[FOM] Convincing Edward Nelson that PA is consistent
Alasdair Urquhart
urquhart at cs.toronto.edu
Wed Jun 20 11:19:10 EDT 2018
Pavel Pudlak has some good results in this area. His paper
"Improved bounds to the length of proofs of finitistic consistency
statements" has a lower bound of n/(log n)^2 for proofs of
Con_T(n), which says "There is no proof of 0=1 of length <=n in T".
This paper appeared in Contemporary Mathematics Volume 65, 1987.
On Tue, 19 Jun 2018, Timothy Y. Chow wrote:
> After sending my last message, I realized that there are a couple of obvious
> additional comments that I should have made. Consider the statement
>
> (*) There is no PA proof of "0=S0" of length < 2^1000.
>
> It would go a long way towards answering the questions I raised if one could
> get good bounds on the length of the shortest proof of (*) in various systems
> S.
>
> Even for S = PA, it's not clear to me that (*) has a proof of length less
> than 2^1000, although I think it is plausible that there might be. Note that
> naively exhausting all PA proofs is going to require something like
> c^(2^1000) steps for some constant c.
>
> There's a pedestrian way to go about proving (*) in PA. The tough part is
> proving that there's no infinite decreasing sequence of ordinals below
> epsilon_0. Since we are given an upper bound of 2^1000, we don't have to go
> all the way up to epsilon_0; I think we can stop with w^w^...^w at worst,
> where the tower has height 2^1000. Constructing a PA proof of this can be
> done by induction on height, but at first glance this might take us
> significantly over 2^1000.
>
> For weaker systems S, say PRA or below, I would conjecture that there isn't
> any proof of (*) shorter than 2^1000. It would be very interesting to me to
> see any kind of provable lower bound, even for S being very weak (say,
> bounded arithmetic).
>
> Such results might show that this avenue towards "convincing Nelson that PA
> is consistent" is doomed, but even a negative result of this sort would be
> interesting in my opinion. Note that Nelson has explicitly rejected PRA and
> has even said things that suggest he might reject EFA.
>
> Tim
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