[FOM] The defence of well-founded set theory

Roger Bishop Jones rbj01 at rbjones.com
Mon Oct 3 04:16:18 EDT 2005

On Thursday 29 September 2005  9:54 pm, Stanislav Barov wrote:

> Historicaly first attempt against Kantorian set theory was
> made by Russel in his Ramified Type Theory where no
> impredicative definitions are avaliable. In case it apear to
> be  wery weak (no ways to define the Dedecind's cuts or free
> group generated by some set, etc.) another way in avoidance of
> reflective paradox formulated in ZF theory as Regularity
> Acsiom. This kind of axiom have some forms of different
> strenght. Mostly minimal conditions of this expresed in
> stratification axiom of NF of W. V. O. Quine.

My interest at present is primarily in the defence of
pure well-founded set theory (as presented in the iterative
conception of set), rather than in other alternatives to
naive set theory such as those offered by Russell and Quine.
Of course, if they were motivated by a critique of well-founded
set theory that critique would be of interest, but Russell's
theory of types was published the same year as Zermelo's set
theory, and long before the articulation of the iterative
conception of set.

Non-well-founded set theories are often motivated by perceived
defects in well-founded set theory, and Quine's reasons for
preferring his systems to Zermelo's are of interest.
>From a brief referral to Quine's "Set Theory and its Logic"
it appears that Quine considered NF primarily as an alternative
liberalisation of Russell's theory of types, rather than as
a response to problems in well-founded set theory.
In this context, stratified comprehension may be thought of
as closer to the constraints on set abstraction which arise
from Russell's type system than is separation.
On the other hand, both Russell's and Zermelo's systems are
well-founded, but Quine's are not.
Did Quine offer a more substantial critique of well-founded
set theories elsewhere?

> H. Wayl
> influented intuitionistic criticue formulated his own version
> of set theory which was precisly described and improved by S.
> Faffermann. As sone in ZF was obvious lack for big sets Geodel
> and Bernais make defenitions for notion of class which isn't
> set and not contained in other class.

Its not clear to me what cogent critique of the iterative
conception these respond to.
(indeed NBG might be, perhaps usually is, considered consistent
with that conception)

> 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?
(posted recently to FOM under the subject "Predicativity")

Roger Jones

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