takes too long to terminate, e.g. 4[k] first reaches 0 for k = 3*(2^402653211-1) ~= 10^121210695 In general, if 'for all n there exists k such that P(n,k)' is provable, then it must be witnessed by a provably computable choice function F: 'for all n: P(n,F(n))'. But the problem is that F(n) grows too rapidly to be provably computable in PA, see [Smo] 1980 for details. Goodstein's theorem was one of the first examples of so-called 'natural independence phenomena', which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Godel. Other finite combinatorial examples were discovered around the same time, e.g. a finite form of Ramsey's theorem, and a finite form of Kruskal's tree theorem, see [KiP], [Smo] and [Gal]. [Kip] presents the Hercules vs. Hydra game which provides an elementary example of a finite combinatorial tree theorem closely related to the Goodstein example above. Kruskal's tree theorem plays a fundamental role in computer science because it is one of the main tools for showing that certain orderings on trees are well-founded. These orderings play a crucial role in proving the termination of rewrite rules and the correctness of the Knuth-Bendix equational completion procedures. See [Gal] for a survey of results in this area. See the references below for further details, especially Smorynski's papers. Start with Rucker's book if you know no logic, then move on to Smorynski's papers, and then the others, which are original research papers. For more recent work, see the references cited in Gallier, especially to Friedman's school of 'Reverse Mathematics', and see [JSL]. -Bill [Gal] Gallier, Jean What's so special about Kruskal's theorem and the ordinal Gamma[0]? A survey of some results in proof theory, Ann. Pure and Applied Logic, 53 (1991) 199-260. [HFR] Harrington, L.A. et.al. (editors) Harvey Friedman's Research on the Foundations of Mathematics, Elsevier 1985. [KiP] Kirby, Laurie, and Paris, Jeff Accessible independence results for Peano arithmetic, Bull. London Math. Soc., 14 (1982), 285-293. [JSL] The Journal of Symbolic Logic, v. 53, no. 2, 1988 This issue contains papers from the Symposium "Hilbert's Program Sixty Years Later". [Kol] Kolata, Gina Does Godel's theorem matter to mathematics?, Science 218 11/19/1982, 779-780; reprinted in [HFR] [Ruc] Rucker, Rudy Infinity and The Mind, 1995, Princeton Univ. Press. [Sim] Simpson, Stephen G. Unprovable theorems and fast-growing functions, Contemporary Math. 65 1987, 359-394. [Smo] Smorynski, Craig (all three papers are reprinted in [HFR]) Some rapidly growing functions, Math. Intell., 2 1980, 149-154. The Varieties of Arboreal Experience, Math. Intell., 4 1982, 182-188. "Big" News from Archimedes to Friedman, Notices AMS, 30 1983, 251-256. [Spe] Spencer, Joel Large numbers and unprovable theorems, Amer. Math. Monthly, Dec 1983, 669-675.This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Oct 26.
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