[FOM] iterative conception/cumulative hierarchy

Nik Weaver nweaver at math.wustl.edu
Sat Feb 25 00:57:35 EST 2012

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).


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.


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