Many thanks for your detailed and thoughtful criticism of my paper,
"Predicativity beyond Gamma_{0}". I give below a point-by-point
response to your comments. However, this gets into a variety of issues of
varying degrees of importance, so perhaps I should first summarize what
I see as the key points of interest.

The three central assertions of my paper are the following:

__1. All previously proposed formal systems for predicativity are too weak
in the sense that anyone who accepted the informal principles motivating
any of these systems would be able to grasp the global validity of that
system and then move beyond it.
__

__
2. All previously proposed formal systems for predicativity are also
too strong, because they all permit a predicatively illegitimate inference
of an assertion of transfinite recursion from an assertion of transfinite
induction --- and the ability to make such inferences is crucial for the
well-ordering proofs which are alleged to establish predicative provability
of all ordinals less than Gamma _{0}.
__

__
3. Using transfinite hierarchies of Tarskian
truth theories it is possible to legitimately establish the predicative
provability of all ordinals less than Gamma _{0}, and indeed of
ordinals significantly larger than Gamma_{0}.__

I think the success of my paper as a whole essentially comes down to the
success of these assertions, and you do contest all three of them, though
in a rather cursory fashion in the first two cases. Indeed, your response
to the first assertion does little more than reiterate an idea which I have
extensively criticized (about *A* coming to accept each member of
*S* step by step, requiring a new exercise of reasoning at that step),
with no further explanation and without even acknowledging my critique of
this idea. You very briefly answer the second assertion in a way that fails
to address the key distinction between (in my terminology) meaningfulness
and intelligibility.

It is only to the third assertion that you offer a
substantial response, with your criticism of the semantics of my systems
Tarski_{<}(*S*). I don't completely understand this criticism,
but it seems to rest on the premise that predicativists are not capable of
reasoning about the concept of intelligibility. This is a bit puzzling
because you evidently do agree that predicativists can reason about such
concepts as truth and second order existence. So I wonder what basis you have
for denying predicativists the ability to reason about intelligibility, if
that actually is your position.

Now to the specific points you raise.

**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 ...*

That's a different question: I was commenting on the fact that virtually
*nobody* is a predicativist nowadays. I suspect the F-S characterization
has made it unacceptable to be one. After all, although predicativism
was always a minority view, there was a definite tradition in the subject
going from Poincare and Russell to Weyl to Wang and Lorenzen. It is hard
to find anyone who openly advocated predicativism after the work on
Gamma_{0} came out in the early 1960's. A likely reason for
this is that "Gamma_{0} is sufficiently tame that it is simply
hard to take seriously any approach to foundations that prevents one from
recognizing ordinals at least this large" (p. 1).

*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
... 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.*

Certainly it has to be taken on its own merits. I wonder where you get the
idea that I think otherwise --- the paragraph from which you have quoted
merely makes the point that if the Gamma_{0} thesis is false, this
fact would have significant philosophical consequences, such as opening up
the possibility that statements like the Cantor-Bendixson theorem and Kruskal's
theorem may turn out to be predicatively provable. That is entirely separate
from my actual criticism of the Gamma_0 thesis. I completely agree that the
predicative unprovability, on the F-S characterization, of these statements
is no argument against the F-S characterization.

**p.2, sec.1, 3d par.** ...
*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.*

I don't think my statement is "not fair" --- you did defend it, forcefully and at length --- but I'll be happy to add a qualification to the effect that you have raised critical questions (which is also true) if you wish. I already do indicate this in the last paragraph on page 11, the second paragraph on page 13, and extensively in footnote 11, but I don't mind saying it here too.

**p.4, 2nd par. up,** *"U(NFA) employs a patently impredicative least
fixed point operator". This is completely wrong.*

(See below, discussion on page 15(b).)

**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 ...*

I don't think you read this sentence carefully enough --- the ambiguity
you ascribe to it is not present. The
premise is not that *A* sees that he has accepted every member of
*S*, the premise is that *we* can see that he would accept every
member of *S*; and the objection is that if we can see this then he
ought to be able to see it too, which would then allow him to use a
reflection principle to go beyond *S*.

