[FOM] The defence of well-founded set theory

[Roger Bishop Jones]

On Monday 03 October 2005  7:56 am, Bill Greenberg wrote:

> I would be particularly interested in your reaction to
> Boolos's  "Must We Believe In Set Theory?" (article no. 8 in
> Boolos's book).

In this article Boolos comes across as a platonist in the
fairly strong sense that he believes in the objective truth
of questions about the existence of abstract objects
in a context devoid of existential presuppositions.

I am not a platonist in this sense.
For me statements asserting the existence of abstract entities
only have meaning 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").

Carnap's distinction between internal and external questions
seems to me important, and, though Boolos does discuss the
distinction (without mention of Carnap) his view of it and
its relevance in the present context is quite different
to my own.

The distinction permits a clean separation between mathematical
questions and metaphysical questions, and allows the sentences
of set theory to be construed as mathematical assertions about
some explicitly declared domain which presuppose, without
asserting, the metaphysical proposition that the entities in
that domain of discourse exist.

Thus the most significant difference between Boolos (in this
paper) and myself, is that we disagree about what the sentences
of ZFC assert.
This is not something which Boolos discusses. It does not
seem to occur to him that anyone might construe ZFC as talking
about "the cumulative heirarchy" rather than about "what sets
exist" in some more absolute sense (though he does make clear
that the difference between these questions is crucial).

Boolos's intuitions about what set existence principles
flow from the iterative conception of sets are of interest,
and I intend to examine more closely his reasons for thinking
that this is only a part of those derivable in ZFC.
(I myself think the iterative conception good enough for all
of ZFC and some large cardinal axioms).
However, his "intuitions" about "what sets exist" in the more
absolute sense which is primary in this article, seem to me
incomprehensible, and irrelevant to set theory
or its justification.

Roger Jones