Re: Impredicativity in Rosennean parlance

[Torkel Franzen <***>]

Judith Rosen says:

>How would you define "factual"?

  Not metaphorical, as conventionally understood. Thus, in the present
contexte, I'm only wondering

1) 

  >Hilbert apparently believed that impredicativities were not
  >formalizable.

  What actual statements of Hilbert's does this refer to?

2)

  >In this sense, syntactic or algorithmic systems, or formalizable
  >systems, are extremely weak in entailment: ....

  Does this statement apply to formal systems as understood in logic?
If so, in what sense, and on what grounds, is it held that "almost
every why question about what is in the system cannot be answered from
within the system". What is an example of a why question that can be
answered from within the system? What is an example of a why question
that cannot be answered from within the system?

3)

  >Within a short time, however, Goedel proved his celebrated
  >Incompleteness Theorem, which in effect show that syntactic rules
  >captured only an infinitesimal part of "real" mathematics--in 
  >effect that Church's Thesis was false, even in this realm.

  Is this a statement about the incompleteness theorem? If so, what
does it mean that the incompleteness theorem showed that only an
infinitesimal part of "real" mathematics is captured by syntactic
rules? Captured in what sense? Infinitesimal how? Does "Church's
Thesis" refer to Church's Thesis as understood in logic? If so, how is
the statement to be reconciled with the fact that Church's Thesis is
not regarded as falsified by Gödel's theorem by anybody in logic, and
in particular not by Gödel himself, who on the contrary emphasized
the importance of Church's Thesis for the applicability of the
theorem?