FOM: goading Harvey
Stephen G Simpson
simpson at math.psu.edu
Mon Sep 7 21:54:48 EDT 1998
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
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