[FOM] Nik Weaver's conceptualism and the correctness of the Schütte-Feferman analysis
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Tue Apr 4 09:32:29 EDT 2006
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
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