[FOM] The Arithmetical Program - An Addendum

[Frode Bjørdal]

Andrey Bovykin has asked me to stress that his joint result with De
Smet does not constitute a genuinely new independence result at the
level of ZFC+Mahlo cardinals. Instead, the result of theirs which I
invoke must be seen as an attempt to achieve maximal concreteness
based upon Proposition E in Friedman's Boolean Relation Theory.

Of these matters I was very much aware, and I am sorry if my
contribution may perhaps be so interpreted as not to take them into
account. For me this was implicitly very clearly present, as I have
studied much of Friedman’s work including important parts of his long
book manuscript with great interest. In such situations one may come
to take important pieces of information for granted, or state
something imprecisely.

Let me emphasize strongly that it in no way has been my intention to
diminish the seminal and profound lifetime achievements of Harvey
Friedman! It is upon such important independence results that Bovykin
and De Smet have based their work.

On the other hand, the arithmetical concreteness, PI(0-2), of the
Bovykin and De Smet sentence, and its equivalence over ACA(0)' with
1-Con(SMAH) was what enabled me to combine ideas to articulate what I
take to be a perhaps feasible Arithmetical Program. The existence of a
PI(0-2) (PI(0-1)) sentence equivalent with 1-Con(SMAH) (CON(SMAH))
over ACA(0)’ and weaker theories has been known for a long time; but
such knowledge would of course not in itself be sufficient for the
Arithmetical Program as I have suggested it. Using one of Friedman’s
recent results, Proposition E in BRT, Bovykin and De Smet are able to
display an example of such a PI(0-2) sentence, a “prefixed polynomial
expression” in their language, equivalent over ACA(0)’ with
1-Con(SMAH). It is important in this connection also to point out that
De Smet and Bovykin have managed to greatly reduce the size of this
arithmetical prefixed polynomial equivalent over ACA(0)’ with
1-Con(SMAH) so that it is at this point writeable on about a page.
Compared with Proposition E their sentence is small in size. But it is
as of yet an open problem whether this work can be carried far enough
so as to enable the Arithmetical Program I envision.


-- 
Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslo
www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html