FOM: power shift axiom/correction

[Harvey Friedman]

As I promised to Forster in 12:27PM 4/2/99, I briefly looked over my claims
of 10:56PM 4/1/99, and found some misstatements. I hope this is more
accurate.

Theorems 2-4 have to be modified. I had the right idea, but I must
strengthen the power shift axiom somewhat. In the process of strengthening
it, I have a better statement of it (in fact, muiltiple versions). At
least, I like these formulations better than the earlier formulations with
schemes. [If we stick to schemes, then the strenghenings amount to
introducing an integer parameter].

I don't know how to prove the Theorems 2,3 of the earlier posting, and
Theorem 4 of the earlier posting is outright refutable by an easy
application of Mathias forcing.

So let's fix this up by starting over. I like two settings for a clearer
and more appropriate statement of the power shift axiom. The first setting
is in class theory, and the second setting is in set theory.

In the class theory setting, the power shift axiom (for class theory) says
the following. Let F:V into omega. There exists sets x,y, where |x| = 2^|y|
and F(x) = F(y). The weak power shift axiom says: Let F:V into {0,1}. There
exists sets x,y, where |x| = 2^|y| and F(x) = F(y).

THEOREM 1. NBG + AxC (Von Neumann Bernays Godel plus every set is well
ordered) refutes the weak power shift axiom (for class theory). [Well
known].

THEOREM 2. NBGC + "there exists a measurable cardinal" is translatable into
NBG + "the power shift axiom (for class theory)."

In the set theory setting, the power shift axiom (for set theory) says the
following. There is an ordinal alpha such that for all F:V(alpha) into
omega there exists x,y such that |x| = 2^|y| and F(x) = F(y). The weak
power shift axiom (for set theory) says: There is an ordinal alpha such
that for all F:V(alpha) into {0,1} there exists x,y such that |x| = 2^|y|
and F(x) = F(y).

In NBG, obviously the power shift axiom (for set theory) immediately
implies the power shift axiom (for class theory). The reverse is unclear.

THEOREM 3. ZFC refutes the weak power shift axiom (for set theory). [Well
known].

THEOREM 4. ZFC + "there exists a measurable cardinal" is translatable into
ZF + "the power shift axiom (for set theory)." [This follows immediately
from Theorem 2.]

We can also consider sequential power shift axioms. In class theory:

$) Let F:V into omega. There exists sets x_1,x_2,... such that for all i,
|x_i+1| >= 2^|x_i|
and F(x_1,x_2,...) = F(x_2,x_3,...).

And

$$) Let V:V into omega. There exists ordinals x_1 < x_2,... such that for
all i,
F(x_1,x_2,...) = F(x_2,x_3,...).

THEOREM 5. NBG proves $) is equivalent to $$). NBG + $), NBGC + $), and
NBGC + "there exists a
measurable cardinal" are all intertranslatable.

Also we consider the sequential power shift axioms in set theory:

#) There is an ordinal alpha such that the following holds. Let F:V(alpha)
into omega. There exists sets x_1,x_2,... in V(alpha) such that for all i,
|x_i+1| >= 2^|x_i| and F(x_1,x_2,...) = F(x_2,x_3,...).

And

##) There is an ordinal alpha such that the following holds. Let F:V into
omega. There exists ordinals x_1 < x_2,... < alpha such that for all i,
F(x_1,x_2,...) = F(x_2,x_3,...).

THEOREM 6. ZF proves #) is equivalent to ##). ZF + #), ZFC + ##), and ZFC +
"there exists a
measurable cardinal" are all intertranslatable.

SOME OPEN QUESTIONS: Is NBG + power shift axiom (for classes) consistent,
or even weak power shift axiom (for classes)? This would imply NF is
consistent and more. Similarly for the set formulations. Can Theorems 2 and
4 be proved for the weak power shift axioms as I originally thought?