[FOM] Query for Martin Davis. was:truth and consistency

Roger Bishop Jones rbj at rbjones.com
Sun Jun 1 15:01:36 EDT 2003

On Sunday 01 June 2003  6:28 am, Bill Taylor wrote:


> So now we come to my query.  Between the totally compelling N
> as described by PA, and the room-to-doubt V as described by
> ZF, there is a middle case, and it is this I would like your
> opinion on, with whatever clarifying comments seem
> appropriate.  I am talking OC about the subject loosely
> referred to as "analysis", whose realm is R, the standard
> reals.

We can put some other things in the middle ground too.

Very close to V (the cumulative heirarchy) there are
the various V(alpha) for suitable ordinals alpha each of
which is a stage in the construction of the cumulative
heirarchy, and some of which may be called "standard"
models of ZFC.

Some people, including myself, have difficulties with V
because the description of the cumulative heirarchy is a
description of an iterative process which of its nature can
never be completed.  The idea of V completed sounds incoherent.
There is no similar problem with the V(alpha).
Some people do still have difficulty in
accepting that the description of V(alpha) is
definite, the stumbling block here being the idea
that the power set contructor gives a set of "all" subsets.
This is also a problem for V so the V(alpha)
remain slightly less problematic than V.

Another stage position between the V(alpha) and N
is the notion of a well-founded model of (say) ZFC.

This fixes an interpretation of N (and lots of other
more complex structures) and so can be no more clearly
defined than than N, but it is arguably more definite
than the notion of a standard model of ZFC.

Possibly there are concepts between that of a standard model
(of ZFC) and that of a well-founded model of ZFC which might
cast light on what it means to call something a standard model,
by being more informative than well-foundedness but less
problematic than the notion of "all subsets".
Such concepts might provide a way to clarify the intended
interpretation(s) of set theory for those who feel that
might be useful.
One tangible benefit which might conceivably arise from
such clarification would be to discover whether or not
CH holds in the standard models (or as some would put it
"is true").

To my bafflement, some respected contributors to FOM appear
to be hostile to discussion of the semantics of set theory.
This contribution will possibly once again draw their ire.
If anyone can explain to me why there should be such
hostility I shall be interested to hear the explanation.

As to R I am inclined to agree that it is conceptually less
definite than N and more definite than either the cumulative
hierarchy as completed or the stages in its contruction.
Perhaps not so definite as the concept of well-foundedness.

Roger Jones

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