2
y
1
...y
r
)).
The x’s and y’s range over On. Note that **) is a sentence in
class theory.
Now consider this transfer principle:
T
0
) for all suitable k,p,q,r,A, *
**.
Unfortunately, it is easy to refute this transfer principle,
even for k = 1 and no constants allowed.
We say that f:N
k
N is weakly regressive iff for all x N
k
,
f(x)
£
min(x). Here min(x) is the least coordinate of x.
Consider the following sentences.
*’) (
"
wr f
1
...f
p
:N
k
N)
(
"
x
1
...x
q
)(
$
y
1
...y
r
)
(A(x
1
...x
q
,y
1
...y
r
))
**’) (
"
wr f
1
...f
p
:N
k
On) (
"
x
1
...x
q
)(
$
y
1
..y
r
)
(A(x
1
...x
q
,y
1
,...y
r
))
Again, the x’s and y’s in the first form range over N, and
the x’s and y’s in the second form range over On.
And the transfer principle:
T
1
) for all suitable k,p,q,r,A, *’
**’.
Our first interesting trans-fer principle T
1
is equivalent to
a large cardinal principle.
Here we use VB + AxC as the base theory.
We can even weaken this transfer principle to
T
1
’) for all suitable k,p,q,r,A, *
**’
and obtain the same results.
We now introduce another modification of T
0
involving
quantification over all functions on N.
Fix E = {2
n
: n N}, and E^ = {2
a
:a On}.