[FOM] The Arithmetical Program

Frode Bjørdal frode.bjordal at ifikk.uio.no
Tue Feb 22 15:25:04 EST 2011

The Arithmetical Program

Hilbert had his Finitist Program, which Bill Tait influentially
identified with Primitive Recursive Arithmetic. In the wake of
Friedman’s seminal and profound research, recent progress in what is
now often known as Unprovability Theory may seem to suggest that
theories like ZFC and stronger ones may be justifiable by quite
arithmetical means. By quite arithmetical means I here include the
means of ACA plus informal uses of the omega rule as when we come to
realize that Gödel's sentence is indeed true.

Andrey Bovykin and Michiel De Smet have recently found a concrete
PI(0-2) sentence equivalent over ACA(0)' with 1-Con(SMAH), where SMAH
is ZFC+{exists a strong n-Mahlo cardinal} for n in Omega. Call this
sentence the BDS-sentence. If written up in its standard form (with
one universal quantifier restricted to N in front), BDS will generate
omega Sigma(0-1) sentences by instantiating with the natural numbers.
Call these the Sigma(0-1) wittnesses. As PA is Sigma(0-1) complete,
all the Sigma(0-1) witnesses are provable in PA, if true.

Given this very impressive recent progress, it seems to this author
that it just might be that we may come to realize that BDS is true on
quite arithmetical grounds - similarly, but somewhat more complexly,
as for Gödels sentence. If so, we should add BDS to our formal
subsystem ACA.  We here call this ACA+BDS. The latter subsystem will
then also prove 1-Con(SMAH), and thence also Con(SMAH) . As ACA+BDS
extends Weak König Lemma, which is equivalent over RCA(0)  to Gödel's
Completeness Theorem, ACA+BDS will thence contain a countable model of
SMAH. So we can, if this carries through, even as skeptics towards the
uncountable, still think that mathematicians who do set theory of
various strengths beyond ZFC at least do not contradict themselves. Of
course, the model of ACA+BDS will be Skolem-strange in the obvious
sense, and will not commit us to the, in this author’s opinion, much
more dubious cumulative hierarchy which inter alia Zermelo so strongly
professed (and in his case even second orderly) against the insights
of Skolem.

I point out that the Arithmetical program has the potential virtue of
great flexibility, as it for all we know is the case that also e.g.
both Solovay's set theory with the Axiom of Determinacy and Quine's NF
may be justified (or perhaps better: explained), even though it is the
case that Solovay's set theory contradicts ZFC.

It would be very nice if many steps of an Arithmetical Program as
described here could be carried through.

Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslo

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