[FOM] The defence of well-founded set theory

barov@mccme.ru barov at mccme.ru
Tue Oct 4 16:05:24 EDT 2005

On Mon, 3 Oct 2005 09:16:18, Roger Jones wrote:

> My interest at present is primarily in the defence of
> pure well-founded set theory (as presented in the iterative
> conception of set)

>> I think mostly obvious
>> failure in fundational motivation and interpretations of set
>> theory contained Lowenghaim-Scoleem theorem. Outside of scope
>> of ZF exist not intended models for evry internal formulation
>> of real analisis.
> This was one of Weaver's "anomalies" which seems to me to have
> little force.
> Do you disagree with my reasons for considering this
> not to be a significant point against set theory?

It is not clear what kind of defence set theory is needed. If we
would like to establish some arguments on consistency of w.f. set theory
it will be differ from defence it as descriptive sciense which explore
some platonic univers (probably in mostly apropriate way), or motivate
it as foundation of all mathematics. Second sense of such defence is not
sucessful, as I think, due to previous arguments (Lowenhgam-Scolem theorem).
Third become superfluous in case some mathematicans view their objects
differently and study metamathematicaly different areas as proffs
or continuous functions, ects. First is qite philosophical question
but may be some kind of "Church Thesis" can be proved on this way.
At last w.f. theory has such unpleasant properties as impossibility
formulate another logics as its internal logics (e.g. sentence F(P_1,...,P_n)
provable in pure predicate logic iff exist set theoretical sentences
Q_1,...,Q_n such that F(Q_1,...,Q_n) provable in ZF(C)), not as logics of
some small models, and especialy incompatability it with other axioms
(e.g. determinancy axiom) which has a sense.
Stanislav Barov

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