1
5.8. ZFC + V = L, indiscernibles, and
Π
0
1
correct arithmetic.
We fix M^ = (C,<,0,1,+,-,•,
↑
,log,
ω
,c
1
,c
2
,...,Y
1
,Y
2
,...) as
given by Lemma 5.7.30. We work entirely within M^. E.g., we
treat C as the universe of points, and regard the elements
of the Y
k
as the internal relations.
In particular, if we say that R is an internal relation,
then we mean that R is an element of some Y
k
. If we say that
R is an M^ definable relation (first and second order
parameters allowed), then we do not necessarily mean that R
is an internal relation. However, by Lemma 5.7.30, vii), if
R is an M^ definable relation which is
bounded
, then R is
an internal relation; i.e., R is an element of some Y
k
. In
fact, Y
k
is the set of all bounded M^ definable relations on
C.
DEFINITION 5.8.1. Functions are always identified with
their graphs. We refer to the elements of Y
1
as the internal
sets.
An important obstacle is that there is no way of showing,
in M^, that the family of all internal subsets of an
internal set is in any sense internal. E.g., no way of
showing that they are all cross sections of some fixed
internal binary relation.
It would appear that this obstacle is fatal, as it
indicates an inability to interpret the power set axiom,
despite bounded comprehension, indiscernibility, and
infinity.
However, in this section, we argue carefully that we can
still construct the constructible universe. Because of the
explicitness of this construction, we can use
indiscernibility to overcome this obstacle within the
constructible universe.
We first have to develop a pairing function. By an
interval, we mean a set [x,y), where x,y
∈
C.
LEMMA 5.8.1. Let k
≥
1 and F be a k-ary M^ definable
function, defined without second order parameters. For all
x, {F(y
1
,...,y
k
): y
1
,...,y
k
< x} is bounded above. For all x,
the restriction of F to [0,x)
k
is an internal function.