FOM: Con(ZF)/its platonistic content/intensional problems.
Robert Tragesser
RTragesser at compuserve.com
Wed Aug 26 08:37:10 EDT 1998
[In part, I want to take a step toward reponding to Steve
Simpson's challenge to bring phenomenology to bear on the question of
whether or not "Con(ZF)?" is essentially a combinatorial problem by
basically siding with Steve against Neil, and doing so by indicating the
as it were Platonistic _Sinn_ of Con(ZF). I am thinking that the case I
make emerges out of Go"del's Platonism as passed through the
phenomenological filter, as Go"del was at least thinking about doing.
I'll avoid being explicit about the frame of phenomenology at issue here,
both because I want to avoid what would be called jargon here and because I
want what I say to stand on its own, as phenomenological con siderations
ought. N.B., see, say, Folledal's remarks on Husserl and Go"del in
Go"del III for very crude pointers here.]
Con(ZF) is the name for an imbroglio of propositions finally
requiring a knowledge of set theory (of which ZF is but a formulation) to
sort out, understand, evaluate:
Neil Tennant is maybe moving a bit too swiftly through the thick
wood of intensional problems rooted in Con( ).
If one is trying to rigorously prove Go"del's second incompleteness
theorem in a general setting (some significantly general class of formal
systems F), then one must struggle to find a canonical form of, or
canonical but in each case (each F) satisfiable conditions on the form of,
Con(F). (As most thoroughly discussed in Sol Feferman's "Arithmetization of
Metamathematics in a General Setting"?)
At the same time, this does not mean that the canonical form of
Con(F) right for proving the second incompleteness theorem in a
siognificantly general setting is best or optimal for framing and thinking
about, understanding, the question of the consistency of, say, ZF.
Indeed, it might be that the best, the most understandable, the most
metamathematically fruitul or salient (I don't mean these all to be
equivalent) choice in this particular case is not the canonical choice. In
particular, it is likely the case that an alternative but equivalent or
stronger formulation of set theory than ZF would yield a better
understanding of Con(ZF).
Indeed, from a Go"delian Platonist point of view, the following as
it were open proposition is rich in sense:
Where Prov.subF.[p] means that p is derivable in the formal
presentation F,
[RP] If Prov.subF.[p], then p
where F is understood to range over (what Weyl called) the "indeterminate
manifold" of formal presentations of set theory ("set theory" understood
Platonistically as a self-standing being partially capturable by formal
presentations F).
[RP] can clearly be understood to any cogent depths only by master
set theorists/logicians.
[RP] points to the platonistic content of Con[ZF].
If I had to rationally reconstruct Simpson's point about needing a
profound understanding of set theory to understand Con(ZF), it would be
along these lines, as making the most and deepest sense of the conviction
that a genuine understanding of Con(ZF) requires a profound understanding
of set theory (of which ZF is one special framing).
robert tragesser
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