[FOM] iterative conception/cumulative hierarchy

[Christopher Menzel]

Am Feb 25, 2012 um 6:57 AM schrieb Nik Weaver:
> Michael Kremer wrote:
> 
>> Here's an old paper by Jim van Aken (RIP) which explains the axioms of
>> ZFC in terms of the idea of one entity presupposing others for its
>> existence (so doing away with the notion of "forming sets" from the
>> get-go).
> 
> Michael,
> 
> This paper looks interesting, but I don't see how you can say it explains
> ZFC.  The basic system he presents, MSU, is extremely weak --- it doesn't
> prove the existence of pairs, or infinite sets, or power sets.
> 
> You only get to ZFC by adding a reflection scheme which the author openly
> acknowledges lacks a compelling informal justification (bottom of p. 1001).
> He points out that "It is known that Ref does not consistently generalize
> to the case of third-order formulas" and "To date, no informal rationale
> for reflection explains why the same rationale does not extend to the
> third-order case."
> 
> So your gloss seems like rather an overstatement.

I'm not sure this gets at what I, at least, took Michael's (and also Richard's) point to be, namely, that the metaphor of set formation is cashed in terms of the idea of a set *dependending on*, or *presupposing*, its members, a relation that is reflected in the (static) structure of the hierarchy — sets of a given rank are dependent upon, or presuppose, sets of lower rank.

Chris Menzel