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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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 tchow@lsa.umich.edu Posts: 1,133 Registered: 12/6/04
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Jul 30, 2006 11:52 AM

In article <eag2eh\$bcp\$1@news.ks.uiuc.edu>,
John Baez <baez@math.removethis.ucr.andthis.edu> wrote:
>I assume that by "etcetera" you mean there's one theory like this
>per ordinal.

[...]
>>If one tries to articulate exactly what
>>is "implicitly" involved in accepting PA in this sense, then one can
>>make a plausibility argument that Gamma_0 is a natural stopping point.

>
>plausibility argument.

Let's look more closely at what the notion of "one theory like this
per ordinal" means. There's no difficulty figuring out what "Con(PA)"
means or how to express that statement in the first-order language
of arithmetic. Ditto with "Con(PA+Con(PA))". However, once you start
ascending the ordinal hierarchy, a difficulty appears. The language
of arithmetic doesn't let you talk about "ordinals" directly---that's a
set-theoretical concept. In order to express a statement like "Con(T)"
for some theory T, you need at minimum to be able to give some sort of
"recursive description" or "recursive axiomatization" of T (where here
I use the word "recursive" in the technical sense of recursive function
theory) in the first-order language of arithmetic. This observation
already yields the intuition that we're not going to be able to ascend
beyond the Church-Kleene ordinal, because we won't even be able to
figure out how to *say* "T is consistent" for a theory T that requires
that many iterations to reach from PA.

There are other problems, though, that potentially get in the way before
we reach the Church-Kleene ordinal. Once we realize that what we need is
a system of "ordinal notations" to "fake" the relevant set theory, we may
(if we are predicavists) worry about issues such as:

1. As we ascend the ordinal hierarchy, isn't it illegitimate to make a jump
to an ordinal alpha unless we've already proved, at the level of some
ordinal beta that we've already reached, that an ordinal of type alpha
exists?

2. And isn't it illegitimate to create sets by quantification over things
other than the natural numbers themselves and sets that we've already
created?

Condition 1 goes by the name of "autonomy" and condition 2 goes by the name
of "ramification." If one formalizes these notions in a certain plausible
manner, then one arrives at Gamma_0 as the least upper bound of theories
that you can get to, starting with (for example) PA.

One can of course wonder whether 1 and 2 above really capture the concept
of "predicativity." Some secondary evidence has accumulated of the
following form: Some argument that intuitively seems to be predicative but
that is not immediately seen to be provable in the Feferman-Schuette
framework is shown, after some work, to indeed be provable below Gamma_0.

It's still possible, of course, for someone---you mentioned Nik Weaver---to
come along and argue that our intuitive notion of predicativism, fuzzy
though it is, can't possibly be identified with the level Gamma_0. The
reason you can't seem to decide immediately whether Weaver's position is
nonsensical or not is probably because the critical questions are not
mathematical but philosophical, and of course it's usually harder to arrive
at definitive answers in philosophy than in mathematics.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences