[FOM] Nik Weaver's conceptualism and the correctness of the Schütte-Feferman analysis

[Aatu Koskensilta]

In his paper Predicativity beyond Gamma_0 Nik Weaver argues that "the 
current understanding of predicativism is fundamentally flawed" and 
that ordinals well beyond Gamma_0 are predicatively recognizable as 
such. He has also expressed these views on this list in which the 
reaction seems to have been that of general indifference, perhaps 
because people feel confident enough in the classical analysis of 
predicativity or because they simply don't find the issue to be 
interesting enough. My own reaction has been largely a combination of 
these two, but I think there's some merit in having a close look at 
Weaver's ideas if for no other reason than that it offers a chance to 
clarify what analyses such as Feferman's actually tell us (or me, in 
any case). Of course, Weaver will see things differently!

First, there's a "general difficulty" Weaver mentions. I think this 
issue is largely a red herring, but I'll say something about it just to 
put it to rest.

Weaver writes

  Suppose A is a rational actor who has adopted some foundational 
stance. Any attempt to
  precisely characterize the limits of A’s reasoning must meet the 
following objection: if
  we could show that A 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. Thus, S
  could not have been a complete collection of all the statements (in a 
given language) that
  A would accept. A similar argument can be made about attempts to 
characterize A’s provable
  ordinals.

and continues to note that there are ways to meet this objection, the 
most relevant of which is that

  [- -] 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 this is exactly what we expect in case of such analyses as 
Feferman's! It's very easy to come up with examples. For example, I 
accept the existence of measurables, and hence e.g. Con(ZFC+a 
measurable exists). Now, from this I can infer that Nik Weaver will in 
the relevant idealized sense accept every claim of form "n is not the 
code of a proof of a contradiction from ZFC+a measurable exists". But 
this hardly gives any reason to suppose Weaver accepts the soundness or 
consistency of the set {"n is not the code of a proof of a 
contradiction from ZFC+a measurable exists" | n in N}. This shows that 
if we know some principles which the rational actor doesn't it's 
perfectly obvious that there will be many cases in which we can tell 
that he will accept every member of a set S without there being any 
reason to suppose he will accept the truth of S or consistency of S or 
whatever, precisely because the principle that leads us to conclude 
that he will accept every member of S is not available for him.

Now, to be fair to Weaver, he is not unaware of this possibility - he 
does greatly exaggerate the "general difficulty", though - but thinks 
that in case of Gamma_0 the reason we see that a predicativist accepts 
every ordinal < Gamma_0 is also available to the predicativist himself. 
This is a specific and on the face of it a reasonable objection, and to 
answer it we must have a close look at the systems used in explicating 
predicative acceptability. Before doing so I'll say something of these 
"rational actors" and "predicativists" and other fictional people and 
then ask Weaver whether I correctly understand what predicativism, at 
least as analyzed by Feferman, is.

Both Feferman and Weaver couch their expositions in terms of what "a 
predicativist" accepts or what a "rational actor" accepts. I think this 
is a bit misleading because it's rather unclear what it means for even 
a hypothetical person to accept an infinite number of statements; 
presumably it means that there is a proof from principles acceptable to 
the person. But in that case we can just dispense with all these 
imaginary people and talk instead of the much clearer concept of 
predicative acceptability, or acceptability of mathematical principles 
on basis of other mathematical principles and concepts in general.

Let's say what exactly Feferman is analysing, i.e. "predicativism given 
natural numbers". The way I've always read this was as Feferman's 
analysis of predicativism as giving the answer to the following 
question:

  What mathematical principles are acceptable on basis of acceptance of 
PA and the concept
  of arithmetical truth as legitimate.

Here by PA I don't mean the formal first order theory but rather a more 
informal version with the induction axiom replaced with something like

  For whatever properties P there are of natural numbers, if P(0) and 
P(n) --> P(n') for all
  n, then P(n) for all n

or

  Whenever P can be seen to make sense, if P(0) and P(n) --> P(n') for 
all
  n, then P(n) for all n

Now, arithmetical P make sense (since arithmetical truth makes sense), 
sentences referring to truth of sentences involving arithmetical truth 
make sense and so forth. So PA as ordinarily understood is 
predicatively acceptable (we can apply induction to arithmetical truth) 
and so forth. What so forth means is the contested question!

(A parenthetical note is in place here. A predicativist *can* 
consistently think that everything provable on basis of acceptance of 
PA and the concept of arithmetical truth as legitimate is true, 2. 
incompleteness theorem withstanding, because this does not lead him to 
knowledge or belief about the consistency of any particular formal 
theory, and thus Con(T) will not be in the formal theory consisting of 
predicatively acceptable.)

Nik, do you think this is a fair informal picture of predicativism?

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus