[FOM] Absolute undecidability

[Joe Shipman]

I don't think your "nontrivial information about an infinite number of bits of Omega" is a clear enough concept.

In the case where MWP is true, then there is an absolutely undecidable true statement of the form "D is the maximal winner" which gives very concrete implications about specific bits of Omega (infinitely many specific finite consecutive subsequences which are not all 1's, one for each length greater than D)..

However, in the case where MWP is false, there is only a finite amount of information that that tells us, and there is no specifiable finite set of bits about which any meaningful information may be learned.

-- JS

Sent from my iPhone

> On Sep 7, 2015, at 7:26 AM, Arne Hole <arne.hole at ils.uio.no> wrote:
> 
> Earlier this summer I posted a link to a draft paper on the subject of absolute undecidability (underdetermination of truth). I have received some useful comments, but I still feel that the main point, which I think is quite significant,  has not come across. Therefore, in an attempt to make things as transparent as possible, I have now made a short PP presentation giving a simple example based on my results. You may find it at
> 
> http://folk.uio.no/arnehole/AbsUndec.pdf
> 
> This should take only some minutes to scan through. It is shown that if all closed formulas in the language L of PA are either true or false in the standard model N, then for each real number r whose base 2 decimals are definable in L, there is a closed formula A_r in L such that if we are able to decide the truth value of A_r in N, then we will have nontrivial information concerning an infinite number of decimal bits in r. As an example, I take r to be the halting probability known as Chaitin's Omega. Other interesting examples include pi, the Euler number e etc. 
> 
> Best, Arne H.
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