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Peter_Smith wrote:
> Oh dear. Another sad rant.
I am sad about the fact that you seem not to know
a universally quantified variable when you see one,
and I am even sadder that ranting is the only thing I
can do about it.
> And in this case an elementary failure to distinguish
I *repeat*, go yourself.
> (i) metalinguistic schemata with "schematic variables"
The prosecution rests. A schematic variable is a variable.
> or placeholders,
Or universally quantified bound variables
> giving us an abbreviated way of specifying an infinite
JEEzus. OF COURSE EVERY specification of ANYThing
infinite MUST be "abbreviated". IT would be infinite TOO if it
weren't. That has nothing to do with whether it does or doesn't
include variables.
> family of objectlanguage wffs containing numerals
> but no quantifiers or variables from
I WAS aware that the object language didn't contain variables
or quantifiers.
(ii) object language quantified sentences.
I HAD NOT failed to make that distinction.
> You might possibly try reading some
> of those books and understanding before
> sounding off so offensively.
Been there, done that, got the diploma (B.A.83, M.S.95).
Why don't YOU try making a relevant distinction in the
question that YOU were trying to answer? YOU are failing
to distinguish between the professor and the student, here.
*I* posed the challenge. It was,
> if you are talking about a theory with no free
> variables, you get something fully quantifier-free and therefore
> negation-complete. But I do NOT agree that it is EVEN POSSIBLE
> to write down "schemata for defining all p.r. functions" under any
> limitation THAT severe.
What is "under" the "limitation" here is
THE WRITING DOWN of the
schemata, NOT just the object language.
Another distinction which I draw but that YOU failed to
is that between the domains of quantification of the
schematic variables and the [possible, though in this
case absent] object-language variables. IF there were
going to be variables in this object-language, they certainly
would've had natural numbers as their domain of instantiation.
In your axiom-schemata, your schematic variables have THIS
EXACT SAME domain. PLEASE KINDLY CONTRAST this
with the more USUAL axiom-schematic case, from, e.g.,
ZFC or PA, where the WHOLE REASON WHY the objective
has to be accomplished via a schematic variable in the meta-
language is because it requires universal quantification
over a domain OF FORMULAS OF A LANGUAGE, or, in any case,
over something ENTIRELY DIFFERENT from, something NOT OCCURRING
IN, the intended domain associated with the object theory.
In the case of ZFC and the axiom schema of replacement,
that would be binary-wff-schemata-instead-of-sets; in the
case of PA and the axiom schema of induction, it would be
wffs-with-1-variable-instead-of-numbers.
My point is simply that if your schematic variables are universally
quantified over a completely different class of entities than any
occurring in your object theory, THEN OF COURSE no one is going
to say that there is simply no important difference between them
and object-language variables. But if they are over the SAME domain
as the objects in the object language, well, you will have invited all
the
abuse you are getting.
The whole question here is really one of how much finite machinery
one might need to specify different kinds of infinite classes (of wffs
or functions or sets or numbers or anything else). The relevant
constraints are on the machinery of specification.
And another thing:
asking anybody to read a book (unless it is introductory) in this
context is stupid. If you can't summarize the relevant
article/argument
FROM the book YOURSELF then you are not smart enough to be
presuming to comment. More to the point, there is likely to be a
treatment in a link somewhere, and since most people reading this
are already doing it over the internet anyway, that is obviously a
better kind of reference than one that is going to require somebody
to log off and travel to a library.
In any case, your version produces a propositional theory of
p.r. functions in which no function definition is statable.
It can't even assert that a constant function is constant.
It can only assert finite pieces of that fact.
The "real" theory that is "really" being looked for here
REALLY DOES HAVE Pi-1 quantification. It just doesn't
have any other kind. It therefore has some strange problems
with negation. That is the branch of the path down which
I was trying to direct the discussion.
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