[FOM] Justifying SRP?
Monroe Eskew
meskew at math.uci.edu
Tue Sep 16 00:56:06 EDT 2014
On Sep 15, 2014, at 2:57 AM, Rupert McCallum <rupertmccallum at yahoo.com> wrote:
> Examining this would probably require a deeper look at why we regard ordinary reflection with second-order parameters as justified in the first place.
I will take a preliminary stab at this. V is built iteratively through an absolute infinity of stages. This absolute infinity outstrips our ability to pin it down with some logical property. At some point we will have built a rich enough initial segment to satisfy a given property. Further, this should also be true if we allow the property to use some arbitrary “oracle” predicate of sets. Eventually we will have built a rich enough structure so that whatever our oracle says about sets in general, is true about those we’ve constructed so far.
Now look at third-order parameters. Because these can talk about unbounded subclasses of V, we are not necessarily approximating them ever more closely in the iterative process. They can talk about some objects which only come at the very end. We cannot hope to say in general that a given property involving such parameters is witnessed at some initial stage, because that parameter cannot in general hope to have the same interpretation as in the original structure. There are third-order predicates that hold of a nonempty collection in the full structure but an empty collection of every initial segment. So we would have to change the interpretation of these parameters in some way to get an analogue to second-order reflection. It’s not clear that the same philosophical intuitions would apply.
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