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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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 baez@math.removethis.ucr.andthis.edu Posts: 446 Registered: 12/13/04
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Jul 29, 2006 12:30 PM

>In article <ea83ig\$qmq\$1@news.ks.uiuc.edu>,
>John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

>>Logicians [...] know a lot about how much induction
>>up to different ordinals buys you. And apparently, induction up to
>>Gamma_0 lets us prove the consistency of a system called "predicative
>>analysis". I don't understand this, nor do I understand the claim
>>I've seen that Gamma_0 is the first ordinal that cannot be defined
>>predicatively - i.e., can't be defined without reference to itself.
>>Sure, saying Gamma_0 is the first solution of
>>
>>phi_x(0) = x
>>
>>is non-predicative. But what about saying that Gamma_0 is the union
>>of all ordinals in the Veblen hierarchy? What's non-predicative

>The situation is somewhat akin to the situation with the Church-Turing
>thesis, in that one is tentatively equating an informal notion
>(predicativity or computability) with a precise mathematical notion.
>himself has articulated potential objections to the "standard view"
>that Gamma_0 marks the boundary of predicativity.

There's also someone named Nik Weaver who has debated Feferman
on this subject:

http://www.cs.nyu.edu/pipermail/fom/2006-April/010472.html
http://www.math.wustl.edu/~nweaver/conceptualism.html

He seems to claim that Gamma_0 and even larger ordinals have predicative
definitions. However, I'm too ignorant to follow this debate.
Usually in physics I have a sense for when people are being reasonable
even if I don't follow the details. In this debate I can't even
do that.

>Having said that, I'll also say that one of the reasons for the standard
>view is that Gamma_0 marks the boundary of "autonomous progressions" of
>arithmetical theories. The book by Torkel Franzen that you cited is
>probably the most accessible introduction to this subject.

This summer I'm in Shanghai without any academic affiliation, so it's
hard to get that book. When I return to Riverside in the fall I'll
try to read it. But my curiosity is burning right now, so I'll take
the liberty of asking some more questions.

>Roughly
>speaking, the idea is that if anyone fully accepts first-order Peano
>arithmetic PA, then implicitly he accepts its consistency Con(PA), as
>well as Con(PA+Con(PA)), etc.

I assume that by "etcetera" you mean there's one theory like this
per ordinal. I browsed a paper by Franzen where he was trying
to explicate how these theories actually let you prove interesting
new stuff.

It's a bit mysterious: I imagine a guy sitting there thinking
"Peano arithmetic is true, so I know it's consistent, and I know
*that's* consistent too, and I know *that's* consistent...", and
so on - and after pondering this way for an transfinite amount of time,
all of a sudden he can do new stuff like prove that Goodstein
sequences approach zero!

I think Franzen was trying to dispel this naive conception.
He said the real action happens at limit ordinals, where
the interpretation of everything changes in some sneaky way.

But, my understanding of his comments like an impressionist
painting of a surreal painting - Dali's "Sacrament of the Last
Supper" as reworked by Monet.

(Hey, I managed to sneak a docahedron into the discussion!)

>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.

>I think you have a better shot at grasping the underlying intuition via
>this approach than by staring at Gamma_0 itself and trying to figure out
>what is non-predicative about its definition.

Okay, I won't try to do that.