I have only just been able to return to your critical examination of the extent of predicativity in this paper. I welcome your extensive and detailed consideration of the issues involved and am impressed by your absorption of the extant literature and your strong philosophical motivations allied to predicativity to see whether existing analyses are satisfactory or require us to go beyond them. But I can do without the supercilious attitude manifested here and there, especially in the last paragraph on p. 16 (of which, more below). Here are some points in response. I did not have time to go over your other two related papers [47] and [48]; it appears that your critique here is largely independent of those.

**p.1, last par.** I disagree completely with the statement that "the
limitation identified by Feferman and Schütte is probably now a primary
reason, possibly *the* primary reason, for predicativism's nearly
universal unpopularity." I think the reasons that it has not achieved the
status of the other big foundational schemes, especially constructivism
and set-theoretical platonism, have nothing to do with its formal
characterization and in fact well predated the work that began in the
1950s in that direction. The main reasons are given in my survey article
"Predicativity" [44] in the section "Predicativity sidelined: 1920-1950,"
and I shall not repeat them here. More to the point is, as you remark,
the predicative unprovability, on the F-S characterization, of such
statements as the Cantor-Bendixson theorem or the (Extended) Kruskal
Theorem or the recognition of Gamma_{0} as an ordinal, which working
mathematicians (do or would) accept without qualm. But that does
not cut any ice one way or another regarding our characterization. If it
is correct, so much the worse for predicativity.
If it is incorrect, and something like what you propose in sec. 2 is
correct, then there will surely be other examples of mathematical theorems
unprovable in such a characterization that working mathematicians would
readily accept and would lead them to ignore your foundational scheme too.
Mathematicians for the most part simply don't like to have their hands
tied and are not persuaded on philosophical grounds to abandon reasoning
that is totally justified on the face of it, such as, to begin with, the
use of the lub principle in analysis. The situation with constructivism
is a bit different; though not accepted as a foundation for mathematics by
99.99% of working mathematicians, they can see the point of doing
mathematics in a way that leads systematically to computable results, even
if in principle only. But the outcome of the Polya-Weyl wager showed that,
for mathematicians, the rejection of impredicative definitions in analysis
is not at all compelling in the same way. So the issue of whether the F-S
characterization, or as you put it, "the Gamma_{0} thesis" is correct has to
be taken on its own merits, if one is interested in it at all.

**p.2, sec.1, 3d par.**
You say that the Gamma_{0} thesis "has been repeatedly
and forcefully defended by its two major figures, Feferman and Kreisel.
Many current authors simply assert it as a known fact." There are two
"facts" at issue here: (UB) Gamma_{0} is an upper bound for the
proof-theoretical strength of predicativity. (LB) It is a lower bound.
Both Kreisel and I have raised critical questions about the proposed
characterization at different stages, so it is not fair to say without
qualification that we forcefully defended it. And others have raised
questions, as you note here and there. And, after all, few logicians have
been concerned with the characterization. We're not talking about the
"passive" involvement of a big "scientific community" here (p.16).
I think most people who have looked at it are persuaded by the autonomy
picture and the result (UB) on the face of it. It has nothing to do with
the complexity of the argument, which is now what one learns in
straightforward Schütte-style infinitary proof theory after Epsilon_{0}
as an upper bound to PA. That's just the bread and butter of the subject.
Your objections to the (UB) result are not novel, as you yourself note.
At the same time (curiously) you object to (LB) not as a result but for
the way it has been proved. Of those thinkers on the subject who have
accepted (UB) as patently correct, few have thought about (LB). I brought
attention to the question of its justification in "A more perspicuous
system for predicativity" [13].

**p.4, 2nd par. up,** "*U*(NFA) employs a patently impredicative least fixed
point operator". This is completely wrong. The point is brought up again
on p. 15 and I will address it there.

**p.5, 1.3, par.1,** "If we could show that *A*
[a rational actor who has accepted some foundational stance]
would accept every member of some set of statements *S*,
then *A* should see this too and then be able to go beyond *S*,
e.g. by asserting its consistency." There is a basic ambiguity here.
It goes without saying that if *A* sees that it has accepted every member
of *S*, then it should accept Con(*S*)--that's just an instance of the
reflection principle for *S*. But this will not be the case if *A* comes to
accept each member of *S* step by step, requiring a new exercise of
reasoning at that step, and does not have a single act of reasoning that
allows it to accept all of *S*. If one follows your line on this, it would
appear that no foundational stance can be characterized by a formal system
*S*, because if one makes it convincing that *A* ought to accept "all" of *S*,
then *A* can go beyond *S*. So, there will be no complete characterization of
finitism, no complete characterization of predicativity, etc. There will
be no ordinal bound that can be established for predicativity.

