In the foreword to J L Bell's Boolean-valued Models and Independence
Proofs in Set Theory (first edition), D S Scott writes

"... in second-order formulations of set theory it [that's CH]
would be decided: only we cannot know which way."

What does that mean?

To start with, what is second order set theory?
My guess is that it's one in which axiom schemata are replaced with
single axioms with a "for all formulae phi" quantifier.

But what would
be a second order formulation of a classes-and-sets theory?

CH is actually equivalent to its being a logical consequence of these axioms,
but since second order logic is not complete the only way to know whether this
is the case is to either prove or disprove CH.

Frederick Williams wrote:
In the foreword to J L Bell's Boolean-valued Models and Independence
Proofs in Set Theory (first edition), D S Scott writes

"... in second-order formulations of set theory it [that's CH]
would be decided: only we cannot know which way."

What does that mean?

It means that either CH or its negation
is a logical consequence of the
axioms of second order set theory.

To start with, what is second order set theory?
My guess is that it's one in which axiom schemata are replaced with
single axioms with a "for all formulae phi" quantifier.

No.
The schemata are replaced by single sentences formulated using
ordinary second order quantifiers, so that the axiom schema of
separation, for example, becomes a single axiom

AxAPEyAz(z \in y iff z \in x and P(z))

where P ranges over all subcollections of the universe of model.

But what would
be a second order formulation of a classes-and-sets theory?

Every axiom schema would be replaced with a single universal second
order axiom.

Frederick Williams wrote:

But what would
be a second order formulation of a classes-and-sets theory?

Every axiom schema would be replaced with a single universal second
order axiom.

Aatu Koskensilta wrote:
Frederick Williams wrote:

But what would
be a second order formulation of a classes-and-sets theory?
Every axiom schema would be replaced with a single universal second
order axiom.

Von Neumann, Bernays, Gödel doesn't have schema, it's finitely
axiomatizable. (Hence my question.)

Frederick Williams wrote:

To start with, what is second order set theory?
My guess is that it's one in which axiom schemata are replaced with
single axioms with a "for all formulae phi" quantifier.

The schemata for the first-order versions of the axioms
ALREADY HAVE a "for all formulae phi" quantifier, or
at least "for all formulae phi with the right number of unbound
variables" quantifier.

Frederick Williams wrote:
But what would
be a second order formulation of a classes-and-sets theory?
.....
Von Neumann, Bernays, Gödel doesn't have schema, it's finitely
axiomatizable. (Hence my question.)

That's true. "Second order NBG" doesn't mean anything determinate

In the foreword to J L Bell's Boolean-valued Models and Independence
Proofs in Set Theory (first edition), D S Scott writes
"... in second-order formulations of set theory it [that's CH]
would be decided: only we cannot know which way."

What does that mean?
To start with, what is second order set theory?
My guess is

Frederick Williams wrote:
But what would
be a second order formulation of a classes-and-sets theory?
....
Von Neumann, Bernays, Gödel doesn't have schema, it's finitely
axiomatizable. (Hence my question.)

Aatu Koskensilta wrote:
That's true. "Second order NBG" doesn't mean anything determinate

In that case, 1st-order ZFC doesn't either.
It's hard to see how ANY 1st-order set theory can be meaningfully
lifted to 2nd-order. The problem is, once you have predicates as
members of your universe of discourse, THEY ARE all sets.
They are ESPECIALLY all OF THE sets, if you are using the standard
semantics of SECOND order logic, which requires that predicate
variables be able to range over "all conceivably/metahphysically
possible"
classes (i.e. subcollections of the universe) of 0th-order objects.
To any set you could conceive of, under THIS semantics, there ALREADY
corresponds A PREDICATE. Therefore, it is simply never relevant to
allege xey. The OBVIOUSLY proper way of phrasing xey, in the context
of a 2nd-order language, is y(x) . You simply cannot draw any
meaningful
distinction between a set y and the corresponding unary predicate ?ey.
If the axioms are not present for the purpose providing a definition of
e, since e is not needed at all, since it is just predication between a
1st-
and 0th- order object of the existing universe of discourse, then how
can
you have a set theory at all?

Please don't guess. There is a standard answer.
Google it if you know how.

Here is an interesting treatment:
web.mit.edu/dmytro/www/NewSetTheory.htm