[FOM] Completability of Cumulative Heirarchy
Roger Bishop Jones
rbj at rbjones.com
Fri Aug 24 03:44:53 EDT 2007
On Thursday 23 August 2007 06:14, Bill Taylor wrote (in a message about
> Earlier, Roger Jones said:
> > ....quote from a message which never reached FOM! ...
> I can see how one might regard the "completed cumulative hierarchy"
> as incoherent; but I cannot see how one might derive a contradiction
> from the supposition thereof.
> You say it is easy. Can you elaborate please?
Informally it is part of this conception that after any completable
collection of stages there will be a successor.
Hence the whole thing, cannot be completable,
Its similar to Burali-Forti.
If there was a set of all the Von-Neuman ordinals then it would itself be a
Von-Neuman ordinal, and hence would be a member of itself.
But the Von-Neumann ordinals are all well-founded.
If our domain consisted of all the pure well-founded sets then it would be
a pure well-founded set, and hence a member of itself.
(You can try fixing this with a limitation of size principle, but to do
that you really do have to limit the size, and you get the conception of
pure-well-founded-sets no greater than a certain size, which is not the
complete cumulative heirachy.)
It is difficult to formalise this contradiction, because the supposition
(that the heirarchy is completable, or that our domain consists of all the
pure-well-founded sets) is not readily formalisable. Axiomatic set theory
may be thought of as various attempts at approximating this conception,
but these approximations can never go beyond what can be interpreted in an
initial segment of the cumulative heirarchy, and therefore fail to capture
the essential ingredient which I am alleging here to be incoherent.
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