FOM: A recent claim of Friedman

[M. Randall Holmes]

Dear FOM'ers:

The following is an excerpt from a recent posting of Friedman's:

<begin excerpt>

13. I's CLASSES, II's CLASSES, SPECIAL COMPREHENSION.

We use variables x1,x2,... over I's classes.

We use variables y1,y2,... over II's classes.

The atomic formulas are of the following forms:

i. u = v, where u,v are variables of either sort.
ii. u in v, where u,v are variables of either sort.

We close under atomic formulas under connectives and quantifiers in the
usual way. We call this language L6. We use the standard 2 sorted predicate
calculus with equality appropriate for L6.

6a. (therexists y1)(x1 = y1).

6b. y1 = y2 iff (forall y3)(y3 in y1 iff y3 in y2).

6c. (forall y1)(phi* implies (therexists x1)(x1 = y1)) implies (therexists
xk+1)(forall x1)(x1 in xk+1 iff (phi and psi)), where phi is a formula of
L6 with all variables among x1,x2,...,xk, phi* is the result of replacing
each bound occurrence of xi by yi and each free occurrence of x1 by y1, and
psi is a formula of L6 in which xk+1 is not free.

THEOREM 13.1. The system 6a-6c is mutually interpretable with ZFC.

<end excerpt>

This appears to be wrong.  What Friedman appears to be doing is
attempting to simplify the axioms of Ackermann set theory, which is
indeed mutually interpretable (or at least equiconsistent) with ZFC.
But he has simplified them too much.

The axioms described above are satisfied, for example, in a nonstandard
model of V_omega constructed in the usual way using an ultrafilter.  Let
the variables yi stand for all elements of the nonstandard model and let
the variables xi range over the standard elements (in the obvious sense).
6a and 6b clearly hold.

A formula phi with all variables restricted to xi's, when transformed
as indicated, will be true only of xi's in our proposed interpretation
just in case it is true of finitely many xi's (a statement true of
infinitely many standard objects would also be true of some
nonstandard objects -- yi's not equal to any xi, in this
interpretation).  Any finite collection of xi's is the extension of
some standard set in V_omega, establishing the truth of 6c in this
interpretation.  Further strengthening the condition phi with psi
doesn't affect this (standard sets have only standard subsets).

The additional assumptions needed are as follows (restoring equivalence
with the Ackermann theory):

for all y1, (y1 in x1 implies for some x2, y1 = x2)  (elements of xi's
are xi's)

for any formula phi of L6 in which y1 is not free, there exists y1
such that (for all x1, x1 in y1 iff phi)  (any class of xi's is a yi)

Both of these are necessary, as can be seen from examination of the proof
of Infinity in Ackermann set theory.

I haven't looked to see to what extent the rest of the posting needs
to be modified to correct this apparent error, but I expect that it
can all be repaired.

                                      --Randall Holmes