[FOM] n-th order ZFC

meskew at math.uci.edu meskew at math.uci.edu
Sat Jul 16 14:08:50 EDT 2011

On Fri, Jul 15, 2011 at 12:56 PM, Roger Bishop Jones <rbj at rbjones.com> wrote:
> But if you did that, I would not understand what question
> you were asking.
> Which is the point.

CH says, "For all sets of reals X either there exists a bijection
between X and R, or X is countable."  Surely you understand the
statement.  Write it in the language of first order ZFC.  Surely you
still understand it.  Now I ask, "Is CH true?" by which I mean,
(following Tarski) "For all X subset of R, does there exist either a
bijection between R and X, or between N and X?"  Surely you understand
the question.

> But of course, "is CH true in V?" is not the same question.
> It is a great step forward to at least attempt to identify a
> single interpretation since if you succeeded in doing that
> you would have made the question definite.

It does mean the same thing.  "in V" just means "in the universe of
all sets."  It's the default domain of quantification, and only
mentioned for emphasis.

> 2. Some philosophers believe that of all the sets which
> might possibly exist only some actually do exist, that V is
> just those sets which actually do exist and that discovering
> which sets actually exist depends on consulting an intuitive
> understanding (of which I am unable to discover any trace in
> myself).

I have no idea what the distinction is between a possibly and actually
existing set.  This is not part of the mathematical language.

> 3 If you were to say that:
>        V is the collection of all (possible)
>        pure well-founded collections
> I should respond that all definite collections of pure
> well-founded collections are themselves pure well-founded
> collections, and that the collection of them all must
> therefore  have no definite extension on pain of
> contradiction.
> This leaves me in some doubt about whether this approach to
> giving a definite meaning to the question is logically
> coherent.

Part of doing set theory is accepting that not all classes are sets.
There is no set of all ordinals, for example.  In 100 years no
contradiction has been discovered.  In fact the set vs. proper class
distinction has become a useful concept to employ in presenting proofs
of basic theorems, such as Zorn's lemma.

I could just as easily question the logical coherence of second order
set theory.  There, one quantifies freely over classes.  But can't you
have classes of classes?  In response, you could go to 3rd-order, or
why not, omega-order set theory.  Then I could ask why can't you have
classes of class-rank omega or higher?  Why the artificial cutoff
point?  Why not just let your collections grow without bound?  But
then the picture looks like first order set theory.

> On the other hand, if you were to ask "Is CH true in second
> order set theory?" I would believe that you had asked me a
> definite question and that I understood its meaning.

All of your worries about the meaningfulness of CH can creep into
second-order set theory.  I can still ask about a model of second
order set theory whether all sets in it are constructable (in L).  Or
I can ask whether PFA is true in the model.  These questions affect
CH.  Although second order logic has a very limited semantics, what
models exist depends entirely on the background theory of sets.


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