Steeple #3: Goodstein s Theorem (glimpse only!) Selmer Bringsjord (with Naveen Sundar G.) Are Humans Rational? v of 12717 RPI Troy NY USA
Back to the beginning
Back to the beginning Main Claim
Back to the beginning Main Claim
And Supporting Main Claim
And Supporting Main Claim
And Supporting Main Claim
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*)!
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!)
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!)
So what would be a specific g*? A truth of arithmetic that you can t prove from the axioms of arithmetic?!?
So what would be a specific g*? A truth of arithmetic that you can t prove from the axioms of arithmetic?!? Here you go:
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!
Goodstein Sequence; Goodstein s Theorem...
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
Grow Function
Example of Grow
Goodstein Sequence For any natural number m
Goodstein Sequence For any natural number m
Sample Values
Sample Values
m Sample Values
Sample Values m 2 2 2 1
Sample Values m 2 2 2 1 3 3 3 3 2 1
Sample Values m 2 2 2 1 3 3 3 3 2 1 4 4 26 41 6 83 19 139... 11327 (96th term)...
Sample Values m 2 2 2 1 3 3 3 3 2 1 4 4 26 41 6 83 19 139... 11327 (96th term)... 5 15 ~1 13 ~1 155 ~1 2185 ~1 3636 1 695975 1 15151337...
Yet, The Gödel-Vindicating Theorems!!
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).
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!
Under the hood
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +!
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 +3 3 +3 1 <!!!! +!! +! 1
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 19 3 =4 444 +3 3 +3 1 <!!!! +4 4 1 <!!!! +!! +! 1 +!! 1
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 19 3 =4 444 19 4 =5 555 +3 3 +3 1 <!!!! +4 4 1 <!!!! +5 +5 1 <!!!! +!! +! 1 +!! 1 +! +! 1
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 19 3 =4 444 19 4 =5 555 +3 3 +3 1 <!!!! +4 4 1 <!!!! +5 +5 1 <!!!! 19 5 =6 666 +6 <!!!! +!! +! 1 +!! 1 +! +! 1 +!
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 19 3 =4 444 19 4 =5 555 +3 3 +3 1 <!!!! +4 4 1 <!!!! +5 +5 1 <!!!! 19 5 =6 666 +6 <!!!! +!! +! 1 +!! 1 +! +! 1 +!...
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 19 3 =4 444 19 4 =5 555 +3 3 +3 1 <!!!! +4 4 1 <!!!! +5 +5 1 <!!!! 19 5 =6 666 +6 <!!!! +!! +! 1 +!! 1 +! +! 1 +!...
Under the hood 19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 19 3 =4 444 19 4 =5 555 +3 3 +3 1 <!!!! +4 4 1 <!!!! +5 +5 1 <!!!! 19 5 =6 666 +6 <!!!! +!! +! 1 +!! 1 +! +! 1 +!... strictly decreasing
19 1 =2 222 +2 2 +2 <!!!! +!! +! 19 2 =3 333 19 3 =4 444 19 4 =5 555 +3 3 +3 1 <!!!! +4 4 1 <!!!! +5 +5 1 <!!!! 19 5 =6 666 +6 <!!!! +!! +! 1 +!! 1 +! +! 1 +!... strictly decreasing
Probably we all agree that a chimp could never get this, but could a future computing machine get this??...
Conjecture (C) (variant of Isaacson s Conjecture )
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.
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!
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