**p.7,** you say that if one accepts the rule of inference (*) one should
accept the corresponding generalized implication (**). A simple example where
this kind of argument fails occurs in modal systems that have a rule of
inference A/NA, where NA (or "box"A) says that A is necessary, but we
don't have A->NA. Or take NA to be Prov('A') in arithmetic.

**p.9(d)**, re my position on the characterizations of predicativity in "A
more perspicuous..." using the systems *P* + Ex/*P*, "Reflecting on
incompleteness" using the system Ref*(PA(*P*)), and "The unfolding of
non-finitist arithmetic" using the system *U*(NFA), vs. the earlier
characterization via progressions: you say that I nowhere openly repudiate
the earlier systems. On the contrary, I said in the first of these
that the latter make use of the prima-facie impredicative concepts of
ordinal or well-ordering in their description, and the point of the work
in [13] as in the succeeding papers is to show how that can be avoided.
I did not repudiate the results of the earlier work which made the (UB) to
predicativity plausible in the context of the history of the subject which
led to the consideration of autonomous ramified progressions.

In view of this, your critiques of the earlier systems are moot, and I shall not deal with them.

**pp.10-11, sec.1.6,**
the linked systems *P* + Ex/*P*. The reason for taking *P*
to be the primary system and Ex/*P* an auxiliary system is that the
existential quantifier over functions and predicates is not to be given
first class status as a logical operator. Ex/*P* only serves to show how
things may be established to exist, either directly or by construction
from a prior existent. This is analogous to the notion of existence in
finitism. See Gödel's early understanding of this in his Collected Works
vol. III, p. 51: "Negatives of general propositions (i.e., existence
propositions) are to have a meaning in our systme only in the sense that
we have found an example but, for the sake of brevity, do not state it
explicitly. I.e., they serve merely as an abbreviation and could be
entirely dispensed with if we wished." In our case, these were dispensed
with in the unfolding paper via the use of definedness as a basic
pre-logical notion.

**p.11, last par.** Your proposed inference to Sigma-1-1 comprehension (and
thence Pi-1-1 comprehension) can't be carried out in *P*, because analytic
formulas are not in the language of *P*.

**pp.12-14, sec. 1.7,** the critique of Ref*(PA(*P*)). This approach was a way
station in trying to see predicativity as one instance of the more general
explanation of what one ought to accept if one has accepted given
principles, and was not completely satisfactory conceptually through its
use of a Kripkean partial truth predicate. But it was important for
stressing the use of open-ended schematic systems and for illustrating the
general program. It was superseded by the unfolding concept, so I shall
not defend it further. But you have not understood (p. 13) how *T*
and *F*
are to be construed relative to an arbitrary definite predicate *P*.

**p.15(b),**
"Way too strong. *U*(NFA) is flatly impredicative in two ways." The
first is supposed to be Ax 7 (p.82) of the unfolding paper. This axiom
simply says that we have a name for each basic predicate symbol *P*; our
operations on names serve to generate names of expressions built up from
*P*, and that are definite if *P* is definite, so this is completely
unproblematic. The second way you criticize,
"the really striking impredicativity...is its use of a least fixed point
operator". But as you point out yourself in fn.10, the leastness is never
used by us, and in any case, the apparent impredicativity here is
completely eliminable, as it is in general for partial recursive functions
by approximation from below; the standard proof is given op.cit. p. 80.
In notes,
Strahm and I have also reworked our approach using an untyped partial
combinatory calculus instead; that is both formally simpler, yields the
needed fixed points in a trivial way, and never requires the leastness
condition. So there is no impredicativity here either, let alone
"flatly".

**p.16, 2nd full par.**
Your principal objection to the three systems is "a basic
impredicativity involving the ability to substitute possibly meaningless
formulas for free set variables." (And this is your objection to the way
(LB) is established.) The essential applications of the substitution rule
rule, to infer from *A(P)* the result of substituting *B(t)* for
*P(t)* are those where *B(t)* simply says that *t* is defined
and of a given type. This
is not meaningless. Or at least you ought to grant it as being
"intelligible" in your sense (sec. 2.3 ff) as distinct from being
"meaningful", and thus something we can reason about.

**p.16, last par.** "...a properly functioning scientific community should be
expected to debate and criticize major ideas, not to passively accept
them, regardless of the stature of their author and the complexity of the
argument. That this was not done in a serious way in the present case
suggests that the community as a whole did not function in the way that
it should have." Really! This reckless, grandiose statement is
offensive and should be stricken. I don't know of any one who passively
accepted the (UB) result for either reason, nor did the few who thought
about the (LB) result and raised questions about it.

**p.17.** The "three basic principles" of your "mathematical conceptualism"
are very vague and do not by themselves dictate the choices you make in
developing your own proposal for what it is predicatively acceptable in
the following pages.

**p.18, 1st par,** "...we do not expect to be able to fully formalize exactly
how far we can go." My comment on p.5 above applies here too.

**p.18, 2.2,** "...intuitionistic logic [is] the appropriate took for general
predicative reasoning." It was pointed out in "A more perspicuous..." and
the unfolding paper, that the results hold equally well for the
intuitionistic versions of the systems introduced there having Gamma_{0} as
both upper and lower bound. I think your argument for the use of
intuitionistic logic for reasoning about formulas which are not recognized
in advance to have a definite meaning is a reasonable one, and so is your
step to marry this to the numerical omniscience scheme (p. 19). Obviously,
our own arguments hold for this logic intermediate between the classical
and strictly intuitionistic logics.

**pp. 25-26.** Given any recursive total order < on omega and an
extension *S* of pure 2nd order arithmetic in this logic, you
define the iterated Tarskian truth theory of *S* along the < relation,
Tarski_{<}*(S)* to have a basic predicate symbol Acc and for
each *a* in omega a symbol *T _{a}* for an

**pp.26-30, sec. 2.8.** Here you iterate the passage from a < relation to
Tarski_{<}(*S*) omega times, taking the < relation to be a standard one for
the ordinals up to Gamma_{0}. Taking *a _{n}* as a standard fundamental sequence
for Gamma

**p.30.** Finally you say that the omega iteration of Tarski theories
just described ought to be accepted as a single theory, and that kind of
step will allow you to capture the ordinal Gamma_{0} itself and far beyond.
But that assumes that the predicativist accepts the general proposition:
"If *S* is acceptable then for any recursive ordering < of omega, so also is
Tarski_{<}(*S*)." This is a higher order notion of acceptability which is not
justified by your interpretations.

**p.34, fn. 1.** I had to laugh when I came to this after trying to wade
through your systems, since here you also blame the failure of the
mathematical community to go predicative on "the awkwardness of
predicative systems [of the kind I have developed] in practice". Wait
till they see these systems of yours! The informal system *J _{2}*
of your
paper [48] seems to me to be hardly more attractive, but any specific
remarks I might want to make about that will have to wait for another day.

**p.37,** [17] and [23], 'Review' -> 'Revue'

**p.37,** the Shapiro volume has appeared, "The Oxford Handbook of the
Philosophy of Mathematics and Logic", Oxford University Press 2005.
My chapter, "Predicativity" is on pp. 590-624.

A final general point. You may not believe that informal notions of proof
such as those of finitism and predicativity can be characterized formally.
But if you do, the case of finitism is instructive. Kreisel characterized
it by a system of strength PA and Tait characterized it by a system of
strength PRA. What accounts for such different results? I have an
informal explanation, namely that Kreisel accepts a concept of finitist
ordinal (or well-ordering) that Tait does not accept. Strahm and I have
set up a basic system FA of finitist arithmetic whose unfolding is
equivalent in strength to PRA. I believe there should be a corresponding
basic system for FA plus a finitist notion of well-ordering, whose
unfolding is equivalent to PA. It may be that our notions of
predicativity differ in much the same way, and that if you explain more
clearly what mathematical conceptualism consists in, one can characterize
it formally as the unfolding of a suitable system, of course much stronger
than *U*(NFA).

Sincerely,

Solomon Feferman