[FOM] Absolute undecidability

[Harvey Friedman]

On Mon, 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.
>
My own take on absolute undecidability, or absolute unprovability goes
like this.

1. It would be nice to have a robust definition of what a reasonable
axiom of set theory is, even if reasonable ones may conflict. Then it
becomes possible to prove that none such reasonable extension settles
a question. I am not aware of such a thing. A good target for this
approach would be to prove that any reasonable axiom of set theory
does not settle the continuum hypothesis. This approach must be
measured by how compelling the notion of "reasonable axiom" really is.

2. One feature of all reasonable axioms that we know are that they are
reasonably short. Even if they are schemes, they are reasonably short.
thus ZFC is reasonably short. But here, the prospects of finding a
single sentence that cannot be settled by any short system of axioms
will still require some handle on "reasonable axioms" beyond being
short, because otherwise we can just add the target sentence or its
negation to ZFC and still have something short. So we are still
confronted with the difficulties in 1.

3. What I succeeded in doing is to identify a real constant c of
nature, that everybody - or everybody I like to talk to (smile) -
likes. And prove that any reasonably short axiom system is not going
to determine what c is. This is the c of the posting

579: Impossible Counting  5/26/15  8:58PM

http://www.cs.nyu.edu/pipermail/fom/2015-May/018742.html

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#88. Impossible Counting, May 26, 2015, 9 pages, draft.

c = THE NUMBER OF BINARY OPERATIONS *:A x A into A, UP TO
12-SIMILARITY - i.e., have the same restrictions to 12 element sets up
to isomorphism. Here A can range over absolutely all sets, or instead
range over countable sets, or instead be fixed at N. The same result
holds.

Harvey Friedman