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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: Aug 8, 2006 2:59 PM

In article <eb9gsf\$l08\$1@glue.ucr.edu>, John Baez <baez@math.ucr.edu> wrote:
>I'd love to hear a bit more of the story, especially if you can tell
>it in a charming and not too rigorous manner. In particular, nothing
>in the paragraph says what's special about Gamma_0. For example,
>suppose I have an ordinal smaller than Gamma_0. How can I give a
>"predicative" proof of induction up to that ordinal? What breaks
>down at Gamma_0?

To get induction up to some ordinal, one first needs to introduce ordinal
notations to speak of the ordinals. The first technical hitch is that
proving that your ordinal notation system "makes sense" (i.e., has all
the self-consistency and uniqueness properties that you want) can in
principle involve an arbitrary amount of arithmetical knowledge. If
you're starting with only PA and the notion of the set of integers, then
this puts an upper bound on how much arithmetical knowledge of this type
you're allowed to assume, and hence an upper bound on the ordinals you
have a right to work with.

The second hitch is that if you're a predicativist, then you're not going
to allow quantification over arbitrary sets but only over the set of
integers and sets that you've already defined. This puts another
restriction on how far your induction can proceed.

Putting these restrictions together gets you up to Gamma_0.
--
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