[FOM] Underdetermination of truth
Arne Hole
arne.hole at ils.uio.no
Tue Jul 14 18:49:01 EDT 2015
I have now updated my draft paper at
http://folk.uio.no/arnehole
slightly (cf. my previous FOM posting cited below). In an attempt to draw a bit more attention to the paper, I will bring a little example of the results the paper gives. This particular example is only briefly mentioned in the current draft version.
In the paper, two recursive functions alpha and beta from the set of positive integers into itself are constructed, with beta strictly increasing and alpha(n)<beta(n) for all n>0. The two functions have the property that the infinite product of all the fractions
alpha(n)/beta(n)
with n ranging from 1 to infinity, converges to a rational number in the open interval (0,1). The main results of the paper are expressed in terms of a recursive function mu from the set of positive integers into itself. For our example, let mu(n) be the value of the (1000n)th decimal of pi when pi is expanded as a decimal number in base beta(n). So, for instance, mu(3) is the value of the 3000th decimal of pi, when pi is written in base beta(3). (The number 1000 is chosen quite arbitrarily.) Then for each n, mu(n) is an integer greater than or equal to 0 and less than beta(n).
We call an integer n>0 a WINNER if mu(n) is at least as big as alpha(n). If n is not a winner, we call it a LOSER. With this choice of mu, the main theorem of the paper implies the following result, which (at least to me) is *very* strange:
Assume that the standard structure N of natural number theory is a model of PA, in the usual sense. Then one of the following two statements is true:
(1) Let gamma be an arbitrary natural number. Then there exist natural numbers phi and delta_0 such that the following holds for all integers delta > delta_0: If U is the set consisting of the integers
phi+1, phi+2, ..., phi+delta,
then the number of winners in U is greater than gamma*E, where E is the expected number of winners in U when for each n, the base beta(n) decimal mu(n) of the number pi is modelled as a random draw from the set of possible base beta(n) digits, with uniform probability.
[Thus the decimals of pi deviate from random distribution by an unbounded factor gamma.]
(2) Order the set of well-formed formulas in the language L(PA) of PA by increasing Gödel numbers, and let A_k be formula number k in the ordering. Then there exists a natural number r such that while the formula D=A_r in L(PA) is closed and true in the standard model of PA, the following holds for all integers n>r-1:
If the theory PA+A_n is consistent and proves D, then n is a loser.
(In fact, the main theorem of the draft paper implies a stronger, arguably even stranger result with this choice of mu. In the paper, the "absurdity" of results similar to the one above is used to argue that the concept of truth in N is underdetermined.)
Regards, Arne H.
>Date: Sun, 21 Jun 2015 14:04:54 +0000
>From: Arne Hole <arne.hole at ils.uio.no>
>To: "fom at cs.nyu.edu" <fom at cs.nyu.edu>
>Subject: [FOM] Underdetermination of truth
>Message-ID: <5725d07d22a04f3fae35f1c72c76663e at mail-ex02.exprod.uio.no>
>Content-Type: text/plain; charset="us-ascii"
>
>I have put a draft paper concerning the concept of truth in the standard model
>of Peano arithmetic at
>
>http://folk.uio.no/arnehole
>
>The paper uses a kind of infinite game construction to argue that the concept
>of truth in the standard model is in fact underdetermined. The draft has a long
>and troublesome history, as some of you may be aware of. Comments are
>most welcome. You may send directly to me at Arne.Hole at ils.uio.no.
>
>Best, Arne H.
>
>
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>End of FOM Digest, Vol 150, Issue 16
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