[FOM] n-th order ZFC
Roger Bishop Jones
rbj at rbjones.com
Fri Jul 15 15:56:16 EDT 2011
On Wednesday 13 Jul 2011 03:23, meskew at math.uci.edu wrote:
> I don't see what the complication is. In first order
> language one can just ask, "Is CH true?"
But if you did that, I would not understand what question
you were asking.
Which is the point.
> The fact that
> there are different models that think different things
> about CH does not prevent you from simply asking whether
> CH is true in V.
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.
1. Some people interested in set theory are willing to
contemplate the possibility that V=L, whereas others have a
conception of V which excludes that possibility.
This is probably not a disagreement about whether CH is true
but a disagreement about what if anything the question "is
CH true in V?" means.
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
I expect that if such a philosopher asked me "is CH true in
V?" and followed that up with his best explanation of what
he meant by that, I would still have to confess that I did
not understand the question, and was inclined to doubt that
it had any objective meaning.
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
This leaves me in some doubt about whether this approach to
giving a definite meaning to the question is logically
There are other examples of the confusing variety of ways in
which philosophers and logicians talk about "V", but enough
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.
You may have had a different question in mind.
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