[FOM] Bound in Gödel 1931
Alasdair Urquhart
urquhart at cs.toronto.edu
Tue Feb 28 19:55:49 EST 2017
I think that Goedel's bound is correct. I argue as follows.
Let a_1, a_2, ..., a_k be a sequence of positive numbers. Then
the Goedel number gn(a_1, ..., a_k) of the sequence is
g = 2^{a_1} . 3^{a_2} . ... . p_k^{a_k},
where p_k is the kth prime number. By induction on k
we can prove the bound
g <= (p_k)^{a_1 + ... + a_k}.
Let a_1, ..., a_k and b_1, ..., b_l be two finite sequences
of positive numbers, x and y their gn's. Then the gn x*y
of their concatenation is bounded by
(p_{k+l})^{a_1 + ... + a_k + b_1 + ... +b_l}.
However,
a_1 + ... + a_k <= 2^{a_1} + 3^{a_2} + ... + (p_k)^{a_k}
<= 2^{a_1} . 3^{a_2} . ... . (p_k)^{a_k}
= x,
since all the terms in the second sum are >= 2. Hence,
gn(a_1, ..., a_k, b_1, ..., b_l) <= (p_{k+l})^{x+y},
which is Goedel's bound.
On Mon, 27 Feb 2017, Richard Heck wrote:
>
> I'm teaching Gödel's classic paper on the incompleteness theorems and have been wondering about the bound he gives in
> definition (8) of the star function. He doesn't explain where he got the bound (as he does for the one in (23)), and I
> don't see any simple way to explain why it works. If I parrot the sort of thing Gödel says in footnote 35, it'd be
> something like this.
>
> Each of the prime factors in x * y is <= Pr(l(x) + l(y)). The exponent on each of these is <= max(x,y) or, if you
> prefer, x+y. And there are l(x) + l(y) factors. So
>
> x*y <= Pr(l(x) + l(y)) ^ { (x+y)(l(x) + l(y) }.
>
> Or, since l(x) + l(y) <= x + y, we could do:
>
> x*y <= Pr(l(x) + l(y)) ^ ((x+y)^2)
>
> Whereas Gödel has
>
> x*y <= Pr(l(x) + l(y)) ^ (x+y).
>
> It's not that I think this bound doesn't work. It probably does. (In general, Gödel's bounds are pretty loose.) But it's
> surprising he doesn't say anything about it, since this is not exactly obvious. Might there by a typo or thinko here?
>
> Any thoughts?
>
> Richard Heck
>
>
> --
> -----------------------
> Richard G Heck Jr
> Professor of Philosophy
> Brown University
>
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