With work, you can formulate Goodstein\'s theorem as a statement in formal arith

ID: 3006488 • Letter: W

Question

With work, you can formulate Goodstein's theorem as a statement in formal arithmetic.

(Quantifiers, logical symbols such as and, or, not, variables, * + 0 1)

Peano wrote down the axioms for arithmetic that we take as standard.

...algebraic relationships like commutive, associative and distributive laws

...all the (ordinary) induction you can eat.

Some time in the 1980's it was shown that Goodstein's Theorem had no proof in

the formal theory of Peano arithmetic.

The proof we just gave involved working with infinities...and this seems necessary

even though Goodstein's Theorem says something purely about finite numbers.

I can definite a function Goodstein(n) which equals the number of steps it takes to

reach 0 if I start with number n and B = 2.

Homework Compute the first few values of Goodstein(n) until you have to give up.

given base = 2

n = n

n = c0bk+c1bk-1+..........+ck-1b+ck

here

c0 >0 aand 0 c1 < b for all 0 i k

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