*... 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.*

Well, that is essentially what I say immediately following the cited
passage: "There are a variety of ways in which this objection could be
overcome ... it may be possible to identify some special limitation in
*A*'s belief system which prevents him from grasping the validity
of all of *S* at once despite his ability to accept each statement
in *S* individually." But I then go on to explain why this would be
a quite extraordinary claim, particularly in the case of the Gamma_{0}
thesis. You offer no comment on this analysis and make no attempt to justify
the claim.

The objection is, as you note that I note, not novel. Howard states it very persuasively in [24], for instance. Kreisel attempts to refute it in several places, in highly cryptic passages which, once one deciphers them, are clearly fallacious; I treat these attempts at some length in sections 1(a), (b), and (c) of my paper. In [11] you dispense with the issue with the comment "The most obvious objection to (1)* was discussed and answered by Kreisel", followed by a reference to [32]. But that answer of Kreisel was abandoned by him in his later paper [33], in favor of a highly speculative suggestion that also does not stand up to scrutiny.

It would be quite remarkable if there were some straightforward sequence of well-ordering proofs such that we could see that a predicativist would accept each proof in the sequence but he could not see this himself. I extensively refute all serious attempts to justify this kind of claim of which I am aware. This is one of the central arguments of my paper and your brief response does not seem to do justice to it.

*If one follows your line on this, it would appear that no foundational
stance can be characterized by a formal system S ...*

This isn't really accurate; I explicitly list several ways in which the objection might be overcome. Whether this can actually be done depends on the details of the case at hand. My "line" is that it has never been convincingly done in the case of predicativism. I haven't thought about finitism.

**p.7,** *you say that if one accepts the rule of inference (*) one should
accept the corresponding generalized implication (**).*

Actually, I say that a predicativist should not accept (*) because of the induction versus recursion problem, and I then ask if one did believe he could accept (*) why one would not also believe he could accept (**). This is followed by two pages of discussion, which you do not address, in which I analyze and refute three separate attempts of Kreisel to justify the (on its face, quite incredible) claim that predicativists can accept (*) but not (**). Your example from modal systems seems not very relevant to this discussion.

There are lots of situations where a deduction rule is valid but the corresponding implication is not. The fact that such cases exist does not go very far toward showing that this particular case is one of them.

**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.*

I wrote: "Feferman nowhere openly repudiates the earlier systems, and I read his remark in [14] as implying that the later systems are merely more `perspicuous' than the earlier ones because they do not assume that predicativists have any understanding of ordinals."

I am unclear as to whether you are complaining that this statement misrepresents you, and if so, how my characterization differs essentially from your own ("On the contrary, I said ...").

I also don't see why, if you do not repudiate the earlier work, my critiques of it are moot. Historically, the autonomous ramified progressions were a major step forward in that they introduced the important distinction between predicative definability and predicative provability of well-orderings. But they are fundamentally unsound in not respecting the predicative distinction between well-ordering in the sense of admitting induction and well-ordering in the sense of admitting recursion.

**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 ...*

This helps explain the intuitive motivation for *P* + Ex/*P* but
does not answer the numerous detailed criticisms I made of that system. For
example, why are we able to introduce a function symbol when the
functional's uniqueness has been proven in *P* but not when its uniqueness
has been proven in Ex/*P*?

You say that "Ex/*P* only serves to show how things may be established
to exist". Does this mean that theorems of Ex/*P* which are not
existence assertions are not to be belived? What if in the course of proving
some existence statement I incidentally prove that some functional is unique
--- should I believe it then? What if this step is essential to the proof of
the existence statement?

I also argued that the predicate substitution rule of *P* + Ex/*P*
is clearly impredicative and that your justification of this rule, if
accepted, would in fact justify full second order comprehension. Your
reponse to this point was the following comment:

**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.*

Of course not. But if one were to accept a substitution rule allowing one to
infer *B(X/C)* from *B(X)* in the context of, say, pure second order
arithmetic, this would imply full comprehension. So you somehow have to
explain why such substitutions are legitimate when *B(X)* is a formula
of *P* but not when *B(X)* is Sigma-1-1.
Your one-sentence justification for the rule ("it may be argued that
the (predicative) provability of *B(X)* establishes its validity also for
properties whose meaning is not understood", [13], p. 92) is not satisfying
because it does not explain why this should be the case only when *B(X)*
is a formula of *P* and not more generally.

