[FOM] Higher Order Set Theory

Aatu Koskensilta aatu.koskensilta at xortec.fi
Wed Mar 9 06:26:09 EST 2005

On Mar 8, 2005, at 10:57 AM, Roger Bishop Jones wrote:

> On Monday 07 March 2005  3:39 pm, Dmytro Taranovsky wrote:
>> Since the universe includes all sets, second order statements
>> about V appear doubtful or meaningless.  However, there is a
>> way to make them meaningful, and to get a reasonable
>> axiomatization of theory, thus resolving how to deal with
>> proper classes, "collections" of proper classes, and proper
>> class categories such as the category of all groups.
> Higher order set theory is a great deal less problematic than
> you make it appear.

In a sense higher order set theory is not problematic at all and has a 
natural axiomatization in Morse-Kelley set theory. However, from a 
conceptual point of view, I believe that higher order set theory is 
*much* more problematic than the universe of sets, V alone. The problem 
is that it's not obvious how there could be a determined totality of 
subcollections of V over which the second order quantifiers were to 
range over. For if such a determined totality exists, why isn't it just 
an another iterative layer on top of those already in V? One could 
argue that the idea of proper classes with any substance is against the 
idea of set theory as the universal foundational framework for 
mathematics, because there is a substantial ontological and conceptual 
involvements beyond those reducible to sets.

Also, if the totality of proper classes is a determined totality, there 
seems to be no problem in talking about the collection of all proper 
classes and so forth. This essentially gives us ZFC + (ZFC for proper 
classes) + (ZFC for proper-proper-classes), ..., i.e. we have just a 
new hierarchy of collections which looks exactly like the cumulative 
hierarchy with a few spots marked (here begin the "proper classes".... 
here begin the "proper-proper-classes" ...).

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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