[FOM] The defence of well-founded set theory

Roger Bishop Jones rbj01 at rbjones.com
Wed Oct 5 17:44:28 EDT 2005

On Wednesday 05 October 2005 11:36 am, A.P. Hazen wrote:
>     Roger Bishop Jones asked  for references relevant to a
> defense of the conception  os sets as  items in a well-founded
> cumulative hierarchy.

Actually slightly more interested in the critiques, but I asked
for references for and against.

>     From  Jones's perspective.  Jones in a couple of replies
> has said he is not a "platonist" and doesn't want  to think
> about the question of what sets REALLY exist (as a matter of
> metaphysical fact), but only questions of what  sets exist
> 	"in the context of some existential presuppositions (i.e.
> 	only in the context of an understanding about the domain
> 	of discourse). ...  I do believe that the sentences of set
> 	theory have a definite objective truth value once the
> 	semantics of the language of set theory has been made
> 	sufficiently definite (for example, but not necessarily,
> 	by identifying the domain of discourse with the sets
> described in "the iterative conception of set")."

The point was actually not about what I want to think about,
but rather about how the sentences of set theory should be
construed if we want to use the language for mathematics
rather than metaphysics.
More specifically it was a rejection of Boolos's presumption
that we always make metaphysical claims when we affirm
mathematical existential propositions.
I can't see why we should construe anyone who says that
there is a prime between 10 and 15 as making a metaphysical
claim, this is surely just maths.

> (((***I personally*** have doubts about trying to distinguish
> in this way between "existence on a conception" and  existence
> in some metaphysically more absolute sense, but)))

Naturally I would be interested to know more (something even)
about your doubts.

> Giaquinto
> gives an interesting exposition of the iterative conception
> viewed in what I think may be a Jonesian way.  He argues that
> both the iterative conception and his "non-absolutist" view of
> it can be found in Zermelo's later work.

So far as I am aware my position on this is not original.
I don't think its much different to Carnap's, so you might
call it "Carnapian", but arguably its a natural consequence
of Hume's positions on mathematics (as "relations between
ideas") and metaphysics (as nonsense), and so I think of the
position as merely "positivistic" (though I don't consider
all metaphysics to be nonesense myself).

Roger Jones

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