Steeple #3: Goodstein s Theorem (glimpse only!)
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1 Steeple #3: Goodstein s Theorem (glimpse only!) Selmer Bringsjord (with Naveen Sundar G.) Are Humans Rational? v of RPI Troy NY USA
2 Back to the beginning
3 Back to the beginning Main Claim
4 Back to the beginning Main Claim
5 And Supporting Main Claim
6 And Supporting Main Claim
7 And Supporting Main Claim
8 Gödel s First Incompleteness Theorem Suppose that elementary arithmetic (i.e., PA) is consistent (no contradiction can be derived in it) and program-decidable (there s a program P that, given as input an arbitrary formula p, can decide whether or not p is in PA). Then there is sentence g* in the language of elementary arithmetic which is such that: g* can t be proved from PA (i.e., not PA - g*)! And, not-g* can t be proved from PA either (i.e., not PA - not-g*)!
9 Gödel s First Incompleteness Theorem Suppose that elementary arithmetic (i.e., PA) is consistent (no contradiction can be derived in it) and program-decidable (there s a program P that, given as input an arbitrary formula p, can decide whether or not p is in PA). Then there is sentence g* in the language of elementary arithmetic which is such that: g* can t be proved from PA (i.e., not PA - g*)! And, not-g* can t be proved from PA either (i.e., not PA - not-g*)! (Oh, and: g* is true!)
10 Gödel s First Incompleteness Theorem Suppose that elementary arithmetic (i.e., PA) is consistent (no contradiction can be derived in it) and program-decidable (there s a program P that, given as input an arbitrary formula p, can decide whether or not p is in PA). Then there is sentence g* in the language of elementary arithmetic which is such that: g* can t be proved from PA (i.e., not PA - g*)! And, not-g* can t be proved from PA either (i.e., not PA - not-g*)! (Oh, and: g* is true!)
11 So what would be a specific g*? A truth of arithmetic that you can t prove from the axioms of arithmetic?!?
12 So what would be a specific g*? A truth of arithmetic that you can t prove from the axioms of arithmetic?!? Here you go:
13 So what would be a specific g*? A truth of arithmetic that you can t prove from the axioms of arithmetic?!? Here you go: That the Goodstein Sequence eventually reaches zero!
14 Goodstein Sequence; Goodstein s Theorem...
15 Pure base n representation of a number r Represent r as only sum of powers of n in which the exponents are also powers of n etc
16 Grow Function
17 Example of Grow
18 Goodstein Sequence For any natural number m
19 Goodstein Sequence For any natural number m
20 Sample Values
21 Sample Values
22 m Sample Values
23 Sample Values m
24 Sample Values m
25 Sample Values m (96th term)...
26 Sample Values m (96th term) ~1 13 ~1 155 ~ ~
27 Yet, The Gödel-Vindicating Theorems!!
28 Yet, The Gödel-Vindicating Theorems!! Theorem 1 (Goodstein s Theorem). For all natural numbers, the Goodstein sequence reaches zero after a finite number of steps. Theorem 2 (Unprovability of Goodstein s Theorem). Goodstein s theorem is not provable in Peano Arithmetic (PA) (or any equivalent theory of arithmetic).
29 Yet, The Gödel-Vindicating Theorems!! Theorem 1 (Goodstein s Theorem). For all natural numbers, the Goodstein sequence reaches zero after a finite number of steps. Theorem 2 (Unprovability of Goodstein s Theorem). Goodstein s theorem is not provable in Peano Arithmetic (PA) (or any equivalent theory of arithmetic). So, Gödel was right, empirically! We have in GT a truth of elementary arithmetic that we can t prove from elementary arithmetic!
30 Under the hood
31 Under the hood 19 1 = <!!!! +!! +!
32 Under the hood 19 1 = <!!!! +!! +! 19 2 = <!!!! +!! +! 1
33 Under the hood 19 1 = <!!!! +!! +! 19 2 = = <!!!! <!!!! +!! +! 1 +!! 1
34 Under the hood 19 1 = <!!!! +!! +! 19 2 = = = <!!!! <!!!! <!!!! +!! +! 1 +!! 1 +! +! 1
35 Under the hood 19 1 = <!!!! +!! +! 19 2 = = = <!!!! <!!!! <!!!! 19 5 = <!!!! +!! +! 1 +!! 1 +! +! 1 +!
36 Under the hood 19 1 = <!!!! +!! +! 19 2 = = = <!!!! <!!!! <!!!! 19 5 = <!!!! +!! +! 1 +!! 1 +! +! 1 +!...
37 Under the hood 19 1 = <!!!! +!! +! 19 2 = = = <!!!! <!!!! <!!!! 19 5 = <!!!! +!! +! 1 +!! 1 +! +! 1 +!...
38 Under the hood 19 1 = <!!!! +!! +! 19 2 = = = <!!!! <!!!! <!!!! 19 5 = <!!!! +!! +! 1 +!! 1 +! +! 1 +!... strictly decreasing
39 19 1 = <!!!! +!! +! 19 2 = = = <!!!! <!!!! <!!!! 19 5 = <!!!! +!! +! 1 +!! 1 +! +! 1 +!... strictly decreasing
40 Probably we all agree that a chimp could never get this, but could a future computing machine get this??...
41 Conjecture (C) (variant of Isaacson s Conjecture )
42 Conjecture (C) (variant of Isaacson s Conjecture ) In order to produce a rationally compelling proof of any true sentence S formed from the symbol set of the language of arithmetic, but independent of PA, it s necessary to deploy concepts and structures of an irreducibly infinitary nature.
43 Conjecture (C) (variant of Isaacson s Conjecture ) In order to produce a rationally compelling proof of any true sentence S formed from the symbol set of the language of arithmetic, but independent of PA, it s necessary to deploy concepts and structures of an irreducibly infinitary nature. If this is right, and computing machines can t use irreducibly infinitary techniques, they re in trouble or: there won t ever be a Singularity!
44
45 slutten
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