**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.*

Well, if *P* is taken to be definite then I suppose I can guess what
you meant by "relativizing *T* and *F* to *P*". But then
the substitution rule of Ref*(PA(*P*)) would be invalid since you
allow substitutions for *P* which are not definite.

The use of open-ended schematic systems is very interesting, but it is also
impredicative on its face in the fact that the schematic predicate symbol
*P* is taken to range over formulas which may themselves contain *P*.
This comment also applies to *U*(NFA).

**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 problem is that elsewhere you use *P* as a schematic predicate symbol,
with the interpretation that statements involving *P* are to be understood
as schemas of true statements, one for each intelligible substitution
instance. Ax 7 is not consistent with this interpretation.

On the other hand, if we interpret *P* in the way you suggest here, as
ranging over definite predicates, this would render the substitution
rule invalid. The same problem appears in [14], and you mention it
there as a difficulty (if somewhat obliquely: "this may involve some
equivocation between the notions of being definite and being determinate",
with no further explanation, [14], p. 42), so I am surprised that
you now feel it is "completely unproblematic". Essentially the same
problem also occurs with regard to the substitution rule in [13]. ("In
*P* we think of `*X*' as ranging over predicates recognized to
have a definite meaning; this would not seem to admit the properties
expressed by formulas of *L _{Ex}*", [13], p. 92.)

__This is the central impredicativity which appears in some form in all of
your systems. In every case you build a hierarchy over an arbitrary set
(or, if you prefer, "definite predicate") and then infer that such hierarchies
exist over arbitrary predicates. Yet the construction of the hierarchy is
not valid for arbitrary predicates.__ In [13] you deal with this problem
by essentially saying that anything provable for arbitrary sets can also be
accepted for arbitrary predicates ([13], p. 92), and in [14] you say that
"this substitution [of predicates for sets] accords with ordinary informal
reasoning" ([14], p. 41). In [19] you pass over the difficulty without comment.

*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 I go on to indicate that this impredicativity could be removed by dropping the minimality condition and switching to intuitionistic logic. As it stands, however, the system is impredicative in this respect.

*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.*

I don't understand this comment. The approximation from below for partial recursive functions involves a simple induction on omega; for your least fixed point operator it involves a generalized inductive definition, and that is impredicative.

*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".*

You now seem to be saying that *U*(NFA) can be modified so as to remove
this impredicative aspect without sacrificing the well-ordering proof. I think
that is true, as I said myself in footnote 10. It would not seem to justify
your earlier comment that my assessment of the least fixed point operator as
patently impredicative is "completely wrong".

I stand on my comment that the system *U*(NFA) as presented in [19]
is patently impredicative in its use of a least fixed point operator (in
addition to its other impredicative aspects).

**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.*

The distinction between "intelligible" (the statement's sense is understood)
and "meaningful" (it has a definite truth value) is crucial. In your
well-ordering proofs you derive a statement about *P(t)* under the
assumption that *P(t)* is meaningful, then substitute a formula
*B(t)* which is merely known to be intelligible.

If we could generally substitute intelligible formulas for meaningful formulas
then we could infer *(exists Y)(forall n)(n in Y <--> A(n))* for any
formula *A* of second order arithmetic, i.e., full second order
comprehension, from the predicatively valid statement
*(forall X)(exists Y)(forall n)(n in Y <--> n in X)*. A
predicativist cannot accept this inference precisely because he cannot
generally substitute intelligible formulas (*A(n)*) for meaningful
ones (*n in X*).

This example is not artificial; it is exactly substitutions of this type which are used in all three systems to pass from induction to recursion.

**p.16, last par.** *Really! This reckless, grandiose statement is
offensive and should be stricken.*

I know that saying "the community as a whole did not function in the way
that it should have" will not make me any friends.
However, the Gamma_{0} thesis has been the
received view for forty years, and this despite not only slightly subtle
problems like the induction versus recursion issue, but also the obvious
problem, never satisfactorily answered, of being able to go beyond any given
characterization --- as well as the numerous other difficulties I identify
in my paper. Why did it require an outsider like me to point
all of this out? Is there any explanation other than the failure of the
foundations community to function properly in this case? If in saying this
I come across as supercilious, or reckless and grandiose,
I am genuinely sorry, but I felt it would have been disingenuous of me
to say nothing.

