[FOM] predicative foundations

Kenny Easwaran easwaran at berkeley.edu
Thu Feb 16 00:45:44 EST 2006

Nik Weaver wrote:

>My preceding message is a response to Harvey Friedman's
>assertion that "The claims that predicativity has some special
>place in the robust hierarchy of logical strengths ranging from
>EFA through j:V into V are unjustified."  As I've explained,
>the predativist prohibition on circular definitions is a simple
>consequence of a disbelief in a platonic universe of sets.
>Friedman has a right to be unaware of this justification but it
>is unwise of him to baldly claim that no justification exists.
>From reading Friedman's postings here, I interpreted him as meaning not
that there is no good argument in favor of predicativity as the stopping
point, but rather that there are good arguments in favor of _many_
different stopping points, and that the argument in favor of
predicativity is not clearly better than all the other arguments. 

In his posting of 2/5/06, Friedman writes:

> Predicativity is one of many not unreasonable "stopping places" in an extensive hierarchy of 
> logical strengths from EFA through j:V into V. Current conventional wisdom is that the
> informal notion of "predicativity" is not sufficiently sharp to correspond to a single place
> in this hierarchy. So there is a serious question as to how natural a place it is to single
> out instead of many other places.

I take the assertion of "many not unreasonable 'stopping places'" to
mean that there are good arguments in favor of many different places. 
Weaver may have a way to sharpen up the notion of predicativity so that
it becomes a particular stopping place.

But if there are good arguments in favor of Weaver's brand of
predicativism, and also good arguments in favor of other stopping
points, then it seems that Weaver has a further burden to bear here,
namely of showing why the arguments in favor of his position are
substantially better than the arguments in favor of each other
position.  Presumably this will involve showing that the arguments in
favor of other positions are somehow flawed.

Until then, it seems that there is no need for a third party (like
myself, and possibly Friedman) to focus on any one stopping point rather
than any other.

Kenny Easwaran

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