[FOM] Gamma_0

[Thomas Forster]

Puzzled of PMMS writes:

I hesitate to ask this question, beco's history teaches me
that the mere act of asking a daft question unblocks the
lobes and renders the query unneccessary, but here goes.

The story is that in order the prove the existence of a worder
of the naturals of length $Gamma_0$ one has to assume that the
second number class is a set.  (I have been thinking about this
because of a question my colleague Imre leader put on a final
year example sheet.)  I can see how to prove the existence of
wellorders of the naturals of all smaller lengths without using
power set (or Hartogs'..) but it seems to me that i can do the
same for Gamma_0 itself.  Why do i need to assume that the second
number class is a set in order to obtain a worder of the naturals
of length $\Gamma_0$ as a concatenation of the \omega-sequence of
worders of N of length \phi(0,0), \phi(\phi(0,0),0) and so on?
Why is this any more difficult than showing that the phis are all
defined below gamma_0?

      tf

URL:  www.dpmms.cam.ac.uk/~tf; DPMMS ph: +44-1223-337981;
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