On Feb 18, 2010, at 1:55 AM, Rob Freeman wrote:
> ...
> On Thu, Feb 18, 2010 at 9:30 AM, Christopher Menzel <cmenzel@xxxxxxxx> wrote:
>> On Wed, 2010-02-17 at 11:32 +1300, Rob Freeman wrote:
>> ...
>>> I'm happy you agree the axiomatic set theories of mathematics are such
>>> incompatible theories.
>>
>> I agree with no such thing
>
> You don't agree the axiomatic set theories of mathematics are incompatible? (01)
I gave you a very clear response to this muddled and vague question. For some
entirely inexplicable reason, you have deleted it and are asking it again.
I'll paste my earlier response back in for the sake of the one or two others
who might care enough to be reading this. I said: The claim in question is
either trivially
false ("Any two distinct set theories are incompatible) or trivally true
("There are incompatible set theories.") In point of fact, virtually
all set theories in common use -- ZF-style theories -- share a robust
axiomatic core. Only a few quirky set theories that are primarily of
theoretical interest -- notably, those like Quine's NF that allow a
universal set -- are incompatible with that core. (02)
>>> Your other arguments are with the authors of my references. As I
>>> understand it you dispute the first author's use of the word
>>> "theories" instead of the word "logics".
>>
>> There isn't really anything to dispute, as if there are two sides to the
>> issue. "logic" is simply the wrong word.
>>
>>> And you dispute second authors their proud claim of precedence for
>>> Thoralf Skolem.
>>
>> They never claimed precedence for anything. And again there is nothing
>> to dispute. The authors simply gave an incorrect informal
>> characterization of the L-S theorem.
>
> So you dispute my use of the words "dispute" and "precedence", as well
> as the first author's use of the word "theories" instead of "logics",
> and the second authors' "characterization of the L-S theorem." (03)
If you like, but of course to put it in these terms suggests that my point was
just a quibble about the meaning of words. I suppose that, if you were to
claim that the Vietnam War was in the 19th century and I pointed out to you
that you were wrong, there is a sense in which I was "disputing" your use of
the words "Vietnam War". But the more correct characterization of the
situation would be that you had simply said something that is provably false.
In the same way, the author's claim that the L-S theorem "[r]oughly speaking
... asserts that there is no complete axiomatization of mathematics" is just
flatly false, at least, if we take "complete" in its usual sense in formal
logic (where a theory T is complete if and only if, for every sentence A of the
language of T, either A or ~A is a theorem of T). The author, Fenstad, is in
fact a very good logician, so I think he was just being careless here (note he
qualifies his remark with "roughly speaking") and was using "complete" to mean
"categorical". A theory is categorical if it has only one model (up to
isomorphism). The Löwenheim-Skolem theorem says that all first-order theories
that have infinite models are non-categorical; in particular, they all have
models the size of the natural numbers. Since it is in fact a theorem of ZF
set theory that there are uncountably many things (the real numbers, for
example), the L-S theorem seems to generate a paradox (the so-called "Skolem
Paradox"): How can a theory that entails there are uncountably many things be
true in a model in which there are only countably many things? As Fenstad
points out, there is no genuine contradiction here, but the theorem does show
that all first-order theories have "unintended" models; somewhat more
colorfully put, no such theory (even one that is not recursively axiomatizable)
can specify the exact mathematical universe of our intuition. So, in a sense,
the non-categoricity of first-order logic that Löwenheim and Skolem first
demonstrated is a kind of "model theoretic incompleteness". I think something
like that is almost surely what Fenstad had in mind. (04)
Chris Menzel (05)
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