[FOM] n-th order ZFC

[Roger Bishop Jones]

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.

However:

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 
myself).
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 
contradiction.
This leaves me in some doubt about whether this approach to 
giving a definite meaning to the question is logically 
coherent.

There are other examples of the confusing variety of ways in 
which philosophers and logicians talk about "V", but enough 
is 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.

Roger Jones