Re: what follows from denying an axiom
- From: george <greeneg@xxxxxxxxxxxxx>
- Date: Sat, 24 Jan 2009 14:30:10 -0800 (PST)
On Jan 23, 5:49 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
It seems to me that the key feature that makes
a language second-order is a comprehension schema.
The KEY difference between the USUAL formulation of "a second-order
language"
and a first-order one is simply that you add predicates that can take
OTHER(first-order)PREDICATES as arguments.
It was already the case that you could quantify over the things
occurring
in argument-positions, so you THEN MUST ALSO have bound(quantified-
over)
variables that can be INSTANTIATED to predicates.
But there are (obviously) any number of RESTRICTIONS OR TRUNCATIONS
you could apply to the range of the predicate variables that would
wind up leaving
this "second" order IN SYNTAX ONLY, my point being that there is an
easy analogy
to one restriction that COULD be made in the first-order case: you
could claim that
domains for your quantified variables HAD to be FINITE. IF you
imposed THAT
restriction then your "first-order language" really would STOP being
first-order because
its EVERY quantified statement would become equivalent to a
propositional one,
to a finite conjunction in the case of a universal quantifier, or a
disjunction in the
case of an existential one. You would just have one -junct per
element of the domain.
Similarly, some kinds of restrictions that you could impose on the
predicate domain,
FAR FROM BEING *MERELY* "non-standard", are simply WRONG. They're
simply INCORRECT. They're simply NOT second-order (just as an upper
bound
on the cardinality of model-domains for a first-order language WOULD
MAKE IT STOP
being first-order). A first-order language is first-order BECAUSE of
what THE
QUANTIFIERS *MEAN*.
THAT is NOT something that you get to re-interpret, not any more than /
\ , \/ , or ~ is.
Similarly, in the case of a second-order language, you DO NOT
*independently*
specify what the domain of the predicate variables is! You specify
ONE domain
for the BASE variables and THEN the domain for the predicate variables
IS
AUTOMAGICALLY the POWERclass of the FIRST-order domain!
Notwithstanding the the locally relevant educated-unto-foolishness
educated fool's
opinion to the contrary, NOT JUST ANY OLD ARBITRARY variation on the
NORMAL
semantics gets to be blessed as "merely" non-standard. That is an
insufferably
sloppy analogy with the first-order case. Non-standard models in the
first-
order case REALLY AREN'T non-standard. The blessing of the standard
model
AS standard is more discrimination than anything else. The standard
model of PA
would less contentiously be dubbed the MINIMAL model or the INITIAL
model
(since it is the initial segment of EVERY model).
.
- References:
- what follows from denying an axiom
- From: george
- Re: what follows from denying an axiom
- From: Chris Menzel
- Re: what follows from denying an axiom
- From: Daryl McCullough
- what follows from denying an axiom
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