FOM: goading Harvey

[Stephen G Simpson]

To: Joseph Shoenfield

Dear Joe,

Concerning your attempt to goad Harvey:

 > but I will state a possible theorem which I hope will clarify what
 > I have in mind: There is a sequence {An} of sentences such that An
 > is strictly pi-1-n+1, and An can be proved from ZFC and the
 > existence of n large cardinals of a certain sort but not (under
 > obvious consistency hypotheses) from n-1 such cardinals.

I don't understand this.  My first question is, what does strictly
pi-1-n+1 mean?  It appears that the purpose of this condition is to
rule out obvious examples such as

   An = Con(ZFC + there exist n inaccessible cardinals)

But what does the condition mean?

 > It would be nice if An was simple in some of the ways that
 > undecidable sentences considered by Harvey and others are simple;
 > but I do not think this essential for the significance of such a
 > result.

Really?  Without some requirement of this nature, wouldn't the result
be obtainable by an easy diagonal argument?

-- Steve