**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.*

The criticism itself is vague. Can you give a specific example of a choice I make which is not dictated by the basic principles? Or can you suggest a way in which they could be made more precise?

**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) ...
You say that if a predicativist accepts S he should also accept
Tarski_{<}(S). What justifies this? The predicativist must
give meaning to all the symbols involved.*

Tarski_{<}(*S*) involves a hierarchy of symbols which are not
initially accepted as intelligible. However, the symbol *T _{a}*
cannot actually be reasoned with in Tarski

Thus, in order to accept Tarski_{<}(*S*) I do not first have
to give meaning to all the symbols involved; I only need to recognize that
any symbol will be given meaning before it is used.

You rhetorically ask "What justifies this?" but may not have noticed that I give a fairly detailed answer to that question on pages 29 and 30.

*With your interpretation, there is a circularity involved: Acc(a) means
that T _{a} is accepted, but we don't know what T_{a} is
until we know which b's are accepted.*

I'm not sure what you mean by "we don't know what *T _{a}* is".

Perhaps your point is just that we have to understand the predicate
Acc before we can start using Tarski_{<}(*S*). That is quite
true. But if you are saying that we don't understand Acc until every
*T _{a}* is intelligible, then again the requirement is too
strict because it is supposed to be through Acc that we reason about the
intelligibility of the

(I should add that if your objection were successful this wouldn't save the
Gamma_{0} thesis --- if we can't even use Tarski_{<}(*S*)
then we seem to be stuck around gamma_{3}, where (gamma_{n})
is the canonical sequence whose supremum is Gamma_{0}.)

*In fact, Acc(a) is
just a stand-in for: the initial segment of the < relation determined by a
is well-ordered, i.e. I(a). And then in order to say what T _{a} is,
we need to say what all the T_{b}'s are for b < a. And in order to do
that we need to
carry out a definition by transfinite recursion up to a. But one of your
main objections to our analysis of predicativity (e.g. p.4, last par.) is
that the step from transfinite induction up to an ordinal notation does
not justify transfinite recursion up to that notation. As we know,
iterated truth theories are equivalent to ramified theories, and so you
are in effect working with here is an autonomous ramified progression,
just what you railed against in an earlier section of the paper.*

I'm not sure what to do with this comment. An essential feature of my
argument is the fact that I prove, in a higher system, the statement
TI(*A*,*a _{n}*) for all formulas

Tarski_{<}(*S*) is indeed analogous to a ramified theory, but
this is not where the well-ordering proof takes place. It takes place in
Tarski_{<}^{omega}(*S*),
which has no analog in the analysis of autonomous ramified progressions.

**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.*

No. There's an important difference between your statement and what is
actually needed. Yours ("the general proposition: `If *S* is acceptable
...')
is indeed impredicative, as it refers to the general concept of acceptability
and this is not a predicatively legitimate concept. If we could formally
discuss acceptability of theories in general, we could set up a system which
refers to its own acceptability in a circular fashion. (See the end of
sections 2.3 and 2.8 for discussion of this point.)

What is actually needed is the ability to discuss, from the outside, the
acceptability of a circumscribed family of theories. In my case I have an
omega sequence of theories, *S _{n}* =
Tarski

**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.*

You seem to have read this footnote hastily. It explicitly refers to
the *J _{2}* paper, not to the systems in the Gamma

You interpret my comment about awkwardness as referring specifically to
systems you developed, but that is not what I had in mind. As you know,
predicative systems intended to be used to formalize core mathematics have
been developed by many people, including Chihara, you, Grzegorczyk, Kondo,
Lorenzen, Takeuti, Wang, Weyl, and Zahn. I was referring to all of these
systems generally. (See line -4 on page 1 of my
*J _{2}* paper for references.)

I do agree that the systems in my Gamma_{0} paper, especially the
two later
ones, are too complicated. I believe that further progress in predicatively
proving stronger well-ordering statements will involve substantially
simplifying my approach in some way.

Sincerely,

Nik Weaver