[FOM] Proli's Question about Set Theory
Andrea Proli
aprol at tin.it
Mon Feb 27 09:57:03 EST 2006
Thank you very much Allen,
I really appreciated your reply.
So, let me see if I got the point: axiomatic set theories are not
interpreted by means of models, so for instance ZF Set theory is a
first-order theory only under the syntactical viewpoint. But, would this
mean that it relies on human intuition in order to check its soundness...?
You say:
"Model theory texts say the domain of a model is a set, and ZF implies
that there is no set of all sets. ((And no set of ordered pairs that can
serve as the interpretation of the membership predicate.)) So it
certainly appears that the "intended interpretation" of ZF is NOT, in the
technical sense, a model of it!"
Two questions:
1) Why does the non-existence of a set of all sets create troubles with
the fact that the domain (I suppose by "domain" you mean the universe of
discourse) of a model is a set?
2) What do you say that ZF implies that there is "no set of ordered pairs
that can serve as the interpretation of the membership predicate"?
I am sorry, but I have to admit that I am really confused and this is
certainly due to my lack of logical/mathematical background. Can you
please, or someone else too, give me some reference to the significant
literature about these issues where I can find the answers to my
elementary doubts, before I bother you all again with questions I do not
even exactly understand?
Thank you in advance, you all are very kind
Andrea
In data Sun, 26 Feb 2006 05:33:42 +0100, A.P. Hazen
<a.hazen at philosophy.unimelb.edu.au> ha scritto:
> Andrea Proli notes that:
>> ... ZF is a first-order theory, and first-order
>> theories have a standard denotational, model-theoretic semantics. In
>> model
>> theory, symbols are given an interpretation in terms of sets and
>> relations
>> (which are also sets). Isn't this a circular definition?
>
> ---There is certainly something funny, at least on the usual
> (sloppy) formulations. Model theory texts say the domain of a model
> is a set, and ZF implies that there is no set of all sets. ((And no
> set of ordered pairs that can serve as the interpretation of the
> membership predicate.)) So it certainly appears that the "intended
> interpretation" of ZF is NOT, in the technical sense, a model of it!
>
> The model theory textbooks are right to define "model" as they do:
> model theory, as a branch of MATHEMATICAL logic, is a branch of
> mathematics, and is typically conducted, these days, in the more or
> less explicit framework of axiomatic set theory. (There are people
> on this forum -- Harvey Friedman and Stephen Simpson come to mind--
> who have elaborated on this. Various statements of model theory turn
> out to require for their proof, and even [[modulo some weak basic
> assumptions]], sometimes, are equivalent to, set-existence axioms.)
> So we are just STUCK with a foundational axiomatic system which, on
> its primary and intended application, is not interpreted in a model.
> (We all hope that it DOES have other, unintended, interpretations in
> models: Gödel's completeness theorem tells us [[modulo some weak
> basic assumptions]] that having a model is equivalent to being
> formally consistent for First-Order theories!)
>
> Set theorists seem not to be bothered by this, but it is a
> curious conceptual situation worth some examination in the philosophy
> of logic. John Etchemendy's book on the "Concept of Logical
> Consequence" (I think that's the title but not quite sure) touches on
> it; his discussion takes off from Kreisel's in "Informal Rigor and
> Completeness Proofs" (in Lakatos, ed., "Problems in the Philosophy of
> Mathematics"; Kreisel's paper is partially reprinted in Hintikka,
> ed., "Philosophy of Mathematics").
> (Précis of central Kreisel-Etchemendy point: if you DEFINE a
> valid First-Order sentence as one that holds in all models,
> it does not IMMEDIATELY follow that all valid sentences in
> the language of set theory are true, since the intended
> interpretation of set theory is not amodel. There is a way
> around this for First-Order logic-- logically provable
> sentences ARE truths of set theory-- but there are hard
> questions for more powerful formal languages: e.g.languages
> with generalized quantifiers.)
>
> And Richard Cartwright's (characteristically clear!) paper "Speaking
> of Everything" ("Nous" vol. 28 (1994) pp. 1-20) gives a forthright
> defense of the position that it is not required, in order for a
> First-Order theory to be
> legitimate/acceptable/comprehensible/understood/true, that its
> variables be interpreted as ranging over a set.
>
> ---
>
> Allen Hazen
> Philosophy Department
> University of Melbourne
>
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