[FOM] iterative conception/cumulative hierarchy

[Christopher Menzel]

Am Feb 28, 2012 um 11:44 PM schrieb Nik Weaver:
> Chris Menzel wrote (quoting me):
>>> So, granting that there is a notion of metaphysical presupposition and
>>> it is well-founded, how does that help us understand which concepts have
>>> extensions?
>> 
>> Again, it's not well-foundedness per se that does the work; it is (as I
>> see it) the unbounded cumulative structure that emerges in a natural way
>> from the assumption of well-foundedness.
> 
> Hold on ... the unbounded cumulative structure doesn't emerge from the
> assumption of well-foundedness.  That came out of van Aken's paper very
> clearly.  You need to separately adopt a reflection principle, one of just
> the right strength, which apparently lacks a good informal justification.

See below.

> Right?  We decided that the iterative conception isn't meant literally.
> It's just a stand-in for some notion of "metaphysical dependence" or
> "presupposition".

Well, I didn't agreed to exactly that. What I said is that the idea of sets being "formed" in stages can be spelled out in terms of the static structure of the cumulative hierarchy, in particular, in terms of the notion of rank.

> But when that notion is axiomatized you get an
> extremely weak system (van Aken's MSU) that captures no unbounded
> cumulative structure. 
> 
> So it turns out that the iterative conception does very little toward
> telling us what concepts have extensions.  All the work is done by a
> reflection principle which has no clear justification.

This, I think, does not follow. True enough, van Aken's basic system MSU is very weak with regard to set existence; he can't even prove the full pairing axiom or the full power set axiom or, more generally, that the hierarchy is itself unbounded in the sense that every set is presuppositionally weaker than some set. However, two things. First, I do not think that MSU axiomatizes the iterative conception. Rather, it axiomatizes what van Aken (reasonably) took to be the key structural element of the cumulative hierarchy, the notion of presuppositional strength — which does in fact turn out to be equivalent to the usual notion of rank. But rank is not all you get when you spell out the iterative conception. Notably, MSU explicitly does not address the "height" of the hierarchy and, hence, does not include any principles that determine its "upward growth". But, by my lights, upward growth is exactly what puts the "iterative" in the iterative conception: Given some urelements, form all the sets that can be formed from them; given the urelements and the sets just formed in the preceding stage, form all the sets that can be formed from them; keep doing that. Note in this regard that Boolos's stage theory — his attempt to axiomatize the iterative conception — includes what he calls "specification axioms" designed explicitly to capture exactly this aspect of the conception; and in conjunction with the other axioms, these axioms entail both pairing and power set and, hence, more generally, the unboundedness of the hierarchy. So I still claim that the fully fledged iterative conception does indeed deliver an unbounded hierarchy and, hence, also the answer given previously to the question of what concepts have (set) extensions, viz., those that are true of sets of arbitrarily high rank.

Second, though, even in van Aken's basic system MSU there is still a related notion of unboundedness in the neighborhood that, as far as it goes, delivers roughly the same answer, despite MSU's inability to prove the unboundedness of the hierarchy. Say that a concept C is unbounded if there is no z such that z has a presuppositional strength (equivalently, a rank) greater than all the things satisfying C.  What concepts have extensions? All and only those that are not unbounded. (This is basically MSU's schema of rank comprehension.) When we add enough to MSU to generate an unbounded hierarchy — and far less than a reflection principle is needed for that, e.g., pairing — the unbounded concepts will be exactly those that are true of sets of arbitrarily high rank.

Chris Menzel

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