This preview shows pages 1–3. Sign up to view the full content.
1
PRIMITIVE INDEPENDENCE RESULTS
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
July 13, 2002
Abstract. We present some new set and class theoretic
independence results from ZFC and NBGC that are particularly
simple and close to the primitives of membership and equality
(see sections 4,5). They are shown to be equivalent to
familiar small large cardinal hypotheses. We modify these
independendent statements in order to give an example of a
sentence in set theory with 5 quantifiers which is
independent of ZFC (see section 6). It is known that all 3
quantifier sentences are decided in a weak fragment of ZF
without power set (see [Fr02a]).
1. SUBTLE CARDINALS.
Subtle cardinals were first defined in a 1971 unpublished
paper of Ronald Jensen and Ken Kunen. The subtle cardinal
hierarchy was first presented in [Ba75]. The main results of
[Ba75] were reworked in [Fr01]. [Fr01] also presents a number
of new properties of ordinals (and linear orderings) not
mentioning closed unbounded sets, which correspond to the
subtle cardinal hierarchy.
The new properties from [Fr01] are not quite in the right
form to be applied directly to this context. We need to use
some new properties  particularly a property called weakly
inclusion subtle.
We follow the usual set theoretic convention of taking
ordinals to be epsilon connected transitive sets.
The following definition is used in [Ba75] and [Fr01]. We say
that an ordinal
l
is subtle if and only if
i)
l
is a limit ordinal;
ii) Let C
l
be closed unbounded, and for each
a
<
l
let A
a
a
be given. There exists
a
,
b
C,
a
<
b
, such that A
a
= A
b
«
a
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
We need the following new definition for present purposes. We
say that an ordinal
l
is inclusion subtle if and only if
i)
l
is a limit ordinal;
ii) Let C
l
be closed unbounded, and for each
a
<
l
let A
a
a
be given. There exists
a
,
b
C,
a
<
b
, such that A
a
A
b
.
LEMMA 1.1. Every inclusion subtle ordinal is an uncountable
cardinal.
Proof: By setting A
0
=
∅
and A
n+1
= {n}, we see that
w
is not
inclusion subtle. Let
l
be inclusion subtle and not a
cardinal. Let h:
l
Æ
d
be oneone, where
d
<
l
is the cardinal
of
l
. Then
d
≥
w
. Define A
a
= {h(
a
)} for
d
<
a
<
l
, A
a
=
∅
otherwise. Let C = (
d
,
l
). This is a counterexample to the
inclusion subtlety of
l
. QED
In light of Lemma 1.1, we drop the terminology “subtle
ordinal” in favor of “subtle cardinal”.
THEOREM 1.2. A cardinal is subtle if and only if it is
inclusion subtle.
Proof: Let
l
be inclusion subtle. Let C
l
be closed
unbounded, and A
a
a
,
a
<
l
, be given. Since
l
is an
uncountable cardinal, we can assume that every element of C
is a limit ordinal. For
a
C, define B
a
= {2
g
:
g
A
a
}
{2
g
+1 <
a
:
g
A
a
}. Let
a
,
b
C,
a
<
b
, B
a
B
b
. Then A
a
= A
b
«
a
. Here 2
g
is
g
copies of 2. QED
For our purposes, we are particularly interested in the
following somewhat technical notion.
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 JOSHUA
 Math

Click to edit the document details