Re: Continuum hypothesis

On Thu, 16 Aug 2007, Peter_Smith wrote:
Well, to be pernickety, since CH can be formulated as a statement of
the language pure second-order logic, it *is* a statement of any
second-order theory whose language includes that of second-order logic
(including second-order arithmetic). Of course, some might well say
that that's just because second-order theories are just set theories
in disguise!

On Aug 16, 4:11 am, William Elliot <ma...@xxxxxxxxxxxxxxxxxx> wrote:
Are you responding to someone or starting a new thread on CH?

He was responding to *you* !
And you would appreciate his actual point:
1) in 1st order set theory, you can quantify over sets;
2) in 2nd-order logic, you can quantify over 1st-order predicates
3) psychologically, however, even if not orthographically, a set
behaves
A LOT LIKE a predicate that is true of and only of the elements of the
set.
So there is a sense in which there is simply no difference between the
two.
In modern treatmens, 2nd-order logic is treated as more powerful than
1st-order set theory, because there is a different convention about
what kinds
of things the domain of quantification can contain. But it is purely
conventional.
The way the theories get written down is not different enough to be a
difference.
If a set S contains an element x then you write xeS, whereas if a
predicate P is
true of an element x, you write Px or P(x), but if S can be an element
of another set
R, so that you write SeR, well then, you could've (if you could have
second-order predicates)
just as easily written R(S) to say the same thing. The linguistic
transformation is SIMPLE
AND EASY. These just are not different ENOUGH to even BE different
(until you invoke
the difference in default semantics for the quantifiers).

Of course, since S could be an element of R, and R could in turn (in
set theory) be an
element of some third set Q, ( SeR & ReQ ), you could say that 1st-
order set theory,
in addition to being able to emulate 2nd-order logic, could also
(simultaneously,
all SEEMINGLY staying at 1st-
order) emulate 3rd & 4th & 5th-order (and all finite higher-order)
logics as well. But people
usually content themselves with saying "2nd-order" because "higher-
order" is in fact
reducible to 2nd; 2nd is in some sense high "enough", i.e., powerful
enough that
adding higher "orders" doesn't actually confer new "powers".

.

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