Unformatted text preview: Minimality, note that at
exact upper bound for ∅, and
incomparable. Thus (D,>) has
elements. But all x Œ D have every stage we introduce an
these exact upper bounds are
a proper class of minimal
a set of strict predecessors. For Dk, note that (Dk,>k) has minimal elements introduced at
every stage a < k. Let x Œ Da, a < k. Then any minimal
element introduced at any stage ≥ a will not be < x. QED
Note also that for every x Œ D, we introduce some y
minimally > x at every stage after x is introduced, and y
remains minimally > x at every later stage. This also
applies to (Dk,>k), for limit ordinals k.
Now fix S to be a nonempty set of limit ordinals, with no
greatest element, whose union is k. We define M[S] to be
(Dk,>k,>>S), where Dk,>k is as above. We define x >>S y if and
only if
x,y Œ Dk Ÿ ($a,b Œ S)(a < b Ÿ y Œ Da Ÿ ("w Œ Db)(x > w)).
LEMMA 7.2. M[S] satisfies Basic.
Proof: x >> y Æ x > y. Let x,y Œ Dk. Let a,b Œ S, a < b, y
Œ Da, ("w Œ Db)(x >k w)). Since y Œ Db, x >k y.
x >> y Ÿ y > z Æ x >> z. Let x,y,z Œ Dk. Let a,b Œ S, a <
b, y Œ Da, ("w Œ Db)(x >k w), y >k z. Then z Œ Da, and so x
>>S z.
x > y Ÿ y >> z Æ x >> z. Let x,y,z Œ Dk. Let a,b Œ S, a <
b, z Œ Da, ("w Œ Db)(y >k w), x >k y. Then ("w Œ Db)(x >k w),
and so x >>S z. 9
($x)(x >> y,z). Let a,b Œ S, a < b, y,z Œ Da. Let x Œ Da+1 be
the exact upper bound of Da introduced in Da+1. Then x >>S
y,z.
x >> y Æ ($z)(x >> z Ÿ z is minimally > y). Let a,b Œ S, a
< b, y Œ Da, ("w Œ Db)(x >k w)). Let y Œ Dg, g < a. Let z Œ
Dg+1 be minimally >k y. Note that x >>S z. QED
LEMMA 7.3. MK proves that there exists a set
ordinals, with no greatest element, with the
indiscernibility property. For all a < b < g
have the same first order properties over V,
any parameters from V(a). S of limit
following
from S, b,g
relative to LEMMA 7.4. Let n < w. ZF proves that there exists a set S
of limit ordinals, with no greatest element, with the
following indiscernibility property. For all a < b < g from
S, b,g have the same first order properties over V, with at
most n quantifiers, relative to any parameters from V(a).
LEMMA 7.5. Let S to be a nonempty set of limit ordinals,
with no greatest element. In M[S], x >> y if and only if
("w Œ Db)(x > w), where b is the least ordinal in S after a,
and a is the least ordinal in S such that y Œ Da.
LEMMA 7.6. Suppose S has the indiscernibility property in
Lemma 7.3. Then M[S] satisfies Basic + Minimal + Existence
+ Amplification.
Proof: By Lemmas 7.1 and 7.2, we need only verify
Amplification. Let y >k x, and j(x) true in M[S], using
>k,=, and >>S x. Let a be the least element of S such that x
Œ Da. Let b be the least element of S after a. Let a’ be the
least element of S such that y Œ Da’. Let b’ be the lea...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Calculus, Set Theory

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