Re: Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)

From: George Greene (greeneg_at_quartet.cs.unc.edu)
Date: 02/22/05 

Date: 22 Feb 2005 15:43:14 -0500

"mitch" <aatuckpointingNOSPAM@sbcglobal.net> writes:
 : The history of "logicism"--as I understand it thus far--ignores
 : "objectification."

I don't see how you can say that, when both Zermelo's original
axiomatization of set theory AND the Tarskian "interpretation"
paradigm in general BOTH insist that logical theories get to be
modeled in DOMAINS containing OBJECTS.

 : Carnap, building on
 : Whitehead and Russell, goes so far as to lament Cantor's definition of set
 : on the basis that Cantor treats sets as objects.

So did Zermelo. Far worse, after having insisted
that there was a domain containing objects, and that among these
were the sets, Zermelo failed to explain why this domain ITSELF
was not (PURELY in virtue of the fact that it "contained" all
these objects, and that's all it did) ALSO a set.

 : Halmos' work is not some nonsense invoking categorial errors as modern
 : advocates of first-order predicate logic seem to think. It arises as
 : consequent to Tarski's work on cylindrical algebras,

As opposed to WHAT other kind of algebras -- BOOLEAN algebras??

 : For my part, I find it a bit strange that logicians are so

Oh, shut up. You do not know enough logicians to be generalizing
(or rather, ignorantly over-generalizing) about what logicians
do or don't do.

 : concerned about
 : consistency and then fail to see that the structures in which they are most
 : interested are fundamentally paraconsistent--or, at least, seem to be.

And what "structures" MIGHT *those* be, grasshoppah?

 : When you make the observation,
 :
 : "Whenever you see an object, you
 : silently add it to the domain of existence.

Nobody makes that observation, least of all logicians.

 : you are expressing the basic presupposition underlying Cohen forcing in
 : set theory. Cohen--or, more correctly, the modern characterization of
 : Boolean-valued models and forcing languages--makes this explicit

Well, thank god you finally conceded that in modern praxis, people
do this explicitly, so they are not doing it "silently".

 : in a way that creates a nearly inconceivable ontology for
 : anyone who takes set theory seriously.

You have yet to explain what it MIGHT mean for ANYone to take
"set theory" seriously. There are a great many DIFFERENT
set theories and obviously they are going to raise DIFFERENT
ontological issues. 

-- 
 --- The history of our nation has demonstrated that separate is seldom, if ever, equal.
 --- (Feb.3,2004) Supreme Judicial Court of Massachusetts (4-3), adv.Sen.#2175

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